Business math

Chapter 1:
Introduction
to Business
Mathematics

What is Business Mathematics? Importance of Business Mathematics
0 Business mathematics 9/24/2015 12:47:00AM
What is mathematics?
Mathematics: Mathematics is the science of order, space,
quantityand relation. It is that science in which unknown
magnitudes and relations are derived from known or assumed
ones by use of operations defined or derived from defined
operations.
Types of mathematics: There are two important types of
mathematics
 Abstract mathematics: Mathematics that is used to do the
general operationsor activities in our daily life is called abstract
mathematics. For example, accounting,financial mathematics.

 Appliedmathematics: Mathematics that is pure which
deals with analysis, observation and experiences of various
facts. For example, applied physics, chemistry and medical
science
What is business mathematics?
Business mathematics: Mathematicsthat is used by
commercial enterprises to record dailytransactions forecast
demand and supply as well as other commercial operations
known as business mathematics.
Importance of business mathematics:
Business mathematics is essential to keep track of an
organization day to day operation. The importance of business
mathematics briefly mentioned below
 To record daily transaction of business: An organization
uses business mathematics to keep record of their daily
transaction. How much product they sold today? How much
profit organization earned from today's operation etc. are
calculatedusing business mathematics.

 To forecast production:Organizations analyzes there
product demandand uses business mathematics to determine
how much production should be done meet up these demand.
 To forecast sales volume: Once an organization determine
their production they calculatetheir sales volume using
business mathematics.
 To calculateprofit or loss: Organizations uses business
mathematics to calculatetheir total cost (TC), total revenue
(TR) and total profit (TP) from the operation.
 To reduce wastage: If a company forecast their production
than they can determine what resources they need such labor,
funds etc. And business mathematics is essential to determine
these resources. This way an organization can reduce wastage
of resources.
you might want to see conceptual framework for financial
reporting
What is prime number?
Prime Number: A number which is not exactly divisibleby any
number except itself and unity (1) is called prime number. Such
as, 2,3,5,7,11,17
Example mathematics of prime number

Problem: How many prime numbers are between 1 to 100?
Solution: There are 25 prime numbers such as,
2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,
83,89,97
Problem: Is the number 97 prime?
Solution: Yes! It is a prime number. An approximate square root
of 97 is 10. The prime less than 10 is 2,3,5,7. 97 is not divisible
by any numbers. So 97 is a prime numbers.
Problem: Is the number 161 prime?
Solution: An approximate square root of 161 is 13. The prime
less than 13 is 2,3,5,7,11. 161 is dvisibleby (161/7)= 23. So 161
is not a prime number.

What is integer (whole number)?
Integer (whole number): Integer are the whole numbers
either positive, negative or zero.
Positive integer: 1,7,8,16
Negative integer: -5,-7,-8
Zero(0) is an even integer. It is neither positive nor negative.
An integer is said to be even (2n) if it is divisibleby 2 otherwise
it is said to be odd (2n+1) or (2n-1)
The formula of consecutive integer is
n, (n+1), (n+2),(n+3)………….Here n is an integer
Example mathematics of integer
Problem: The difference between the square of two
consecutive integer is 47. Find them
Solution: Let the numbers be n and (n+1)

According to the question,
(n+1)² - n² = 47
or, n²+2n+1-n² = 47
or, 2n+1 = 47
or, 2n = 46
or, n = 46÷2
or, n = 23
So (n+1) = (23+1) = 24
Therefore the required numbers = 23 and 24
Problem: The difference of the square of two consecutive
integer is 53. Find them
Solution: Let the numbers n and (n+1)
According to the question,
(n+1)² - n² =53
or, n²+2n+1+n² = 53
or, 2n+1 = 53

or, 2n = 52
or, n = 52÷2
or, n = 26
So (n+1) = (26+1) = 27
Therefore, the required numbers are 26 and 27
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Simple Definition of Business Mathematics And
Advantages of Business Mathematics
"
Business mathematics may be define different
mathematical formulas or mathematical steps which are taken for
development in business . Large number of business theories
which is used to solve business problems are included in business
mathematics .
"

Explanation of Definition of Business Mathematics
From this simple definition , we find that business mathematics is nothing
more than different formulas and theories like interest rate , annuity rate ,
matrix theory , linear programming theory and probability theory and many
more . With these formulas and theories business can calculate many
solutions of different problems .
Advantages of Business Mathematics
 With interest rate of math’s, businessman can calculate the interest on debt
, loan or bonds .
 With matrix , businessman can calculate salary bill of different department
and branches .
 With Linear programming , business can determine the quantity of two
products at which profit is maximize or cost is minimize .
 With discounting and factoring technique businessman can calculate present
value of bill , rat of discounting the bill and banker's gain
 With currency translation formula of math’s, businessman can cost of
currency between two countries .
 With Assignment solution technique businessman can solve transportation
problems.
 Large numbers of business estimations are done on the basis of probability
theory of business mathematics.
 Market research bureau can use consistency data theory of business
mathematics for calculating estimating profit , capital and sale of business .
The Importance of Basic Mathin Business
by Carol Deeb, DemandMedia
Business ownership requires morethan skill in creating a productor talent at
providing a service. Overseeing the finances of your company is key to survival
and success. Understanding basic business math is necessary for profitable

operations and accurate record keeping. Knowing how to add, subtract, multiply,
divide, round and usepercentages and fractions is the minimum you need to
price your productand meet your budget. If math is not your strength, partner
with someonewho can take over that role or hire a trusted employee to help
your operation stay in the black and grow responsibly.
Calculate Production Costs
Before you formally establish your business, you mustestimate the costto
manufactureor acquireyour product or performyour service. Adding all expenses
associated with making or buying items helps you realize if you can be
competitive with other companies and profitable enough to sustain your business
and make a reasonableincome. In addition to the standard costs of production,
such as materials and machinery, add accompanying expenses, such as shipping,
labor, interest on debt, storage and marketing. The basis to your business plan is
an accurate representation of how much you will spend on each item.
Determine Pricing
To ensureyou can operate your business and produceenough cash flow to invest
into your enterprise, you mustcharge enough for your productto be profitable.
Markup is the difference between your merchandisecost and the selling price,
giving you gross profit. If your operations requirea large markup, such as 70
percent, you may not be competitive in your industry if other companies sell the
same items for less. Once you have determined your markup, oneway to
calculate the retail price is to divide using percents or decimals. For example, if a
productcosts $10 to produceand your markup is 35 percent, subtract.35 from1
(or 100 percent), which gives you .65, which is 65 percent. To calculate the price
of your product, divide 10 by .65, which rounds to $15.38.
If you want to determine the net profitfor a certain time period, you will need to
subtractreturns, costs to producean item and operating expenses fromyour
total amount of sales, or gross revenue, during that time. Discounts on products,
depreciation on equipment and taxes also must be calculated and subtracted
fromrevenue. To arriveat your net profit, add any interest you earned from
credit extended to customers, which is reflected as a percent of the amount each
person owes. Your net profithelps you understand if you are charging enough for
your productand selling an adequate volume to continue to operate your
business or even expand.

AnalyzeFinances
To analyze the overall financial health of your business, you willneed to project
revenue and expenses for the future. It's important to understand the impact to
your accounting records when you changea number to reflect an increaseor
decrease in futuresales. Estimating how much an employee affects revenue will
indicate if you can afford to add to your staff and if the profits realized will be
worth the expense. If a competitor starts selling a cheaper product, you may need
to calculate the amount by which your volume must increaseif you reduce prices.
You may need to know if you can afford to expand your operations to improve
sales. Using basic business math to understand how these types of actions impact
your overall finances is imperative before taking your business to the next level.
The importance of mathematics
Mathematical thinking is important for all members of a modern society as a habit of
mind for its use in the workplace, business and finance; and for personal decision-
making. Mathematics is fundamental to national prosperity in providing tools for
understanding science, engineering, technology and economics. It is essential in public
decision-making and for participation in the knowledge economy.
Mathematics equips pupils with uniquely powerful ways to describe, analyse and
change the world. It can stimulate moments of pleasure and wonder for all pupils when
they solve a problem for the first time, discover a more elegant solution, or notice
hidden connections. Pupils who are functional in mathematics and financially capable
are able to think independently in applied and abstract ways, and can reason, solve
problems and assess risk.
Mathematics is a creative discipline. The language of mathematics is international. The
subject transcends cultural boundaries and its importance is universally recognised.
Mathematics has developed over time as a means of solving problems and also for its
own sake.
Scope and importance of business mathematics?
Best Answer: Mathematics is an important subject and knowledge of it enhances a
person's reasoning, problem-solving skills, and in general, the ability to think. Hence it is
important for understanding almost every subject whether science and technology,
medicine, the economy, or business and finance. Mathematical tools such as the theory
of chaos are used to mapping market trends and forecasting of the same. Statistics and
probability which are branches of mathematics are used in everyday business and

economics. Mathematics also form an important part of accounting, and many
accountancy companies prefer graduates with joint degrees with mathematics rather
than just an accountancy qualification. Financial Mathematics and Business
Mathematics form two important branches of mathematics in today's world and these
are direct application of mathematics to business and economics. Examples of applied
maths such as probability theory and management science, such as queuing theory,
time-series analysis, linear programming all are vital maths for business.

