This document discusses different types of number systems, including natural numbers, integers, rational numbers, real numbers, and complex numbers. It provides details on key concepts for each system. Natural numbers are the counting numbers starting from 1. Integers add negative whole numbers. Rational numbers are fractions with integer numerators and denominators. Real numbers include rational and irrational numbers, which can be represented by decimals. Complex numbers consist of real numbers combined with imaginary numbers using the imaginary unit i, where i^2 = -1. Each number system forms a proper subset of the next largest system.

1. English For Math
Suryakancana University, Second Lecture

2. Numeral system
• A numeral system (or system of numeration) is a
writing system for expressing numbers, that is, a
mathematical notation for representing numbers of a
given set, using digits or other symbols in a consistent
manner.
Numerical digit
• A digit is a numeric symbol (such as "2" or "5") used in
combinations (such as "25") to represent numbers
(such as the number 25) in positional numeral systems.
For example, the decimal system (base 10) has ten
digits (0 through to 9), whereas binary (base 2) has two
digits (0 and 1).

3. Positional systems in detail
• In a positional base-b numeral system (with b a natural
number greater than 1 known as the radix or base), b basic
symbols (or digits) corresponding to the first b natural
numbers including zero are used. For example, for the
decimal system (the most common system in use today) the
radix is ten, because it uses the ten digits from 0 through 9.
• the numeral (4327)10 means (4 × 103) + (3 × 102) +
(2 × 101) + (7 × 100), noting that 100 = 1.
• Another example is (100)10 (in the decimal system)
represents the number one hundred, whilst (100)2, (in the
binary system with base 2) represents the number four.
1 × 22 + 0 × 21 + 0 × 20 = 4.

4. Radix or Base

5. Number
• A number is a mathematical object used to count,
measure and label. The original examples are the
natural numbers 1, 2, 3, and so forth. A notational
symbol that represents a number is called a numeral. In
addition to their use in counting and measuring,
numerals are often used for labels (as with telephone
numbers), for ordering (as with serial numbers), and
for codes (as with ISBNs). In common usage, the term
number may refer to a symbol, a word or a
mathematical abstraction.

6. Number
Venn Diagram Subsets of the complex numbers.

7. Main classification
Different types of numbers have many different uses. Numbers can be classified
into sets, called number systems, such as the natural numbers and the real
numbers.

8. 1. Natural numbers (ℕ)
• In mathematics, the natural numbers (sometimes
called the whole numbers if 0 is included) The most
familiar numbers are the natural numbers or counting
numbers: 1, 2, 3, and so on. Traditionally, the sequence
of natural numbers started with 1. The mathematical
symbol for the set of all natural numbers is ℕ.
• In the base 10 numeral system, in almost universal use
today for mathematical operations, the symbols for
natural numbers are written using ten digits. It consists
of 10 symbols or digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9,
which are used to make our numbers. Each digit in a
number has a place value.

9. 1. Natural numbers (ℕ)
• the rightmost digit of a natural number has a place
value of 1, and every other digit has a place value ten
times that of the place value of the digit to its right.

10. 2. Integers (ℤ)
• we used the set of whole numbers which consists of the numbers
0, 1, 2, 3, 4, 5, . . . . In algebra we extend the set of whole numbers
by adding the negative numbers -1, -2, -3, -4, -5, . . . . The numbers
. . . -5, -4, -3, -2, -1, 0 1, 2, 3, 4, 5, . . . are called integers, written as
ℤ. The number zero is called the origin.
• Each integer has an opposite. The opposite of a given integer is the
corresponding integer, which is exactly the same distance from the
origin as the given integer. For example, the opposite of -4 is +4 or
4. The opposite of 0 is 0.
• The positive distance any number is from 0 is called the absolute
value of the number. The symbol for absolute value is | |. Hence,
| − 6| = 6 and | + 10| = 10. In other words, the absolute value of
any number except 0 is positive. The absolute value of 0 is 0, i.e.,
|0| = 0.

11. 2. Integers (ℤ)
• The natural numbers form a subset of the integers. the
natural numbers without zero are commonly referred to as
positive integers, and the natural numbers with zero are
referred to as non-negative integers.
• When the same number is multiplied by itself, the indicated
product can be written in exponential notation. For
example, 3 × 3 can be written as 32
, there the 3 is called
the base and the 2 is called the exponent.
• 32 is read as ‘‘three squared’’ or ‘‘three to the second
power’’, 33
is read as ‘‘three cubed’’ or ‘‘three to the third
power’’, 34
is read as ‘‘three to the fourth power,’’ etc.

12. 2. Order of Operations Integers (ℤ)
In mathematics, we have what is called an order of operations
to clarify the meaning when there are operations and
grouping symbols (parentheses) in the same problem.
The order of operations is :
1. Parentheses
2. Exponents
3. Multiplication or Division, left to right
4. Addition or Subtraction, left to right
Multiplication and division are equal in order and should be
performed from left to right. Addition and subtraction are
equal in order and should be performed from left to right.
• Math Note: The word simplify means to perform the
operations following the order of operations.

