2. 2.1.3 Building the Modern Number System
The Natural Numbers
We build the modern number system beginning with
numbers used for counting.
Counting numbers, or natural numbers, comprise the set
{1,2,3,4,…}.
Natural numbers are further categorized according to their
factors (or divisors). Natural numbers are either prime or
composite.
The Fundamental Theorem of Arithmetic: every composite
number can be uniquely expressed as a product of prime
numbers.
4. The Integers
Negative numbers came out of subtracting natural
numbers.
Uses of negative numbers:
1. In commerce, where debts and losses are
represented by negative numbers.
2. In temperature and elevation measurements
The set of all numbers that we can make by
adding or subtracting natural numbers is called
the set of integers.
5. The integers include the natural numbers,
also called positive integers, zero, and the
negatives of all natural numbers, or negative
integers.
The set of whole numbers comprise zero and
the positive integers.
Properties of Integers:
1. Every integer, except 0, has a sign
which indicates whether it is positive
(+) or negative (-). 0 is neither positive nor
negative.
2. Every integer has a magnitude
(or absolute value) which indicates
how far it lies from 0 on the number
line.
6. The Rational Numbers
The set of all possible outcomes of dividing
integers (except dividing by 0) is called the set
of rational numbers.
Rational (from the word ratio which refers to
the division of two numbers)
The set of rational numbers is the set of all
numbers that can be expressed in the form x/y
where both x and y are integers and y ≠ 0.
The set of integers is a subset of the set of
rational numbers.
7. At one time in ancient Greece, all numbers were
believed to be rational numbers.
A secret society of followers of Pythagoras (500
B.C.) believed that numbers had special and
mystical meanings. Examples:
1 was considered divine.
Even numbers were considered feminine.
Odd numbers besides 1 were considered masculine.
The number 5, sum of the first feminine and masculine numbers,
represented marriage.
7 represented the seven “planets” known to the Greeks; the belief
that 7 was a “lucky number” probably came from them.
8. The motto of the Pythagoreans: “All is number.”
Their sacred belief was that all numbers were either “whole”,
by which they meant the natural numbers (they did not
recognize zero or negative numbers), or fractions made by
the division of “whole” numbers.
But using the Pythagorean Theorem, they realized that a
right triangle with two sides of length 1 has a third side of
length equal to the square root of 2 , which they could not
express as a fraction.
Eventually they proved that the square root of 2 cannot be
expressed by dividing two whole numbers, that is, it is an
irrational number.
They attempted to keep this as a secret because their
fundamental beliefs may be challenged, and even killed one
of their members, Hippasus, for telling others of their
discovery.
9. The Real Numbers
The combination of the rational and irrational
numbers is called the real numbers.
Another way to describe real numbers is as the
rational numbers and “everything in between”.
Each point on the number line has a
corresponding real number, and vice versa.
10. Imaginary and Complex Numbers
Finding a real number that is a square root of a
negative number is impossible. Thus another type of
“non-real” numbers, called imaginary numbers, was
invented to solve this problem.
Imaginary numbers are numbers that represent the
square root of negative numbers.
A special number called i (for “imaginary”) is defined
to be the square root of negative 1.
Imaginary numbers cannot be shown on a real number
line because they are not real numbers.
The complex numbers are numbers that include all the
real numbers and all the imaginary numbers.
11. Complex Numbers
Real Numbers Imaginary Numbers
Irrational Rational
Integers Other fractions
Negative 0 Positive
Prime 1 Composite
12. Numbers are Beautiful
If you don’t see why,
No one can tell you.
If they aren’t beautiful,
Nothing is.
-Paul Erdös
12
13. 13
NameName U.S. meaningU.S. meaning
MillionMillion 1,000,000 (6 zeros)1,000,000 (6 zeros)
BillionBillion 1,000,000,000 (9 zeros)1,000,000,000 (9 zeros)
TrillionTrillion 1,000,000,000,000 (12 zeros)1,000,000,000,000 (12 zeros)
QuadrillionQuadrillion 1 followed by 15 zeros1 followed by 15 zeros
QuintillionQuintillion 1 followed by 18 zeros1 followed by 18 zeros
SextillionSextillion 1 followed by 21 zeros1 followed by 21 zeros
SeptillionSeptillion 1 followed by 24 zeros1 followed by 24 zeros
OctillionOctillion 1 followed by 27 zeros1 followed by 27 zeros
Names and values of numbers
14. 14
Metric Prefixes for small values
prefixprefix Abbrev.Abbrev. valuevalue
decideci dd 1010-1-1
centicenti cc 1010-2-2
millimilli mm 1010-3-3
micromicro µµ 1010-6-6
nanonano nn 1010-9-9
picopico pp 1010-12-12
femtofemto ff 1010-15-15
attoatto aa 1010-18-18
zeptozepto zz 1010-21-21
yoctoyocto yy 1010-24-24
=
ns
s
1
1µ
s
s
9
6
10
10
−
−
.,000110 3
== −
15. 15
Metric Prefixes for large values
prefixprefix Abbrev.Abbrev. valuevalue
decadeca dada 101011
hectohecto hh 101022
kilokilo kk 101033
megamega MM 101066
gigagiga GG 101099
teratera TT 10101212
petapeta PP 10101515
exaexa EE 10101818
zettazetta ZZ 10102121
yottayotta YY 10102424
16. 2.1.4 Prime Numbers: Mysteries and
Applications
It is difficult to generate the sequence of prime
numbers.