History of Numbers
Numbers and counting have become an integral part of our everyday life,
especially when we take into account the modern computer.These words you
are reading have been recorded on a computer using a code of ones and
zeros. It is an interesting story how these digits have come to dominate our
world.
Numbers Around the World
Presently, the earliest known archaeological evidence of any form of writing
or counting are scratch marks on a bone from 150,000 years ago. But the
first really solid evidence of counting,in the form of the number one, is from
a mere twenty-thousand years ago. An ishango bone was found in the
Congo with two identical markings of sixty scratches each and equally
numbered groups on the back.These markings are a certain indication of
counting and they mark a defining moment in western civilization.1
Zoologists tell us that mammals other than humans are only able to count
up to three or four, while our early ancestors were able to count
further.They believed that the necessity for numbers became more apparent
when humans started to build their own houses, as opposed to living in
caves and the like.
Anthropologists tell us that in Suma, in about 4,000 BCE, Sumerians used
tokens to represent numbers, an improvement over notches in a stick or
bone. A very important development from using tokens to represent
numbers was that in addition to adding tokens you can also take away,
giving birth to arithmetic, an event of major significance.The Sumerian’s
tokens made possible the arithmetic required for them to assess wealth,
calculate profit and loss and even more importantly, to collect taxes, as well
as keep permanent records. The standard belief is that in this way numbers
became the world’s first writings and thus accounting was born.
More primitive societies, such as the Wiligree of Central Australia, never
used numbers, nor felt the need for them.We may ask, why then did the
Sumerians on the other side of the world feel the need for simple
mathematics? The answer of course, was because they lived in cities which
required organizing. For example, grain needed to be stored and
determining how much each citizen received required arithmetic.
Egyptians loved all big things, such as big buildings, big statues and big
armies. They developed numbers of drudgery for everyday labor and large
numbers for aristocrats, such as a thousand, ten thousand and even a

million.The Egyptians transformation of using “one” from counting things to
measuring things was of great significance.
Their enthusiasm for building required accurate measurements so they
defined their own version of “one.” A cubit was defined as the length of a
mans arm from elbow to finger tips plus the width of his palm. Using this
standardized measure of “one” the Egyptians completed vast construction
projects, such as their great pyramids, with astonishing accuracy.
Two and a half thousand years ago, in 520 BCE, Pythagorus founded his
vegetarian school of math in Greece. Pythagorus was intrigued by whole
numbers,noticing that pleasing harmonies are combinations of whole
numbers. Convinced that the number one was the basis of the universe, he
tried to make all three sides of a triangle an exact number of units, a feat
which he was not able to accomplish. He was thus defeated by his own
favorite geometrical shape, one for which he would be forever famous.
His Pythagorean theorem has been credited to him, even though ancient
Indian texts, the Sulva Sutras (800 BCE) and the Shatapatha Brahmana (8th
to 6th centuries BCE) prove that this theorem was known in India some two
thousand years before his birth.
Later in the third century BCE, Archimedes, the renowned Greek scientist,
who loved to play games with numbers, entered the realm of the
unimaginable, trying to calculate such things as how many grains of sand
would fill the entire universe. Some of these intellectual exercises proved to
be useful, such as turning a sphere into a cylinder. His formula was later
used to take a globe and turn it into a flat map.
Romans invading Greece were interested in power, not abstract
mathematics. They killed Archimedes in 212 BCE and thereby impeded the
development of mathematics. Their system of Roman numerals was too
complicated for calculating, so actual counting had to be done on a counting
board, an early form of the abacus.
Although the usage of the Roman numeral system spread all over Europe
and remained the dominant numeral system for more than five hundred
years, not a single Roman mathematician is celebrated today. The Romans
were more interested in using numbers to record their conquests and count
dead bodies.
Numbers in Early India

In India, emphasis was not on military organization but in finding
enlightenment. Indians, as early as 500 BCE, devised a system of different
symbols for every number from one to nine, a system that came to be called
Arabic numerals, because they spread first to Islamic countries before
reaching Europe centuries later.
What is historically known goes back to the days of the Harappan civilization
(2,600-3,000 BCE). Since this Indian civilization delved into commerce and
cultural activities, it was only natural that they devise systems of weights
and measurements. For example a bronze rod marked in units of 0.367
inches was discovered and points to the degree of accuracy they demanded.
Evidently,such accuracy was required for town planning and construction
projects.Weights corresponding to units of 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10,
20, 50, 100, 200 and 500 have been discovered and they obviously played
important parts in the development of trade and commerce.
It seems clear from the early Sanskrit works on mathematics that the
insistent demand of the times was there, for these books are full of problems
of trade and social relationships involving complicated calculations. There
are problems dealing with taxation, debt and interest, problems of
partnership, barter and exchange, and the calculation of the fineness of
gold. The complexities of society, government operations and extensive
trade required simpler methods of calculation.
Earliest Indian Literary and Archaeological References
When we discuss the numerals of today’s decimal number system we usually
refer to them as “Arabian numbers.” Their origin, however, is in India, where
they were first published in the Lokavibhaga on the 28th of August 458
AD.This Jain astronomical work, Lokavibhaga or “Parts of the Universe,” is
the earliest document clearly exhibiting familiarity with the decimal system.
One section of this same work gives detailed astronomical observations that
confirm to modern scholars that this was written on the date it claimed to be
written: 25 August 458 CE (Julian calendar). As Ifrah2 points out, this
information not only allows us to date the document with precision, but also
proves its authenticity. Should anyone doubt this astronomical information,
it should be pointed out that to falsify such data requires a much greater
understanding and skill than it does to make the original calculations.
The origin of the modern decimal-based place value system is ascribed to
the Indian mathematician Aryabhata I, 498 CE. Using Sanskrit numeral
words for the digits, Aryabhata stated “Sthanam sthanam dasa gunam” or
“place to place is ten times in value.”The oldest record of this value place

assignment is in a document recorded in 594 CE, a donation charter of
Dadda III of Sankheda in the Bharukachcha region.
The earliest recorded inscription of decimal digits to include the symbol for
the digit zero, a small circle, was found at the Chaturbhuja Temple at
Gwalior, India, dated 876 CE.This Sanskrit inscription states that a garden
was planted to produce flowers for temple worship and calculations were
needed to assure they had enough flowers. Fifty garlands are mentioned
(line 20), here 50 and 270 are written with zero. It is accepted as the
undisputed proof of the first use of zero.
The usage of zero along with the other nine digits opened up a whole new
world of science for the Indians. Indeed Indian astronomers were centuries
ahead of the Christian world.The Indian scientists discovered that the earth
spins on its axis and moves around the sun, a fact that Copernicus in Europe
didn’t understand until a thousand years later—a discovery that he would
have been persecuted for, had he lived longer.
From these and other sources there can be no doubt that our modern
system of arithmetic—differing only in variations on the symbols used for the
digits and minor details of computational schemes—originated in India at
least by 510 CE and quite possibly by 458 CE.
The first sign that the Indian numerals were moving west comes from a
source which predates the rise of the Arab nations. In 662 AD Severus
Sebokht, a Nestorian bishop who lived in Keneshra on the Euphrates river,
wrote regarding the Indian system of calculation with decimal numerals:
“ ... more ingenious than those of the Greeks and the Babylonians, and of
their valuable methods of calculation which surpass description...” 3
This passage clearly indicates that knowledge of the Indian number system
was known in lands soon to become part of the Arab world as early as the
seventh century. The passage itself, of course, would certainly suggest that
few people in that part of the world knew anything of the system. Severus
Sebokht as a Christian bishop would have been interested in calculating the
date of Easter (a problem to Christian churches for many hundreds of
years). This may have encouraged him to find out about the astronomy
works of the Indians and in these, of course, he would find the arithmetic of
the nine symbols.
The Decimal Number System

The Indian numerals are elements of Sanskrit and existed in several variants
well before their formal publication during the late Gupta Period (c. 320-540
CE). In contrast to all earlier number systems, the Indian numerals did not
relate to fingers, pebbles, sticks or other physical objects.
The development of this system hinged on three key abstract (and certainly
non-intuitive) principles: (a) The idea of attaching to each basic figure
graphical signs which were removed from all intuitive associations, and did
not visually evoke the units they represented; (b) The idea of adopting the
principle according to which the basic figures have a value which depends on
the position they occupy in the representation of a number; and (c) The idea
of a fully operational zero, filling the empty spaces of missing units and at
the same time having the meaning of a null number. 4
The great intellectual achievement of the Indian number system can be
appreciated when it is recognized what it means to abandon the
representation of numbers through physical objects. It indicates that Indian
priest-scientists thought of numbers as an intellectual concept, something
abstract rather than concrete. This is a prerequisite for progress in
mathematics and science in general, because the introduction of irrational
numbers such as “pi,” the number needed to calculate the area inside a
circle, or the use of imaginary numbers is impossible unless the link between
numbers and physical objects is broken.
The Indian number system is exclusively a base 10 system, in contrast to
the Babylonian (modern-day Iraq) system, which was base 60; for example,
the calculation of time in seconds, minutes and hours. By the middle of the
2nd millennium BC, the Babylonian mathematics had a sophisticated
sexagesimal positional numeral system (based on 60, not 10). Despite the

invention of zero as a placeholder, the Babylonians never quite discovered
zero as a number.
The lack of a positional value (or zero) was indicated by a space between
sexagesimal numerals.They added the “space” symbol for the zero in about
400 BC. However, this effort to save the first place-value number system did
not overcome its other problems and the rise of Alexandria spelled the end
of the Babylonian number system and its cuneiform (hieroglyphic like)
numbers.
It is remarkable that the rise of a civilization as advanced as Alexandria also
meant the end of a place-value number system in Europe for nearly 2,000
years. Neither Egypt nor Greece nor Rome had a place-value number
system, and throughout medieval times Europe used the absolute value
number system of Rome (Roman Numerals). This held back the
development of mathematics in Europe and meant that before the period of
Enlightenment of the 17th century, the great mathematical discoveries were
made elsewhere in East Asia and in Central America.
The Mayans in Central America independently invented zero in the fourth
century CE.Their priest-astronomers used a snail-shell-like symbol to fill
gaps in the (almost) base-20 positional ‘long-count’ system they used to
calculate their calendar. They were highly skilled mathematicians,
astronomers, artists and architects. However, they failed to make other key
discoveries and inventions that might have helped their culture survive. The
Mayan culture collapsed mysteriously around 900 CE. Both the Babylonians
and the Mayans found zero the symbol, yet missed zero the number.
Although China independently invented place value, they didn’t make the
leap to zero until it was introduced to them by a Buddhist astronomer from
India in 718 CE.
Zero becomes a real number
The concept of zero as a number and not merely a symbol for separation is
attributed to India where by the 9th century CE practical calculations were
carried out using zero, which was treated like any other number, even in the
case of division.
The story of zero is actually a story of two zeroes: zero as a symbol to
represent nothing and zero as a number that can be used in calculations and
has its own mathematical properties.
It has been commented that in India, the concept of nothing is important in
its early religion and philosophy and so it was much more natural to have a

symbol for it than for the Latin (Roman) and Greek systems. The rules for
the use of zero were written down first by Brahmagupta, in his book
“Brahmasphutha Siddhanta” (The Opening of the Universe) in the year 628
CE. Here Brahmagupta considers not only zero, but negative numbers, and
the algebraic rules for the elementary operations of arithmetic with such
numbers.
“The importance of the creation of the zero mark can never be
exaggerated.This giving to airy nothing, not merely a local habitation and a
name, a picture, a symbol, but helpful power, is the characteristic of the
Hindu race from whence it sprang. It is like coining the Nirvana into
dynamos. No single mathematical creation has been more potent for the
general on-go of intelligence and power.” - G. B. Halsted 5
A very important distinction for the Indian symbol for zero, is that, unlike
the Babylonian and Mayan zero, the Indian zero symbol came to be
understood as meaning nothing.
As the Indian decimal zero and its new mathematics spread from the Arab
world to Europe in the Middle Ages, words derived from sifr and zephyrus
came to refer to calculation, as well as to privileged knowledge and secret
codes. Records show that the ancient Greeks seemed unsure about the
status of zero as a number.They asked themselves,“How can nothing be
something?” This lead to philosophical and, by the Medieval period, religious
arguments about the nature and existence of zero and the vacuum.
The word “zero” came via the French word zéro, and cipher came from the
Arabic word safira which means “it was empty.” Also sifr, meaning “zero” or
“nothing,” was the translation for the Sanskrit word sunya, which means
void or empty.
The number zero was especially regarded with suspicion in Europe, so much
so that the word cipher for zero became a word for secret code in modern
usage. It is very likely a linguistic memory of the time when using decimal
arithmetic was deemed evidence of dabbling in the occult, which was
potentially punishable by the all-powerful Catholic Church with death.6
WHAT IS THE ORIGIN OF
NUMBERS?