13. 3. Rational numbers (ℚ)
• A rational number is a number that can be expressed
as a fraction with an integer numerator and a positive
integer denominator. Fractions are written as two
integers, the numerator and the denominator, with a
dividing bar between them. The fraction
𝑚
𝑛
represents
m parts of a whole divided into n equal parts, where n
is not zero. m is also called dividend, and n is also called
divisor. Two different fractions may correspond to the
same rational number; for example
1
2
and
2
4
are equal.
• The symbol for the rational numbers is ℚ (for quotient).

14. 3. Rational numbers (ℚ)
• A fraction whose numerator is less than its
denominator is called a proper fraction. For example,
5
8
,
2
3
, 𝑎𝑛𝑑
1
6
. A fraction whose numerator is greater than
or equal to its denominator is called an improper
fraction. For example,
5
3
,
6
6
, 𝑎𝑛𝑑
10
4
. A number that
consists of a whole number and a fraction is called a
mixed number. For example, 6
5
8
, 8
2
3
, 𝑎𝑛𝑑 3
1
6
. When
the numerator of a fraction is zero, the value of the
fraction is zero. For example,
0
6
= 0.
• A real number that is not rational is called irrational.
Irrational numbers include 2, 𝜋, ℮, and 𝜑.

15. 4. Real numbers (ℝ)
• A real number may be either rational or irrational; either
algebraic or transcendental; and either positive, negative,
or zero (Integers). They may be expressed by decimal
representations that have an infinite sequence of digits to
the right of the decimal point, in which a decimal point is
placed to the right of the digit with place value 1. Each digit
to the right of the decimal point has a place value one-
tenth of the place value of the digit to its left. For example,
123.456 represents
123456
1000
, or, in words, one hundred, two
tens, three ones, four tenths, five hundredths, and six
thousandths.
• The symbol for the real numbers is ℝ.

16. 4. Real numbers (ℝ)
• One half is 0.5, one fifth is 0.2, one tenth is 0.1, and one
fiftieth is 0.02. To represent the rest of the real numbers
requires an infinite sequence of digits after the decimal
point. Since it impossible to write infinitely many digits,
real numbers are commonly represented by rounding or
truncating this sequence, or by establishing a pattern,
such as 0.333..., with an ellipsis to indicate that the
pattern continues. Thus 123.456 is an approximation of
any real number between
123456
1000
or
123456
1000
(rounding) or
any real number between
123456
1000
or
123456
1000
(truncation).

17. 4. Real numbers (ℝ)
• The real numbers include all the rational numbers, such as the
integer and the fraction, and all the irrational numbers such as
2, 𝜋, ℮, and 𝜑.
• Real numbers can be thought of as points on an infinitely long
line called the number line or real line, where the points
corresponding to integers are equally spaced. The real line can
be thought of as a part of the complex plane, and complex
numbers include real numbers.
Number Line : Real numbers can be thought of as points on an infinitely long number line

18. 5. Complex numbers (ℂ)
• Moving to a greater level of abstraction, the real
numbers can be extended to the complex numbers.
This set of numbers arose historically from trying to
find closed formulas for the roots of cubic and quartic
polynomials. This led to expressions involving the
square roots of negative numbers, and eventually to
the definition of a new number: a square root of −1,
denoted by i, a symbol assigned by Leonhard Euler, and
called the imaginary unit.
• The complex numbers consist of all numbers of the
form 𝑎 + 𝑏𝑖, where a and b are real numbers, and i is
the imaginary unit, that satisfies the equation 𝑖2
= −1.

19. 5. Complex numbers (ℂ)
• Complex numbers allow for solutions to certain equations
that have no solutions in real numbers. For example, the
equation (𝑥 + 1) 2
= −9 has no real solution, since the
square of a real number cannot be negative.
• Complex numbers provide a solution to this problem. The
idea is to extend the real numbers with the imaginary unit i
where 𝑖2
= −1. so that solutions to equations like the
preceding one can be found. In this case the solutions are
(−1 + 3𝑖) and (−1 − 3𝑖).
• −1 + 3𝑖 + 1
2
= (3𝑖)2
= (3)2
(𝑖)2
= 9 −1 = −9
• −1 − 3𝑖 + 1
2
= (−3𝑖)2
= (−3)2
(𝑖)2
= 9 −1 = −9

20. Number
Each of the number systems mentioned above is a
proper subset of the next number system. Symbolically,
ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂ.

21. Thank you For your attention
Individual Task :
1. Find more about Subclasses of the integers and make an explanation
about it:
a. Even and odd numbers
b. Prime numbers
c. Fibonacci numbers
d. perfect numbers
2.Give 5 example of :
a. Reducing Fractions to the lowest term,
b. Changing Fractions to Higher Terms,
c. Changing Improper Fractions to Mixed Numbers,
d. Changing Mixed Numbers to Improper Fractions.
3. Find the place value of Decimal digits and give 5 different example of
naming the decimal.
 The task be collected next week. You have to work in foolscap and
write with your hand only.