Basic question: How many primes are there?
Euclid (c. 300 B.C.) proved that there are infinitely
many primes.
Erathosthenes, Greek mathematician who lived in
the third century B.C., devised a systematic
method for generating primes, called the Sieve of
Erathosthenes.
17. The Search for the Largest Prime
As of 2008, the largest prime is
with 12978189 digits. This is also the
largest prime so far.
17
1243112609
−
12 −p
• This is called a Mersenne prime, named
after F. Marin Mersenne. For a prime p, a
Mersenne prime is any prime of the form
18. Other Interesting Primes
Twin primes are two primes of the form p and p+2. The
largest as of 2009 is
with 100355 digits.
18
1256551646835 333333
±⋅
122
+
n
• A Fermat prime is a prime expressible in the
form
where .n 0≥
19. The Sieve of Erathosthenes (An
Algorithm)
Given a list of natural numbers from 1 to n.
Cross out 1 because it is neither prime nor composite.
The next number 2 is prime; cross out all subsequent multiples of
2 because they are composite. (We call 2 a sieve number because
it helps us “sift through” or remove other numbers in the list.)
The next number 3 is prime; then cross out all subsequent
multiples of 3.
Move to the next number that has not been crossed out. Use 5 as
a sieve number and cross out all multiples of 5 that have not been
crossed out yet.
Continue this process until we reach the end of the list.
The numbers that remain after all the “crossings out” are the
primes on the list.
21. In principle, the sieve of Erathosthenes could be used
on a list of numbers of any length.
The method, however, is extremely tedious.
Kulik, a 19th century Austrian astronomer, spent 20
years using this method to find all primes between 1
and 100 million!
However, the library to which he gave his manuscripts
lost the sections containing the primes between
12,642,000 and 22,852,800.
22. A formula to produce primes could generate
lists with much less effort.
Mathematicians have searched in vain for such
a formula for more than 2,000 years!
A few formulas work over a limited range of
numbers before failing.
Example: the expression n2
- n + 41 successfully
produces primes for small values of n. This
formula fails after n = 41, that is, it generates
only 41 primes before it produces a composite
number. In addition, the formula misses all
primes less then 41.
Other formulas also fail. Mathematicians
believe that a suitable formula does not exist.
23. The Search is On!
Great Internet Mersenne Prime Search
A project pioneered by George Woltman in 1997
A free software
So far has discovered 13 Mersenne primes
The 47th
Mersenne prime is the largest prime, which has been
discovered before the 45th
and 46th
.
23
24. 24
Divisibility rules for 2,3,4,5,9,10
All even numbers are divisible by 2.
A number is divisible by 3 if the sum of its
digits is divisible by 3.
A number is divisible by 4 if the sum of its last
2 digits is divisible by 4.
Example: 3451 is divisible by 3 since
(3+4+5+1) = 12 is divisible by 3.
A number is divisible by 5 if its last digit is 0 or
5.
25. 25
A number is divisible by 9 if the sum of its
digits is divisible by 9.
Example: 23454 is divisible by 9 since
2 + 3 +4 + 5 + 4 = 18 is divisible by 9.
A number is divisible by 10 if its last digit is 0.
26. Applications of prime numbers
In cryptography, wherein messages are written in code
to protect privacy and maintain security.
A security system can use large composite numbers as
a lock. The two primes multiplied to make the
composite represent the keys.
Because there is no efficient way to find the prime
factorization, the lock can be opened only by people
who hold the keys.
Research seeks efficient methods of factoring large
numbers, and computers are getting faster. But as
larger and larger primes are found, more inviolable
locks can be designed.
27. 2.1.5 Infinity
Infinity may be the most astonishing aspect of the
concept of numbers.
Georg Cantor (1845-1918) began a serious study of
infinity over a century ago. His results shocked
the mathematical world at that time.
Cardinality is another term for the number of
elements of a set.
28. How can you determine whether two sets have the
same cardinality?
One way is by counting the elements of each set
and see whether the count is the same for both
sets.
Another way is to determine if there is a one-to-
one correspondence between the members of the
two sets.
29. The Paradox of Infinite Sets
The set of natural numbers and the set of even integers
have the same cardinality.
1 2 3 4 5 6 …
2 4 6 8 10 12 ...
Similarly, there are as many natural numbers as odd
numbers, and as many natural numbers as multiples of
3, etc.
Galileo in 1638 considered these as unexplainable
paradoxes and chose not to work with infinity further.
30. The Arithmetic of Infinity
After 250 years, Cantor took these paradoxes as
starting points for further work and invented a new
arithmetic, called transfinite arithmetic, that applies to
infinity.
The cardinality of the natural numbers is symbolized
by ℵo (pronounced “aleph naught” or aleph null”).