The numbers that are used worldwide nowadays are basically Arabic numbers. They
were actually developed by Arabic Muslim scientists who revised the Indian version of
numbers that contains only nine numbers. That took place during the 8th century (771
A.D) when an Indian Gastronomist came to the Almansour royal palace with a book –
famous at that time – about astronomy and mathematics called “Sod hanta” written by
Brahma Jobta around 626 A.D. Almansour ordered to translate the book into Arabic and
explore more sciences.
There were different forms in the Indian version of numbers, Arabs kept some of these
forms and changed others to create their own vision of numbers which was used in the
Middle East and mainly in Baghdad. Thanks to Al Khawarizmi (Algoritmi), Arabic
numbers took their final form. At the beginning, they were not widely spread but they
became known in the Maghreb and Andalusia. Europe then adopted these numbers
because of their practicality in comparison with Roman numbers. Now they are used
worldwide.
1 -Indian numbers:
2-Arabic numbers:
1, 2, 3, 4, 5, 6, 7, 8, 9
In designing the Arab numbers, Al Khawarizmi based his choice of a particular form on
the number of angles that each number should contain. For instance, the number one
contains only one angle, number two has two angles, and number three includes three
angles, ects…
This picture clarifies the original forms of the Arab numbers, in each angle contains a
dot:

These numbers were later modified until they reached the present forms in which we
use them now. But the genius invention that Muslim scientists brought to us is the zero
(as it contains no angles).
The first usage of the zero dates back to 873 A.D, but the first Indian zero was
registered around 876 A.D.
The numbers used worldwide nowadays are all Arabic numbers not only because of
their beautiful forms, but also because of their practicality . Indeed, unlike Indian
numbers, Arabic numbers make a clear distinction between the zero and the dot so that
no confusion would be made while reading numbers.
Classification of Numbers
This classification of Numbers represents the most accepted elementary classification,
and is useful in computing sense.
Class Symbol Description

Natural Number Natural numbers are defined as non-negative counting
numbers: = { 0, 1, 2, 3, 4, ... }. Some exclude 0
(zero) from the set: * = {0} = { 1, 2, 3, 4, ...
}.
Integer Integers extend by including the negative of
counting numbers:
= { ..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ... }.
The symbol stands for Zahlen, the German word for
"numbers".
Rational Number A rational number is the ratio or quotient of an integer
and another non-zero integer:
= {n/m | n, m ∈ , m ≠ 0 }.
E.g.: -100, -20¼, -1.5, 0, 1, 1.5, 1½ 2¾, 1.75, &c
Irrational Number Irrational numbers are numbers which cannot be
represented as fractions.
E.g.: √2, √3;, π, e.
Real Number Real numbers are all numbers on a number line. The set
of is the union of all rational numbers and all
irrational numbers.
Imaginary Number An imaginary number is a number which square is a
negative real number, and is denoted by the symbol i,
so that i2
= -1.
E.g.: -5i, 3i, 7.5i, &c.
In some technical applications, j is used as the symbol
for imaginary number instead of i.
Complex Number A complex number consists of two part, real number
and imaginary number, and is also expressed in the
form a + bi (i is notation for imaginary part of the
number).
E.g.: 7 + 2i
Definition of Real Numbers

Properties, Examples, and Non Examples
A Real Definition :
The definition in math text books of real numbers is often not helpful to the average
person who is trying to gain an introductory and intuitive sense of what a real
number.
Real numbers are just the numbers on the number line.
It is the easiest way to think of them. Basically, if you can put the number in
question on an infinitely big number line, then it is a real number. Also, you have to
be add ,subtract ,multiply, divide that number in a way that is consistent with the
number line. They include many types of numbers:
Types of Real Numbers with examples
 Rational Numbers -- in other words all integers , fractions and decimals
(including repeating decimals)
o ex: 2,3 -2, ½, -¾ , .34
 Irrational Numbers
o , , yes, irrational numbers can be ordered and put on a
number line, we know that comes before
Properties of Real Numbers
 Real numbers can be ordered (this is not true, for instance, of imaginary
numbers )
 They can be added, subtracted , multiplied and divided by nonzero numbers
in an ordered way. So what does that mean? Basically it means that
comes before on the number line and that they both come
before . We know that this fact is true for rational and irrational
numbers. Think about the rational numbers 3 and 5, we know that we can
order 3 and 5 as follows. 3 comes before 5 and both numbers come before
8(3+5) .
Even and Odd Numbers

Even numbers can be divided evenly into groups of two. The number four can be
divided into two groups of two.
Odd numbers can NOT be divided evenly into groups of two. The number five can
be divided into two groups of two and one group of one.
Even numbers always end with a digit of 0, 2, 4, 6 or 8.
2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30 are even numbers.
Odd numbers always end with a digit of 1, 3, 5, 7, or 9.
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31 are odd numbers.
1
Odd
2
Even
3
Odd
4
Even
5
Odd
6
Even
7
Odd
8
Even
9
Odd
10
Even
11
Odd
12
Even
1
Odd
3
Odd
5
Odd
7
Odd
9
Odd
11
Odd
2
Even
4
Even
6
Even
8
Even
10
Even
12
Even
Odd Number
An odd number is an integer of the form , where is an integer.The odd numbers are therefore ...,
, , 1, 3, 5, 7, ... (OEIS A005408), which are also the gnomonic numbers.Integers which are notodd are
called even.
Odd numbers leave a remainder of1 when divided by two, i.e., the congruence holds for odd . The
oddness ofa number is called itsparity,so an odd number has parity1, while an even number has parity 0.
The generating function for the odd numbers is
The productof an even number and an odd number is always even, as can be seen by writing
which is divisible by 2 and hence is even.
Prime and Composite Numbers

A prime number is a whole number that only has two factors which are itself and
one. A composite number has factors in addition to one and itself.
The numbers 0 and 1 are neither prime nor composite.
All even numbers are divisible by two and so all even numbers greater than two
are composite numbers.
All numbers that end in five are divisible by five. Therefore all numbers that end
with five and are greater than five are composite numbers.
The prime numbers between 2 and 100 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31,
37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97.
http://gradestack.com/Complete-CAT-Prep/Number-System/Prime-Numbers-and/19133-3880-35700-
study-wtw

Chapter 3:
Indices,
Surds &
Logarithm

Exponents
The exponent of a number says how many times to use the number in a
multiplication.
In 82
the "2" says to use 8 twice in a multiplication,
so 82
= 8 × 8 = 64
In words: 82
could be called "8 to the power 2" or "8 to the second power", or
simply "8 squared"
Exponents are also called Powers or Indices.
Some more examples:
Example: 53
= 5 × 5 × 5 = 125
 In words: 53
could be called "5 to the third power", "5 to the power 3" or
simply "5 cubed"
Example: 24
= 2 × 2 × 2 × 2 = 16
 In words: 24 could be called "2 to the fourth power" or "2 to the power 4" or
simply "2 to the 4th"
Exponents make it easier to write and use many multiplications
Example: 96
is easier to write and read than 9 × 9 × 9 × 9 × 9 × 9

You can multiply any number by itself as many times as you want
using exponents.
Try here:
3 4
34
= 3 × 3 × 3 × 3 = 81
© 2015 MathsIsFun.com v0.81
In General
So in general:
an tells you to multiply a by itself,
so there are n of those a's:
Other Way of Writing It
Sometimes people use the ^ symbol (above the 6 on your keyboard), as it is
easy to type.
Example: 2^4 is the same as 24
 2^4 = 2 × 2 × 2 × 2 = 16
Negative Exponents
Negative? What could be the opposite of multiplying?
Dividing!

A negative exponent means how many times to divide one by the number.
Example: 8-1
= 1 ÷ 8 = 0.125
You can have many divides:
Example: 5-3
= 1 ÷ 5 ÷ 5 ÷ 5 = 0.008
But that can be done an easier way:
5-3
could also be calculated like:
1 ÷ (5 × 5 × 5) = 1/53
= 1/125 = 0.008
In General
That last example showed an easier way to handle negative
exponents:
 Calculate the positive exponent (an)
 Then take the Reciprocal (i.e. 1/an)
More Examples:
Negative Exponent Reciprocal of Positive Exponent Answer
4-2
= 1 / 42
= 1/16 = 0.0625
10-3
= 1 / 103
= 1/1,000 = 0.001
(-2)-3
= 1 / (-2)3
= 1/(-8) = -0.125

What if the Exponent is 1, or 0?
1
If the exponent is 1, then you just have the number itself
(example 91
= 9)
0 If the exponent is 0, then you get 1 (example 90 = 1)
But what about 00
? It could be either 1 or 0, and so people say it
is "indeterminate".
It All Makes Sense
My favorite method is to start with "1" and then multiply or divide as many
times as the exponent says, then you will get the right answer, for example:
Example: Powers of 5
.. etc..
52
1 × 5 × 5 25
51
1 × 5 5
50
1 1
5-1 1 ÷ 5 0.2
5-2
1 ÷ 5 ÷ 5 0.04
.. etc..

If you look at that table, you will see that positive, zero or negative exponents
are really part of the same (fairly simple) pattern.
Be Careful About Grouping
To avoid confusion, use parentheses () in cases like this:
With () : (-2)2
= (-2) × (-2) = 4
Without () : -22
= -(22
) = - (2 × 2) = -4
With () : (ab)2
= ab × ab
Without () : ab2
= a × (b)2
= a × b × b
Indices & the Law of Indices
Introduction
Indices are a useful way of more simply expressing large numbers. They also present us with many
useful properties for manipulating them using what are called the Law of Indices.
What are Indices?
The expression 25
is defined as follows:
We call "2" the base and "5" the index.
Law of Indices
To manipulate expressions, we can consider using the Law of Indices. These laws only apply to
expressions with the same base, for example, 34
and 32
can be manipulated using the Law of
Indices, but we cannot use the Law of Indices to manipulate the expressions 35
and 57
as their base
differs (their bases are 3 and 5, respectively).
Six rules of the Law of Indices
Rule 1:

Any number, except 0, whose index is 0 is always equal to 1, regardless of the value of the base.
An Example:
Simplify 20
:
Rule 2:
An Example:
Simplify 2-2
:
Rule 3:
To multiply expressions with the same base, copy the base and add the indices.
An Example:
Simplify : (note: 5 = 51
)
Rule 4:
To divide expressions with the same base, copy the base and subtract the indices.
An Example:
Simplify :
Rule 5:
To raise an expression to the nth index, copy the base and multiply the indices.
An Example:
Simplify (y2
)6
:
Rule 6:
An Example:
Simplify 1252/3
:

You have now learnt the important rules of the Law of Indices and are ready to try out some
examples!
Unit 3 Section 2 : Laws of Indices
There are three rules that should be used when working with indices:
When m and n are positive integers,
1. am × an = am + n
2. am ÷ an = am – n or
am
an
= am –n (m ≥ n)
3. (am)n = am × n
These three results are logical consequences of the definition of an , but really need a
formal proof. You can 'verify' them with particular examples as below, but this is not
a proof:
27
× 23
= (2 × 2 × 2 × 2 × 2 × 2 × 2) × (2 × 2 × 2)
= 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
= 210
(here m= 7, n = 3 and m + n = 10)

or,
27
÷ 23
=
2 × 2 × 2 × 2 × 2 × 2 × 2
2 × 2 × 2
= 2 × 2 × 2 × 2
= 24
(againm = 7, n = 3 and m – n = 4)
Also,
(27
)3
=
27
× 27
×
27
= 221
(usingrule 1) (againm = 7, n = 3 and m × n = 21)
The proof of the first rule is given below:
Proof
am
× an
= a × a × ... × a
m of these
× a × a × ... × a
n of these
= a × a × ... × a × a × a × ... × a
(m+n) of these
= am+n
The second and third rules can be shown to be true forall positive
integers m and n in a similar way.
We can see an important result using rule 2:

xn
xn
= xn – n
=
x0
but
xn
xn
= 1, so
x0
= 1
This is true for any non-zero value of x, so, for example, 30 = 1, 270 = 1 and 10010 =
1.
Example 1
Fill in the missing numbers in each of the following expressions:
(a) 24
× 26
= 2
(b) 37
× 39
= 3
(c) 36
÷ 32
= 3
(d) (104
)3
= 10
Example 2
Simplify each of the following expressions so that it is in the form an, where n is a
number:
(a) a6
× a7
(b)
a4
× a2
a3
(c) (a4
)3

Exercises
Work out the answers to the questions below and fill in the boxes. Click on
the button to find out whether you have answered correctly. If you are right
then will appear and you should move on to the next question.
If appears then your answer is wrong. Click on to clear your
original answer and have another go. If you can't work out the right answer then
click on to see the answer.
Question 1
Fill in the missing numbers:
(a) 23
× 27
= 2
(b) 36
× 35
= 3
(c) 37
÷ 34
= 3
(d) 83
× 84
= 8
(e) (32
)5
= 3
(f) (23
)6
= 2
(g)
36
32
= 3
(h)
47
42
= 4
Question 2

(a) a3
× a2
= a
(b) b7
÷ b2
= b
(c) (b2
)5
= b
(d) b6
× b4
= b
(e) (z3
)9
= z
(f)
q16
q7
= q
Question 3
Explain why 94
= 38
.
( ) = ×
=
Question 4
Calculate:
(a) 30
+ 40
(b) 60
× 70
(c) 80
– 30
(d) 60
+ 20
– 40
Question 5

(a) 36
× 3 = 317
(b) 46
× 4 = 411
(c)
a6
a
= a4
(d) (z )6
= z18
(e) (a19
) = a95
(f) p16
÷ p = p7
(g) (p )8
= p40
(h) q13
÷ q = q
Question 6
Calculate:
(a)
23
22
+ 30
(b)
34
33
– 30
(c)
54
52
+
62
6

(d)
77
75
–
59
57
(e)
108
105
–
56
53
(f)
417
414
–
413
411
Question 7
(a) 82
= 2
(b) 813
= 9 = 3
(c) 256
= 5
(d) 47
= 2
(e) 1254
= 5
(f) 10006
= 10
(g) 81 = 4
(h) 256 = 4
= 8
Question 8

(a) 8 × 4 = 2 × 2 = 2
(b) 25 × 625 = 5 × 5 = 5
(c)
243
9
=
3
3
= 3
(d)
128
16
=
2
2
= 2
Question 9
Is eachof the following statements true or false?
(a) 32
× 22
= 64
(b) 54
× 23
= 107
(c)
68
28
= 38
(d)
108
56
= 22
Question 10
Complete each expression:
(a) (26
× 23
)4
= (2 )4
= 2

(b)
36
32
5
= (3 )5
= 3
(c)
23
× 24
27
4
= (2 )4
= 2
(d)
32
× 9
33
4
= (3 )4
= 3
(e)
62
× 68
63
4
= (6 )4
= 6
(f)
78
72
× 73
5
= (7 )5
= 7
Uses of Index Numbers
The main uses of index numbers are given below:
o Index numbers are used in the fields of commerce, meteorology, labour, industrial, etc.
o The index numbers measure fluctuations during intervals of time,
group differences ofgeographical position of degree etc.
o They are used to compare the total variations in the prices of different commodities in
which the unit of measurements differs with time and price etc.
o They measure the purchasing power of money.
o They are helpful in forecasting the future economic trends.
o They are used in studying difference between the comparable categories of
animals,persons or items.
o Index numbers of industrial production are used to measure the changes in the level of
industrial production in the country.

o Index numbers of import prices and export prices are used to measure the changes in
the trade of a country.
o The index numbers are used to measure seasonal variations and cyclical variations in a
time series.
Surds
When we can't simplify a number to remove a square root (or cube root etc)
then it is a surd.
Example: √2 (square root of 2) can't be simplified further so it is a surd
Example: √4 (square root of 4) can be simplified (to 2), so it is not a surd!
Have a look at some more examples:
Number Simplified As a Decimal
Surd or
not?
√2 √2 1.4142135...(etc) Surd
√3 √3 1.7320508...(etc) Surd
√4 2 2 Not a surd
√¼ ½ 0.5 Not a surd
3
√11 3
√11 2.2239800...(etc) Surd
3
√27 3 3 Not a surd
5
√3 5
√3 1.2457309...(etc) Surd
The surds have a decimal which goes on forever without repeating, and
are Irrational Numbers .

In fact "Surd" used to be another name for "Irrational",
but it is now used for a root that is irrational.
How did we get the word "Surd" ?
Well around 820 AD al-Khwarizmi (the Persian guy who we get the name
"Algorithm" from) called irrational numbers "'inaudible" ... this was later
translated to the Latin surdus ("deaf" or "mute")
Conclusion
 When it is a root and irrational, it is a surd.
 But not all roots are surds.
Surds
Introduction
Surds are numbers left in root form (√) to express its exact value. It has an infinite number of non-
recurring decimals. Therefore, surds are irrational numbers.
There are certain rules that we follow to simplify an expression involving surds. Rationalising the
denominatoris one way to simplify these expressions. It is done by eliminating the surd in the
denominator. This is shown in Rules 3, 5 and 6.
It can often be necessary to find the largest perfect square factor in order to simplify surds. The
largest perfect square factor is found by looking at any possible factors of the number that is being
square rooted. Lets say that you are looking at the square root of 242. Can you simplify this? Well, 2
x 121 is 242 and we can take the square root of 121 without leaving a surd (because we get 11).
Since we cannot take the square root of a larger number that can be multiplied by another to give
242 then we say that 121 is the largest perfect square factor.
Six Rules of Surds
Rule 1:
An Example:
Simplify :

Since , as 9 is the largest perfect square factor of 18.
Rule 2:
An Example:
Simplify :
Rule 3:
By multiplying both the numberator and denominator by the denominator you can rationalise the
denominator.
An Example:
rationalise :
Rule 4:
An Example:
Simplify :

Rule 5:
Following this rule enables you to rationalise the denominator.
An Example:
Rationalise :
Rule 6:
Following this rule enables you to rationalise the denominator.
An Example:
Rationalise :
Base b Logarithm
The base
b
logarithm of
x,
logbx,
is the power to which you need to raise
b
in order to get
x.

Symbolically,
logbx = y means by = x.
Logarithmic form Exponential form
Notes
1.
logbx
is only defined if
b
and
x
are both positive, and
b ≠ 1.
2.
log10x
is called the common logarithm of
x,
and is sometimes written as
log x.
3.
logex
is called the natural logarithm of
x
and is sometimes written as
ln x.
Examples
The following table lists some exponential equations and their equivalent logarithmic form.
Exponentia
l Form
103
= 1,000 42
= 16 33
= 27 51
= 5 70
= 1 4−2
= 1/16 251/2
= 5
Logarithmi
c Form
log10 1,000 =
3
log416 =
2
log327 =
3
log55 =
1
log71 =
0
log4(1/16) = −
2
log25 5 = 1/
2
Here are some for you to try
Exponential Form 102
= 100 3−2
= 1/9
Logarithmic Form
log = log =

Exponential Form ^ = ^ =
Logarithmic Form log31 = 0 log5(1/125) = − 3
Example 1 Calculating Logarithms by Hand
(a) log28 =
Power to which you need to raise
2
in order to get
8
=
3 Since
23
= 8
(b) log41 =
4
in order to get
1
=
0
Since
40
= 1
(c) log10 10,000 =
10
in order to get
10,000
=
4
Since
104
= 10,000
(d) log10 1/100 =
10
in order to get
1/100
=
−2
Since
10−2
= 1/100
(e) log327 =
(f) log93 =
(g) log3(1/81) =
Algebraic Properties of Logarithms

The following identities hold for any positive
a ≠ 1
and any positive numbers
x
and
y.
Identity Example
(a)
loga(xy) = logax + logay
log216 = log28 + log22
(b)
loga
(
x
y
)
= logax − logay
log2
(
5
3
)
= log25 − log23
(c)
loga(xr
) = rlogax
log2(65
) = 5log26
(d)
logaa = 1
loga1 = 0
log22 = 1
log31 = 0
(e)
loga
(
1
x
)
= − logax
log2
(
1
3
)
= − log23
(f)
logax =
log25 =

log x
log a
=
ln x
ln a
log 5
log 2
≈ 2.3219
Example 2 Using the Properties of Logarithms
Let
a = log 2,
b = log 3,
and
c = log 5.
Write the following in terms of
a,
b,
and
c.
Note If any answer you give is not simplified -- for instance, if you say
a + a
instead of
2a
-- it will be marked wrong.
Answer
(a)
log 6
log 2 + log 3 = a + b
(b) log 15
(c)
log 30
log 2 + log 3 + log 5 = a + b+c
(d) log 12
(e)
log 1.5
log 3 − log 2 = b − a
(f) log(1/9)
(g) log 32 log 25
= 5log 2 = 5a

(h) log(1/81)
Q Where do the identities come from?
A Roughly speaking, they are restatements in logarithmic form of the laws of
exponents.
Q Why is
logaxy = logax + logay
?
A Let
s = logax,
and
t = logay.
In exponential form, these equations say that
as
= x
and
at
= y.
Multiplying these two equations together gives
as
at
= xy,
that is,
as + t
= xy.
Rewriting this in logarithmic form gives
loga(xy) = s + t = logax + logay
as claimed.
Here is an intuitive way of thinking about it: Since logs are exponents, this identity
expresses the familiar law that the exponent of a product is the sum of the exponents.
The second logarithmic identity is shown in almost the identical way, and we leave it
for you for practice.
Q Why is
loga(xr
) = rlogax
?
A Let
t = logax.
Writing this in exponential form gives
at
= x.