We have the following results:
ℵo+ 1= ℵo and ℵo+ ℵo= ℵo
31. Consider the set of positive
rational numbers. What is the
cardinality of this set?
The infinite array above contains all positive rational numbers. We haveThe infinite array above contains all positive rational numbers. We have
the one-to-one correspondence:the one-to-one correspondence:
11 22 33 44 55 66 77 88 ......
1/11/1 1/21/2 2/12/1 3/13/1 2/22/2 1/31/3 1/41/4 2/32/3 ......
1 2 3 4 5 …
1 1/1 1/2 1/3 1/4 1/5
2 2/1 2/2 2/3 2/4 2/5
3 3/1 3/2 3/3 3/4 3/5
4 4/1 4/2 4/3 4/4 4/5
5 5/1 5/2 5/3 5/4 5/5
.
32. Thus, there are as many natural numbers as
there are rational numbers!
And we have the result:
ℵox ℵo= ℵo and (ℵo)2
= ℵo
Similarly, (ℵo)3
= ℵo and so on.
33. The Bane of Pythagoreans
Cantor showed that some infinite sets have cardinality
greater than ℵo.
He showed that the real numbers cannot be put into
one-to-one correspondence with the natural numbers,
and that there are more irrationals than rationals.
In fact, between any two points in the number line, the
number of irrationals is greater than the number of all
rational numbers.
The cardinality of this new, higher infinity is
designated ℵ1.
34. Thus the symbol ∞ cannot be used because
infinity has more than one “level”.
Outline of Cantor’s argument that there are more
irrationals than rationals:
Irrational numbers cannot be written exactly in decimal form
because they are non-terminating decimals. Suppose now that
there is a scheme for matching the irrationals to the natural
numbers, as in the list:
1 → 0.142678435…
2 → 0.383902892…
3 → 0.293758778…
4 → 0.563856365…
:
35. Regardless of the method used for matching,
we will always be able to write another
irrational number that is not already on the
list.
To do so, for the first digit of the new number,
we choose something other than the first digit
of the first number on the list, that is, anything
other than 1. For the second digit, we choose
something other than the second digit of the
second number on the list, or something other
than 8. And so on to infinity.
36. The resulting irrational number will differ in at
least one digit from every number on the list.
In other words, we will have found a number
that was “missed” by the matching scheme.
Thus, the natural numbers cannot be put in
one-to-one correspondence with the
irrationals.
Our conclusion: the cardinality of the
irrationals is greater than that of either the
natural or rational numbers.
37. Higher Orders of Infinity
Does a level of infinity exist between
ℵo and ℵ1?
The answer is unknown, but a set with such
cardinality has never been found.
The “continuum hypothesis” says that no set with
such cardinality exists.
38. Does a set with higher cardinality than the
reals exist?
Yes, as Cantor proved.
In fact, he showed that an infinite number of
higher levels of infinity exist, and their cardinality
might be designated
ℵo,ℵ1, ℵ2, ℵ3, ℵ4,...
But no one has ever been able to describe a set
with an infinity higher than ℵ2.
39. 2.1.6 Putting Numbers in Perspective
In ancient times, there was no way to express
extremely large or small numbers; in fact it was
unnecessary.
Today, these seemingly incomprehensible numbers
are dealt with in the real world.
Goal: learn to think quantitatively by developing
methods for interpreting such numbers.
40. tens or hundreds of billions of pesos of
spending and taxation
the collective impact of six billion people on
the environment
a nuclear weapon with one megaton of
explosive power
a computer with gigabytes of memory and
processing times measured in nanoseconds,
microseconds or milliseconds
Can you assess the values of these numbers?Can you assess the values of these numbers?
41. Survival and prosperity in the modern world
depend on decisions that involve numbers that
may, at first, seem incomprehensibly large or
small.
To make wise decisions, you must find ways of
putting such numbers into perspective.
Our task: to learn how to make extremely large or
small numbers comprehensible by relating them
to numbers which we are already familiar with.
42. 2.1.7 Writing Large or Small Numbers
Consider the following numbers:
The diameter of the Galaxy is about
1,000,000,000,000,000,000 kilometers
The nucleus of a hydrogen atom has a
diameter of about 0.000000000000001 meters
These numbers are difficult to read and most
people will just skip right over them. There is a
better way of expressing such numbers.
43. The Scientific Notation
Dealing with large and small numbers is much
easier with a special notation.
Numbers written with a number between
1 and 10 multiplied by a power of 10 are said to
be in scientific notation.
A number written in scientific notation can be
quickly converted to ordinary notation.
There is no shortcut for adding or subtracting
numbers in scientific notation.
44. Advantages of the Scientific Notation
The scientific notation simplifies writing extremely
large or small numbers.
Rounding and expressing numbers in scientific
notation allow quick approximations of the exact
answers.
Example: Estimate the product of 5795
and 326.
45. The danger of scientific notation
The scientific notation makes extremely large or
small numbers deceptively easy to write.
Example:
1026
does not look much different from 1020
, when
written, but is a million times larger