Raising this equation to the
rth
power gives
art
= xr
.
Rewriting in logarithmic form gives
loga(xr
) = rt = rlogax,
as claimed.
Identity (d) we will leave for you to do as practice.
Q Why is
loga(1/x) = − logax
?
A This follows from identities (b) and (d) (think about it). <>
Q Why is
logax =
log x
log a
=
ln x
ln a
?
A Let
s = logax.
In exponential form, this says that
as
= x.
Take the logarithm with base b of both sides, getting
logbas
= logbx,
then use identity (c):
slogba = log bx,
so
s =
logbx
logba
Since logarithms are exponents, we can use them to solve equations where the
unknown is in the exponent.
Example 3 Solving for the Exponent

Solve the following equations for
x.
(a)
4−x2
= 1/64.
(b)
5(1.12x + 3
) = 200
Solution We can solve both of these equations by translating from exponential form
to logarithmic form.
(a) Write the given equation in logarithmic form:
4−x2
= 1/64 Exponential Form
log4(1/64) = − x2
Logarithmic Form
Thus, −x2
= log4(1/64) = − 3
giving x = ±31/2
.
(b) Before converting to logarithmic form, first divide both sides of the equation by 5:
5(1.12x + 3
) = 200
1.12x + 3
= 40 Exponential Form
log1.140 = 2x + 3 Logarithmic Form
This gives 2x + 3 = ln 40/ln 1.1 ≈ 38.7039, Identity (e)
so that x ≈ 17.8520.
You can now either go on and try the exericses in the exercise set for this topic.
PROPERTIES OF LOGARITHMS
Property 1: because .

Example 1: In the equation , the base is 14 and the exponent is 0.
Remember that a logarithm is an exponent, and the corresponding logarithmic
equation is where the 0 is the exponent.
Example 2: In the equation , the base is and the exponent is 0.
Remember that a logarithm is an exponent, and the corresponding logarithmic
equation is .
Example 3: Use the exponential equation to write a logarithmic
equation. The base x is greater than 0 and the exponent is 0. The corresponding
logarithmic equation is .
Example 4: In the equation , the base is 3, the exponent is 1, and the
answer is 3. Remember that a logarithm is an exponent, and the
corresponding logarithmic equation is .
Example 5: In the equation , the base is 87, the exponent is 1, and
the answer is 87. Remember that a logarithm is an exponent, and the
corresponding logarithmic equation is .
equation. If the base p is greater than 0, then .
Example 7: Since you know that , you can write the logarithmic
equation with base 3 as .
Example 8: Since you know that , you can write the logarithmic
equation with base 13 as .

equation with base 4. You can convert the exponential equation
to the logarithmic equation . Since the 16 can be written
as
, the equation can be written .
The above rules are the same for all positive bases. The most common bases are the
base 10 and the base e. Logarithms with a base 10 are called common logarithms,
and logarithms with a base e arenatural logarithms. On your calculator, the base 10
logarithm is noted by log, and the base e logarithm is noted by ln.
There are an infinite number of bases and only a few buttons on your calculator. You
can convert a logarithm with a base that is not 10 or e to an equivalent logarithm with
base 10 or e. If you are interested in a discussion on how to change the bases of a
logarithm, click on Change of Base.
For a discussion of the relationship between the graphs of logarithmic functions and
exponential functions, click on graphs.
Properties of Logarithms
1. loga (uv) = loga u + loga v 1. ln (uv) = lnu + lnv
2. loga (u / v) = loga u - loga v 2. ln (u/ v) = ln u - lnv
3. loga un
= n loga u 3. ln un
= n ln u
The properties on the left hold for any base a.
The properties on the right are restatements of the general properties for the natural
logarithm.
Many logarithmic expressions may be rewritten, either expanded or condensed, using
the three properties above. Expanding is breaking down a complicated expression into
simpler components. Condensing is the reverse of this process.
Example 2.
Expanding an expression.

rewrite usingexponentialnotation
property3
property1
Example 3.
Expanding an expression.
property2
property1
property3
Example 4.
Condensing an expression.
property3
property1
property2
Common Mistakes
 Logarithms break products into sums by property 1, but the logarithm of a
sum cannot be rewritten. For instance, there is nothing we can do to the
expression ln( x2
+ 1).
 log u - log v is equal to log (u / v) by property 2, it is not equal to log u / log v.
Exercise 3:
(a) Expand the expression . Answer
(b) Condense the expression 3 log x + 2 log y - (1/2) log z. Answer

4.3 - Properties of Logarithms
Change of Base Formula
One dilemma is that your calculator only has logarithms for two bases on it.
Base 10 (log) and base e (ln). What is to happen if you want to know the
logarithm for some other base? Are you out of luck?
No. There is a change of base formula for converting between different bases.
To find the log base a, where a is presumably some number other than 10
or e, otherwise you would just use the calculator,
Take the log of the argument divided by the log of the base.
loga x = ( logb x ) / ( logb a )
There is no need that either base 10 or base e be used, but since those are
the two you have on your calculator, those are probably the two that you're
going to use the most. I prefer the natural log (ln is only 2 letters while log is 3,
plus there's the extra benefit that I know about from calculus). The base that
you use doesn't matter, only that you use the same base for both the
numerator and the denominator.
loga x = ( log x ) / ( log a ) = ( ln x ) / ( ln a )
Example: log3 7 = ( ln 7 ) / ( ln 3 )
Logarithms are Exponents
Remember that logarithms are exponents, so the properties of exponents are
the properties of logarithms.
Multiplication
What is the rule when you multiply two values with the same base together
(x2
* x3
)? The rule is that you keep the base and add the exponents. Well,
remember that logarithms are exponents, and when you multiply, you're going
to add the logarithms.
The log of a product is the sum of the logs.

loga xy = loga x + loga y
Division
The rule when you divide two values with the same base is to subtract the
exponents. Therefore, the rule for division is to subtract the logarithms.
The log of a quotient is the difference of the logs.
loga (x/y) = loga x - loga y
Raising to a Power
When you raise a quantity to a power, the rule is that you multiply the
exponents together. In this case, one of the exponents will be the log, and the
other exponent will be the power you're raising the quantity to.
The exponent on the argument is the coefficient of the log.
loga xr
= r * loga x
Melodic Mathematics
Some of the statements above are very melodious. That is, they sound good.
It may help you to memorize the melodic mathematics, rather than the
formula.
 The log of a product is the sum of the logs
 The sum of the logs is the log of the products
 The log of a quotient is the difference of the logs
 The difference of the logs is the log of the quotient
 The exponent on the argument is the coefficient of the log
 The coefficient of the log is the exponent on the argument
Okay, so the last two aren't so melodic.
Common Mistakes
I almost hesitate to put this section in here. It seems when I try to point out a
mistake that people are going to make, that more people make it.

 The log of a sum is NOT the sum of the logs. The sum of the logs is the
log of the product. The log of a sum cannot be simplified.
loga (x + y) ≠ loga x + loga y
 The log of a difference is NOT the difference of the logs. The difference
of the logs is the log of the quotient. The log of a difference cannot be
simplified.
loga (x - y) ≠ loga x - loga y
 An exponent on the log is NOT the coefficient of the log. Only when the
argument is raised to a power can the exponent be turned into the
coefficient. When the entire logarithm is raised to a power, then it can
not be simplified.
(loga x)r
≠ r * loga x
 The log of a quotient is not the quotient of the logs. The quotient of the
logs is from the change of base formula. The log of a quotient is the
difference of the logs.
loga (x / y) ≠ ( loga x ) / ( loga y )
Logarithm
Logarithm is reverse of exponentiation. Exponential and logarithms are inverse functions of each
other. The relation between exponential function and logarithm is given below:
Here, a is known as base and x is exponent. We pronounce logayas "log of y at base a".
For example: Exponential expression 23=8 will be equivalent to logarithmic expression log28=3.
Types of Logarithms
There are two types of logarithms:
 Common Logarithm: A logarithm with base 10 is known as common logarithm. For
example: log104. We see log without any base very often. It means that the base is 10. For
 example: log104 or log 4 are same.
 Natural Logarithm: A logarithm with base e is known as natural logarithm. For
example: loge4. e is a constant whose value is approximately 2.178. Natural logarithms are
also represented as ln x.

Change of Base Formula
We can convert a logarithmic expression of one base into that of another base by using following
formula:
Following are the rules for operations of logarithms:
 loga(mn)=logam+logan
 $log_{a}(mn)=log_{a}m-log_{a}n$
 logamn=nlogam
Exponents follow following rules:
 loga1=0 as a0=1
 logaa=1 as a1=a
 logaax=x as ax=ax
Logarithms can be divided into two types:
 Common logarithms:
Logarithms of the base 10 are called common logarithms.
o log1025
o log1010
o log1016
 Natural logarithms:
Logarithms of the base eare called natural logarithms.
o loge10
o loge400
o loger
Did you know that logea can be represented as ln a?
https://brilliant.org/discussions/thread/types-of-exponents-2/
Exponential Functions: Introduction (page 1 of 5)

Exponential functions look somewhat similar to functions you have seen before, in that
they involve exponents, but there is a big difference, in that the variable is now the
power, rather than the base. Previously, you have dealt with such functions as f(x) = x2,
where the variable x was the base and the number 2 was the power. In the case of
exponentials, however, you will be dealing with functions such as g(x) = 2x, where the
base is the fixed number, and the power is the variable.
Let's look more closely at the function g(x) = 2x. To evaluate this function, we operate as
usual, picking values of x, plugging them in, and simplifying for the answers. But to
evaluate 2x, we need to remember how exponents work. In particular, we need to
remember that negative exponents mean "put the base on the other side of the fraction
line".
So, while positive x-values give us values like these:

...negative x-values give us values like these:
Copyright © ElizabethStapel 2002-2011 All RightsReserved
Putting together the "reasonable" (nicely
graphable) points, this is our T-chart:
...and this is our graph:

You should expect exponentials to look
like this. That is, they start small —very
small, so small that they're practically
indistinguishable from "y = 0", which is
the x-axis— and then, once they start
growing, they grow faster and faster, so
fast that they shoot right up through the
top of your graph.
You should also expect that your T-chart
will not have many useful plot points. For
instance, forx = 4 and x = 5, the y-values
were too big, and for just about all the
negative x-values, the y-values were too
small to see, so you would just draw the line right along the top of the x-axis.
Note also that my axis scales do not match. The scale on the x-axis is much wider than
the scale on the y-axis; the scale on the y-axis is compressed, compared with that of
the x-axis. You will probably find this technique useful when graphing exponentials,
because of the way that they grow so quickly. You will find a few T-chart points, and
then, with your knowledge of the general appearance of exponentials, you'll do your
graph, with the left-hand portion of the graph usually running right along the x-axis.
You may have heard of the term "exponential growth". This "starting slow, but then
growing faster and faster all the time" growth is what they are referring to. Specifically,
our function g(x) above doubled each time we incremented x. That is, when x was
increased by 1 over what it had been, y increased to twice what it had been. This is the
definition of exponential growth: that there is a consistent fixed period over which the
function will double (or triple, or quadruple, etc; the point is that the change is always a
fixed proportion). So if you hear somebody claiming that the world population is
doubling every thirty years, you know he is claiming exponential growth.
Exponential growth is "bigger" and "faster" than polynomial growth. This means that, no
matter what the degree is on a given polynomial, a given exponential function will
eventually be bigger than the polynomial. Even though the exponential function may
start out really, really small, it will eventually overtake the growth of the polynomial,
since it doubles all the time.

For instance, x10 seems
much "bigger" than 10x, and
initially it is:
But eventually 10x (in blue
below) catches up and
overtakes x10 (at the red
circle below, where x is ten
and y is ten billion), and it's
"bigger" than x10 forever
after:
Exponential functions always have some positive number other than 1 as the base. If
you think about it, having a negative number (such as –2) as the base wouldn't be very
useful, since the even powers would give you positive answers (such as "(–2)2 = 4") and
the odd powers would give you negative answers (such as "(–2)3 = –8"), and what would
you even do with the powers that aren't whole numbers? Also, having 0 or 1 as the base
would be kind of dumb, since 0 and 1 to any power are just0 and 1, respectively; what
would be the point? This is why exponentials always have something positive and other
than 1 as the base.
Logarithmic Series
Infinite series ofvarious simple functions ofthe logarithm include
(1)
(2)

(3)
(4)
where is the Euler-Mascheroni constant and is the Riemann zeta function.Note that the first two of these are
divergent in the classical sense,butconverge when interpreted as zeta-regularized sums.
SEE ALSO:
Logarithmic Series
In mathematics, the logarithmic function is main division. Now we are going to explain about how to help
to the students about logarithmic series. Basic logarithmic function is defined as function
of , in logarithmic series there is no series about logarithm function like ln(x), but there
is simple series in
In mathematics, the logarithmic function is main division. Now we are going to explain about how to help
to the students about logarithmic series. Basic logarithmic function is defined as function
of , in logarithmic series there is no series about logarithm function like ln(x), but there
is simple series in
That is
Here are two things that are sign change to plus and minus alternating for the logarithmic series.
Now we are going see about help to the students based on logarithmic series.
If a > 0, by Exponential Theorem
+ ............to infinity
putting a = 1 + x

If , by Exponential Theorem
............... to infinity
putting a = 1 + x
.. .. ...........................................to infinity.
By Binomial Theorem for any index
................... to infinity
Equating these two series
Equating coifficients of y on both sides,
..................... to infinity
...................................to infinity
so
...................................to infinity
this series is called Logarithmic Series

Some Basic Logarithmic Series
Back to Top
Logarithmic series for help to the students online:
1.
2.
3.
4.
5.
6.
These are the basic important logarithmic series.
Examples on Logarithmic Series
Back to Top
Below are some examples based on logarithmic series
Problem 1: To solve the logarithmic function of
Solution: Given function is
We would rewrite the function depended upon logarithm series.
Like,
Now the function is
Here just taking the series,

Answer:
Problem 2: To solve the logarithmic function of
Solution: Given function is
Here we can apply logarithmic law, that is
Now the function is
Here just taking the series,
Answer:
Logarithmic Series Practice Problems
Back to Top
Prepare some problems about logarithmic series for help:
Problem 1: To solve the logarithmic function of using the logarithmic series.
Answer
Problem 2: To solve the logarithmic function of ln(1+3x)^-2 using the logarithmic series.
Answer

Problem 3: To solve the logarithmic function of ln(1+x)^-5 using the logarithmic series.
Answer
Exponential Series
In this page, we are going to discuss about exponential series concept. If an exponential
function ex can be expressed as an infinite exponent series, then it is an exponent
series function, which is shown below. The exponential function in math is defined as ex,
where e is an any integer. For example, let us assume ex as an exponential function. Let
us take the value of x as zero, x = 0. Then, the solution is of the form of e0 = 1.
Exponential Series Expansion
Exponential series formula is of the form,
ex=1+x1!+x22!+x33!+......=∑∞n=0xnn!
From the given exponential series formula, we can port value for the exponent 'x' as an
infinite series like the given series of ex. If the exponent function of x is n, then the
coefficient function of x is 1n!.
Some of the exponential series formulas are as follows,
1. e−x=1−x1!+x22!−x33!+......
2. ex+e−x2=1+x22!+x44!+.........
3. ex−e−x2=x+x33!+x55!+.........
Exponential Fourier Series
If D is the amplitude, ω0 is the radian frequency [rad/s], ω02π is the frequency [hertz]
and T0=2πω0 is the period [sec.], then the exponential fourier series is defined as:
f(t)=∑∞n=−∞Dnejnω0t
Where, Dn=1T0∫T0f(t)ejnω0tdt

ω0=2πT0
and D0 = C0 = a0
Taylor Series Exponential
If we have a function y = f(x) = ex, then the taylor series of the function at any point x = a
for the given function is as follows:
f(x)=ex=ea+ea(x−a)+ea2!(x−a)2+ea3!(x−a)3+......
If a = 0, then e0 = 1. Then, the exponential series is ex = 1+x+x22!+x33!+......
Some expansion of exponential series with the help of Taylor series is as follows:
1. e−x2=1−x21!+x42!−x63!+....
2. esinx=1+x+x22−x48−x515+.....
3. etanx=1+x+x22+x32+3x48+...... where IxI<π2
Exponential Theorem
If a > 0, then prove that ax=1+x(logea)+x22!(logea)2+......
Proof:
Exponential series is ex=1+x1!+x22!+x33!+...... ...................(i)
Now, let us assume that if logea=b, then eb=a
Put x = bx in equation (i), then we get
ebx=1+bx+(bx)22!+......
eb(x)=1+bx+(bx)22!+......
Replace ebx by ax, then
ax=1+x(logea)+x22!(logea)2+......
Exponential Series Examples
Given below are some of the examples on exponential series.

Solved Examples
Question 1: Expand the power series of given exponential function e2x.
Solution:
Given exponential function is e2x
We know power series formula for ex:
ex=1+x+x22!+x33!+........
Now, applying the above formula, we get
e2x=1+(2x)+(2x)22!+(2x)33!+......
= 1+2x+4x22!+8x36+16x424+.......
= 1+2x+2x2+43x3+23x4+......
Thus, the power series of e2x is 1+2x+2x2+43x3+23x4+......
Question 2: Expand the power series of the exponential function e- x.
Solution:
The exponential function is e- x
We know power series formula for ex:
ex=1+x+x22!+x33!+.........
Now, applying the above formula, we get
e−x=1+−x1!+(−x)22!+(−x)33!+........
= 1+−x1+(−x)22+(−x)36+........
= 1−x+x22−x36+.......
Thus, the power series of e- x
is 1−x+x22−x36+.......

Set theory
Written by: Robert R. Stoll
Set theory, branch of mathematics that deals with the properties of well-defined
collections of objects, which may or may not be of a mathematical nature, such as
numbers or functions. The theory is less valuable in direct application to ordinary
experience than as a basis for preciseand adaptable terminology for
the definitionof complex and sophisticated mathematical concepts.
Between the years 1874 and 1897, theGerman mathematician and logician Georg
Cantor created a theory of abstract sets of entities and made it into a
mathematical discipline. This theory grew out of his investigations of some
concrete problems regarding certain types of infinite sets of real numbers. A set,
wroteCantor, is a collection of definite, distinguishableobjects of perception or
thought conceived as a whole. The objects are called elements or members of the
set.
The theory had the revolutionary aspectof treating infinite sets as mathematical
objects that are on an equal footing with those that can be constructed in a finite
number of steps. Since antiquity, a majority of mathematicians had carefully
avoided the introduction into their arguments of the actual infinite (i.e., of sets
containing aninfinity of objects conceived as existing simultaneously, at least in
thought). Since this attitude persisted until almost the end of the 19th century,
Cantor’s work was thesubjectof much criticism to the effect that it dealt with
fictions—indeed, that it encroached on the domain of philosophers and violated
the principles of religion. Once applications to analysis began to be found,
however, attitudes began to change, and by the 1890s Cantor’s ideas and results
were gaining acceptance. By 1900, settheory was recognized as a distinct branch
of mathematics.
At justthat time, however, severalcontradictions in so-called naive set theory
were discovered. In order to eliminate such problems, an axiomatic basis was
developed for the theory of sets analogous to that developed for
elementary geometry. The degree of success that has been achieved in this

development, as well as the presentstature of set theory, has been well
expressed in the Nicolas BourbakiÉléments de mathématique (begun 1939;
“Elements of Mathematics”): “Nowadays itis known to be possible, logically
speaking, to derive practically the whole of known mathematics from a single
source, The Theory of Sets.”
Introduction to naive set theory
Fundamentalset concepts
In naive set theory, a set is a collection of objects (called members or elements)
that is regarded as being a single object. To indicate that an object x is a member
of a setA one writes x ∊ A, while x ∉ A indicates that x is not a member of A. A set
may be defined by a membership rule (formula) or by listing its members within
braces. For example, the set given by the rule “prime numbers less than 10” can
also be given by {2, 3, 5, 7}. In principle, any finite set can be defined by an explicit
list of its members, but specifying infinite sets requires a rule or pattern to
indicate membership; for example, the ellipsis in {0, 1, 2, 3, 4, 5, 6, 7, …} indicates
that the list of natural numbers N goes on forever. Theempty (or void, or null)
set, symbolized by {} or Ø, contains no elements at all. Nonetheless, it has the
status of being a set.
A set A is called a subsetof a set B (symbolized by A ⊆ B) if all the members
of A are also members of B. For example, any set is a subsetof itself, and Ø is a
subsetof any set. If both A ⊆ B and B ⊆ A, then A and B haveexactly the same
members. Part of the set concept is that in this case A = B; that is, A and B are the
same set.
OPERATIONSON SETS
The symbol∪ is employed to denote the union of two sets. Thus, the set A ∪ B—
read “A union B” or “the union of A and B”—is defined as the set that consists of
all elements belonging to either set A or set B (or both). For example, suppose
that Committee A, consisting of the 5 members Jones, Blanshard, Nelson, Smith,
and Hixon, meets with Committee B, consisting of the 5 members Blanshard,
Morton, Hixon, Young, and Peters. Clearly, the union of Committees A and B must

then consistof 8 members rather than 10—namely, Jones, Blanshard, Nelson,
Smith, Morton, Hixon, Young, and Peters.
The intersection operation is denoted by the symbol∩. The set A ∩ B—read
“Aintersection B” or “the intersection of A and B”—is defined as the set
composed of all elements that belong to both A and B. Thus, the intersection of
the two committees in the foregoing example is the set consisting of Blanshard
and Hixon.
If E denotes the set of all positive even numbers and O denotes the set of all
positive odd numbers, then their union yields the entire set of positive integers,
and their intersection is the empty set. Any two sets whoseintersection is the
empty set are said to be disjoint.
When the admissible elements are restricted to some fixed class of objects U, U is
called the universalset (or universe). Then for any subset A of U,
the complement ofA (symbolized by A′ or U − A) is defined as the set of all
elements in the universeUthat are not in A. For example, if the universeconsists
of the 26 letters of the alphabet, the complement of the set of vowels is the set of
consonants.
In analytic geometry, the points on a Cartesian grid are ordered pairs (x, y) of
numbers. In general, (x, y) ≠ (y, x); ordered pairs are defined so that (a, b) = (c, d)
if and only if both a = c and b = d. In contrast, the set {x, y} is identical to the set
{y, x} because they have exactly the samemembers.
The Cartesian productof two sets A and B, denoted by A × B, is defined as the set
consisting of all ordered pairs (a, b) for which a ∊ A and b ∊ B. For example,
ifA = {x, y} and B = {3, 6, 9}, then A × B = {(x, 3), (x, 6), (x, 9), (y, 3), (y, 6), (y, 9)}.
RELATIONS IN SET THEORY
In mathematics, a relation is an association between, or property of, various
objects. Relations can be represented by sets of ordered pairs (a, b)
wherea bears a relation to b. Sets of ordered pairs are commonly used to
representrelations depicted on charts and graphs, on which, for example,
calendar years may be paired with automobile production figures, weeks with
stock market averages, and days with averagetemperatures.

A function f can be regarded as a relation between each objectx in its domain and
the value f(x). A function f is a relation with a special property, however: each x is
related by f to one and only one y. That is, two ordered pairs (x, y) and (x, z)
in fimply that y = z.
A one-to-one correspondencebetween sets A and B is similarly a pairing of each
object in A with one and only one object in B, with the dual property that each
object in B has been thereby paired with one and only one object in A. For
example, ifA = {x, z, w} and B = {4, 3, 9}, a one-to-one correspondencecan be
obtained by pairing x with 4, z with 3, and w with 9. This pairing can be
represented by the set {(x, 4), (z, 3), (w, 9)} of ordered pairs.
Many relations display identifiable properties. For example, in the relation “is the
same colour as,” each object bears the relation to itself as well as to someother
objects. Such relations are said to be reflexive. The ordering relation “less than or
equal to” (symbolized by ≤) is reflexive, but “less than” (symbolized by <) is not.
The relation “is parallel to” (symbolized by ∥) has the property that, if an object
bears the relation to a second object, then the second also bears that relation to
the first. Relations with this property aresaid to be symmetric. (Note that the
ordering relation is not symmetric.) These examples also havethe property that
whenever one object bears the relation to a second, which further bears the
relation to a third, then the first bears that relation to the third—e.g.,
if a < b and b < c, then a < c. Such relations are said to be transitive.
Relations that have all three of these properties—reflexivity, symmetry,
andtransitivity—arecalled equivalence relations. In an equivalence relation, all
elements related to a particular element, say a, are also related to each other,
and they formwhat is called the equivalence class of a. For example, the
equivalence class of a line for the relation “is parallel to” consists of the set of all
lines parallel to it.
Essential features of Cantorian set theory
At best, the foregoing description presents only an intuitive concept of a set.
Essential features of the concept as Cantor understood it include: (1) that a set is
a grouping into a single entity of objects of any kind, and (2) that, given an

object x and a set A, exactly one of the statements x ∊ A and x ∉ A is true and the
other is false. The definite relation that may or may not exist between an object
and a set is called the membership relation.
A further intent of this description is conveyed by what is called the principle of
extension—a set is determined by its members rather than by any particular way
of describing the set. Thus, sets A and B are equal if and only if every element
in A is also in B and every element in B is in A; symbolically, x ∊ A implies x ∊ B and
vice versa. There exists, for example, exactly one set the members of which are 2,
3, 5, and 7. Itdoes not matter whether its members are described as “prime
numbers less than 10” or listed in some order (which order is immaterial)
between small braces, possibly {5, 2, 7, 3}.
The positive integers {1, 2, 3, …} are typically used for counting the elements in a
finite set. For example, the set {a, b, c} can be put in one-to-one correspondence
with the elements of the set {1, 2, 3}. The number 3 is called the cardinal number,
or cardinality, of the set {1, 2, 3} as well as any set that can be put into a one-to-
one correspondencewith it. (Becausethe empty set has no elements, its
cardinality is defined as 0.) In general, a set A is finite and its cardinality is n if
there exists a pairing of its elements with the set {1, 2, 3, … , n}. A set for which
there is no such correspondenceis said to be infinite.
To define infinite sets, Cantor used predicate formulas. The phrase“x is a
professor” is an example of a formula; if the symbol x in this phraseis replaced by
the name of a person, there results a declarative sentence that is true or false.
The notation S(x) will be used to represent such a formula. The phrase“x is a
professor atuniversity y and xis a male” is a formula with two variables. If the
occurrences of x and y are replaced by names of appropriate, specific objects, the
result is a declarative sentence that is true or false. Given any formula S(x) that
contains the letter x (and possibly others), Cantor’s principleof abstraction asserts
the existence of a set A such that, for each object x, x ∊ A if and only if S(x) holds.
(Mathematicians later formulated a restricted principle of abstraction, also known
as the principle of comprehension, in which self-referencing predicates, or S(A),
are excluded in order to prevent certain paradoxes.See below Cardinality and
transfinite numbers.) Becauseof the principle of extension, the
set A corresponding to S(x) must be unique, and it is symbolized by {x | S(x)},

which is read “The set of all objects x such that S(x).” For instance, {x | x is blue} is
the set of all blue objects. This illustrates the fact that the principle of abstraction
implies the existence of sets the elements of which are all objects having a certain
property. Itis actually more comprehensive. For example, it asserts the existence
of a set B corresponding to “Either x is an astronautor x is a naturalnumber.”
Astronauts haveno particular property in common with numbers (other than
both being members of B).
EQUIVALENT SETS
Cantorian set theory is founded on the principles of extension and abstraction,
described above. To describesome results based upon these principles, the
notion ofequivalence of sets will be defined. The idea is that two sets are
equivalent if it is possibleto pair off members of the firstset with members of the
second, with no leftover members on either side. To capture this idea in set-
theoretic terms, the set Ais defined as equivalent to the set B (symbolized
by A ≡ B) if and only if there exists a third set the members of which are ordered
pairs such that: (1) the firstmember of each pair is an element of A and the
second is an element of B, and (2) each member of A occurs as a firstmember and
each member of B occurs as a second member of exactly one pair. Thus,
if A and B are finite and A ≡ B, then the third set that establishes this fact provides
a pairing, or matching, of the elements of A with those of B. Conversely, if it is
possibleto match the elements of A with thoseof B, thenA ≡ B, becausea set of
pairs meeting requirements (1) and (2) can be formed—i.e., ifa ∊ A is matched
with b ∊ B, then the ordered pair (a, b) is one member of the set. By thus defining
equivalence of sets in terms of the notion of matching, equivalence is formulated
independently of finiteness. As an illustration involving infinite sets, Nmay be
taken to denote the set of natural numbers 0, 1, 2, … (someauthors exclude 0
fromthe natural numbers). Then {(n, n2
) | n ∊ N} establishes the seemingly
paradoxical equivalence of N and the subsetof N formed by the squares of the
natural numbers.
As stated previously, a set B is included in, or is a subsetof, a set A (symbolized
byB ⊆ A) if every element of B is an element of A. So defined, a subsetmay

possibly include all of the elements of A, so that A can be a subsetof itself.
Furthermore, the empty set, because it by definition has no elements that are not
included in other sets, is a subsetof every set.
If every element of set B is an element of set A, but the converseis false
(henceB ≠ A), then B is said to be properly included in, or is a proper subset
of, A(symbolized by B ⊂ A). Thus, if A = {3, 1, 0, 4, 2}, both {0, 1, 2} and
{0, 1, 2, 3, 4} are subsets of A; but {0, 1, 2, 3, 4} is not a proper subset. A finite set
is nonequivalent to each of its proper subsets. This is not so, however, for infinite
sets, as is illustrated with the set N in the earlier example. (The equivalence
of N and its proper subsetformed by the squares of its elements was noted
by Galileo Galilei in 1638, who concluded that the notions of less than, equal to,
and greater than did not apply to infinite sets.)
CARDINALITY AND TRANSFINITENUMBERS
The application of the notion of equivalence to infinite sets was first
systematically explored by Cantor. With N defined as the set of natural numbers,
Cantor’s initial significantfinding was that the set of all rational numbers is
equivalent to N but that the set of all real numbers is not equivalent to N. The
existence of nonequivalentinfinite sets justified Cantor’s introduction of
“transfinite” cardinal numbers as measures of sizefor such sets. Cantor defined
the cardinalof an arbitrary set A as the concept that can be abstracted
fromA taken together with the totality of other equivalent sets. Gottlob Frege, in
1884, and Bertrand Russell, in 1902, both mathematical logicians, defined the
cardinal number of a set A somewhatmore explicitly, as the set of all sets that
are equivalent to A. This definition thus provides a place for cardinalnumbers as
objects of a universewhoseonly members are sets.
The above definitions are consistentwith the usageof natural numbers as
cardinal numbers. Intuitively, a cardinal number, whether finite (i.e., a natural
number) or transfinite(i.e., nonfinite), is a measure of the size of a set. Exactly
how a cardinal number is defined is unimportant; whatis important is that if
and only if A ≡ B.

To compare cardinalnumbers, an ordering relation (symbolized by <) may be
introduced by means of the definition if A is equivalent to a subset
of B and B is equivalent to no subsetof A. Clearly, this relation is irreflexive
and transitive: and imply .
When applied to naturalnumbers used as cardinals, the relation < (less than)
coincides with the familiar ordering relation for N, so that < is an extension of that
relation.
The symbolℵ0 (aleph-null) is standard for the cardinalnumber of N (sets of this
cardinality are called denumerable), and ℵ (aleph) is sometimes used for that of
the set of real numbers. Then n < ℵ0 for each n ∊ N and ℵ0 < ℵ.
This, however, is not the end of the matter. If the power set of a set A—
symbolizedP(A)—is defined as the set of all subsets of A, then, as Cantor
proved, for every set A—a relation that is known as Cantor’s theorem. It
implies an unending hierarchy of transfinitecardinals: .
Cantor proved that and suggested that there are no cardinal numbers
between ℵ0 and ℵ, a conjectureknown as the continuum hypothesis.
There is an arithmetic for cardinal numbers based on naturaldefinitions
of addition,multiplication, and exponentiation (squaring, cubing, and so on), but
this arithmetic deviates from that of the natural numbers when transfinite
cardinals are involved. For example, ℵ0 + ℵ0 = ℵ0 (becausethe set of integers is
equivalent to N), ℵ0 · ℵ0 = ℵ0 (because the set of ordered pairs of natural numbers
is equivalent to N), and c + ℵ0 = c for every transfinite cardinal c (becauseevery
infinite set includes a subsetequivalent to N).
The so-called Cantor paradox, discovered by Cantor himself in 1899, is the
following. By the unrestricted principle of abstraction, the formula “x is a set”
defines a set U; i.e., it is the set of all sets. Now P(U) is a set of sets and so P(U) is a
subsetof U. By the definition of < for cardinals, however, if A ⊆ B, then it is not
the casethat . Hence, by substitution, . But by Cantor’s theorem,
. This is a contradiction. In 1901 Russelldevised another contradiction of
a less technical nature that is now known as Russell’s paradox. The formula “x is a
set and (x ∉ x)” defines a set R of all sets not members of themselves.
Using proof by contradiction, however, it is easily shown that (1) R ∊ R. But then

by the definition of R it follows that (2) (R ∉ R). Together, (1) and (2) form a
contradiction.
Set theory
Written by: Robert R. Stoll
Axiomatic set theory
In contrastto naive set theory, the attitude adopted in an axiomatic development
of set theory is that it is not necessary to know whatthe “things” are that are
called “sets” or what the relation of membership means. Of sole concern are the
properties assumed about sets and the membership relation. Thus, in an
axiomatic theory of sets, set and the membership relation ∊ are undefined terms.
The assumptions adopted about these notions are called the axioms of the
theory. Axiomatic set theorems are the axioms together with statements that can
be deduced fromthe axioms using the rules of inference provided by a systemof
logic. Criteria for the choice of axioms include: (1) consistency—itshould be
impossibleto derive as theorems both a statement and its negation; (2)
plausibility—axioms should be in accord with intuitive beliefs about sets; and (3)
richness—desirableresults of Cantorian set theory can be derived as theorems.
The Zermelo-Fraenkelaxioms
The first axiomatization of set theory was given in 1908 by ErnstZermelo, a
German mathematician. Fromhis analysis of the paradoxes described abovein
the sectionCardinality and transfinite numbers, heconcluded that they are
associated with sets that are “too big,” such as the set of all sets in Cantor’s
paradox. Thus, the axioms that Zermelo formulated are restrictive insofar as the
asserting or implying of the existence of sets is concerned. As a consequence,

there is no apparent way, in his system, to derive the known contradictions from
them. On the other hand, the results of classicalset theory shortof the paradoxes
can be derived. Zermelo’s axiomatic theory is here discussed in a formthat
incorporates modifications and improvements suggested by later
mathematicians, principally Thoralf Albert Skolem, a Norwegian pioneer
in metalogic, and AbrahamAdolf Fraenkel, an Israelimathematician. In the
literature on set theory, it is called Zermelo-Fraenkelset theory and abbreviated
ZFC (“C” because of the inclusion of the axiom of choice).
SCHEMAS FOR GENERATING WELL-FORMED FORMULAS
The ZFC “axiom of extension” conveys the idea that, as in naive set theory, a set is
determined solely by its members. Itshould be noted that this is not merely a
logically necessary property of equality but an assumption about the membership
relation as well.
The set defined by the “axiom of the empty set” is the empty (or null) set Ø.
For an understanding of the “axiom schema of separation”
considerableexplanationis required. Zermelo’s original systemincluded the
assumption that, if a formula S(x) is “definite” for all elements of a set A, then
there exists a set the elements of which are precisely those elements x of A for
which S(x) holds. This is a restricted version of the principle of abstraction, now

known as the principle of comprehension, for it provides for the existence of sets
corresponding to formulas. Itrestricts that principle, however, in two ways: (1)
Instead of asserting the existence of sets unconditionally, it can be applied only in
conjunction with preexisting sets, and (2) only “definite” formulas may be used.
Zermelo offered only a vague description of “definite,” but clarification was given
by Skolem (1922) by way of a precise definition of what will be called simply a
formula of ZFC. Using tools of modern logic, the definition may be made as
follows:
 I. For any variables x and y, x ∊ y and x = y are formulas (such formulas are
called atomic).
 II. If S and T are formulas and x is any variable, then each of the following is a
formula: If S, then T; S if and only if T; S and T; S or T; not S; for all x, S; for
some x,T.
Formulas areconstructed recursively (in a finite number of systematic steps)
beginning with the (atomic) formulas of (I) and proceeding via the constructions
permitted in (II). “Not(x∊ y),” for example, is a formula (which is abbreviated
tox ∉ y), and “There exists an x such that for every y, y ∉ x” is a formula. A
variable isfreein a formula if it occurs at least once in the formula without being
introduced by one of the phrases “for some x” or “for all x.” Henceforth, a
formula S in which xoccurs as a free variable will be called “a condition on x” and
symbolized S(x). The formula “For every y, x ∊ y,” for example, is a condition on x.
Itis to be understood that a formula is a formalexpression—i.e.,
a term without meaning. Indeed, a computer can be programmed to generate
atomic formulas and build up fromthem other formulas of ever-increasing
complexity using logical connectives (“not,” “and,” etc.) and operators (“for all”
and “for some”). A formula acquires meaning only when an interpretation of the
theory is specified; i.e., when (1) a nonempty collection (called the domain of the
interpretation) is specified as the rangeof values of the variables (thus the term
set is assigned a meaning, viz., an object in the domain), (2) the membership
relation is defined for these sets, (3) the logical connectives and operators are
interpreted as in everyday language, and (4) the logical relation of equality is
taken to be identity among the objects in the domain.

The phrase“a condition on x” for a formula in which x is free is merely suggestive;
relative to an interpretation, such a formula does impose a condition on x. Thus,
the intuitive interpretation of the “axiom schema of separation” is: given a
set A and a condition on x, S(x), thoseelements of A for which the condition holds
forma set. Itprovides for the existence of sets by separating off certain elements
of existing sets. Calling this the axiom schema of separation is appropriate,
because it is actually a schema for generating axioms—one for each choice of S(x).
AXIOMS FOR COMPOUNDING SETS
Although the axiom schema of separation has a constructivequality, further
means of constructing sets from existing sets must be introduced if someof the
desirable features of Cantorian set theory are to be established. Three axioms—
axiom of pairing, axiom of union, and axiom of power set—are of this sort.
By using five of the axioms (2–6), a variety of basic concepts of naive set theory
(e.g., the operations of union, intersection, and Cartesian product; the notions of
relation, equivalence relation, ordering relation, and function) can be defined
with ZFC. Further, the standard results about these concepts that were attainable
in naive set theory can be proved as theorems of ZFC.
AXIOMS FOR INFINITE AND ORDERED SETS
If I is an interpretation of an axiomatic theory of sets, the sentence that results
froman axiom when a meaning has been assigned to “set” and “∊,” as specified
by I, is either true or false. If each axiom is true for I, then I is called a model of the
theory. If the domain of a model is infinite, this fact does not imply that any
object of the domain is an “infinite set.” An infinite set in the latter senseis an
object d of the domain D of I for which there is an infinity of distinct objects d′
in D such that d′Edholds (Estanding for the interpretation of ∊). Though the
domain of any model of the theory of which the axioms thus far discussed are
axioms is clearly infinite, models in which every set is finite have been devised.
For the full development of classicalset theory, including the theories of real

numbers and of infinite cardinal numbers, the existence of infinite sets is needed;
thus the “axiom of infinity” is included.
The existence of a unique minimal set ω having properties expressed in the axiom
of infinity can be proved; its distinct members are Ø, {Ø}, {Ø, {Ø}}, {Ø, {Ø}, {Ø,
{Ø}}}, … . These elements aredenoted by 0, 1, 2, 3, … and are called natural
numbers. Justification for this terminology rests with the fact that the Peano
postulates (five axioms published in 1889 by the Italian mathematician Giuseppe
Peano), which can serveas a base for arithmetic, can be proved as theorems in
set theory. Thereby the way is paved for the construction within ZFC of entities
that have all the expected properties of the real numbers.
The origin of the axiom of choice was Cantor’s recognition of the importance of
being able to “well-order” arbitrary sets—i.e., to define an ordering relation for a
given set such that each nonempty subsethas a least element. The virtue of a
well-ordering for a set is that it offers a means of proving that a property holds for
each of its elements by a process (transfiniteinduction) similar to mathematical
induction. Zermelo (1904) gavethe firstproof that any set can be well-ordered.
His proof employed a set-theoretic principle that he called the “axiom of choice,”
which, shortly thereafter, was shown to be equivalent to the so-called well-
orderingtheorem.
Intuitively, the axiom of choice asserts thepossibility of making a simultaneous
choice of an element in every nonempty member of any set; this guarantee
accounts for its name. The assumption is significant only when the set has
infinitely many members. Zermelo was the firstto state explicitly the axiom,
although it had been used but essentially unnoticed earlier (see also Zorn’s
lemma). Itsoon became the subjectof vigorous controversybecauseof its
nonconstructivenature. Some mathematicians rejected it totally on this ground.
Others accepted it but avoided its usewhenever possible. Some changed their
minds about it when its equivalence with the well-ordering theorem was proved
as well as the assertion that any two cardinal numbers c and d are comparable
(i.e., that exactly one of c < d, d < c, c = d holds). There are many other equivalent
statements, though even today a few mathematicians feel that the use of the
axiom of choice is improper. To the vastmajority, however, it, or an equivalent
assertion, has become an indispensableand commonplace tool. (Because of this

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