mathematics

1. Real Number System

2. 𝑪𝒐𝒎𝒑𝒍𝒆𝒙 𝑵𝒖𝒎𝒃𝒆𝒓𝒔
𝑹𝒆𝒂𝒍 𝑵𝒖𝒎𝒃𝒆𝒓𝒔 𝑰𝒎𝒂𝒈𝒊𝒏𝒂𝒓𝒚 𝑵𝒖𝒎𝒃𝒆𝒓𝒔
𝑹𝒂𝒕𝒊𝒐𝒏𝒂𝒍 𝑵𝒖𝒎𝒃𝒆𝒓𝒔 𝑰𝒓𝒓𝒂𝒕𝒊𝒐𝒏𝒂𝒍 𝑵𝒖𝒎𝒃𝒆𝒓𝒔
𝑰𝒏𝒕𝒆𝒈𝒆𝒓𝒔 𝑵𝒐𝒏 − 𝑰𝒏𝒕𝒆𝒈𝒆𝒓𝒔
𝑵𝒆𝒈𝒂𝒕𝒊𝒗𝒆 𝑰𝒏𝒕𝒆𝒈𝒆𝒓𝒔 𝑾𝒉𝒐𝒍𝒆 𝑵𝒖𝒎𝒃𝒆𝒓𝒔
𝑵𝒂𝒕𝒖𝒓𝒂𝒍 𝑵𝒖𝒎𝒃𝒆𝒓𝒔 𝒁𝒆𝒓𝒐

3. Rational Numbers
The set of numbers that can be
expressed as the ratio of two integers.
Can be written in the form
𝑎
𝑏
.

4. Irrational Numbers
The set of numbers that cannot be
expressed as the ratio of two integers.
Irrational numbers have non-repeating
decimals that are non-terminating.
2 = 1.41442113562…
3
5 = 1.709975947…
𝜋 = 3.141592654…

5. Integers
It comprises the negative integers,
positive and zero.
𝑍 = {… − 6, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5 … }
Negative Integers
𝑍−
= {… − 6, −5, −4, −3, −2, −1}
Positive Integers
𝑍+
= {1, 2, 3, 4, 5, 6 … }
Zero
0

6. Non-Integers
Fractions and Decimals
(Terminating and Non-Terminating)
1
4
= 0.25
1
3
= 0.333333 … 𝑜𝑟 0. 3

7. Natural Numbers (Counting Numbers)
𝑁 = {1, 2, 3, 4, 5, 6, 7, 8, 9 … }
Whole Numbers
𝑊 = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9 … }

8. Properties of Addition

9. The Commutative Property of Addition states
that the two Real numbers can be added in any
order to get the same result.
If a and b represent Real Numbers ,
then a + b = b + a.
Examples:
𝟕 + 𝟔 = 𝟔 + 𝟕

10. The Associative Property of Addition.
Illustrates that it doesn’t matter how we group
or associate numbers in addition.
If a, b, c represent Real Numbers , then
(a + b)+c = a + (b + c).
Example: 𝟕 − 𝟖 + 𝟑
(𝟕 − 𝟖) + 𝟑 𝟕 + (−𝟖 + 𝟑)
(−𝟏) + 𝟑
= 𝟐
𝟕 + (−𝟓)
= 𝟐

11. The Closure Property of Addition
states that the sum of any two Real numbers is
also a real number .
If a and b represent Real Numbers ,
then a + b = c where c ∈ 𝑹.

12. The Identity/Zero Property of Addition.
states that when we add zero to a number, the
number remains the same. Hence, zero is referred as
the additive identity.
If a represents a number, then
𝒂 + 𝟎 = 𝟎 + 𝒂 = 𝒂.
Example:
𝟕 + 𝟎 = 𝟕 𝟎 + 𝟕 = 𝟕

13. The Inverse Property of Addition.
illustrates those two numbers that are the
same distance away from the origin, but on
opposite directions are called opposites or
additive inverses of each other.
If a represents a number, then (-a) is its
opposite or negative or its additive inverse
so that 𝒂 + (−𝒂) = (−𝒂) + 𝒂 = 𝟎.
Example:
𝟕 + (−𝟕) = 𝟎 (−𝟕) + 𝟕 = 𝟎

14. Properties of Multiplication

15. The Commutative Property of Multiplication
states that the two Real numbers can be
multiplied in any order to get the same result.
If a and b represent Real Numbers ,
then (a) (b) = (b) (a).
Example:
(𝟕)(𝟔) = (𝟔)(𝟕)

16. The Associative Property of Multiplication.
Illustrates that it doesn’t matter how we group
or associate numbers in multiplication.
If a, b, c represent Real Numbers , then
𝒂 ∙ 𝒃 ∙ 𝒄 = 𝒂 ∙ 𝒃 ∙ 𝒄 .
Example: (𝟐) ∙ (−𝟖) ∙ (𝟑)
[ 𝟐 −𝟖 ] ∙ 𝟑
(−𝟏𝟔)(𝟑)
= −𝟒𝟖
(𝟐)(−𝟐𝟒)
= −𝟒𝟖
𝟐 ∙ [ −𝟖 ∙ 𝟑]

17. The Distributive Property of Multiplication Over
Addition.
demands multiplying a number to every number
inside a parenthesis, then combine the results by
addition
If a, b, c represent Real Numbers , then 𝐚 ∙ 𝒃 + 𝒄 =
(𝒃 + 𝒄) ∙ 𝒂.
Example:
(𝟐) ∙ (𝟑 + −𝟖 )
(𝟐) ∙ (−𝟓)
= −𝟏𝟎
(𝟐) ∙ (𝟑 + −𝟖 )
(𝟐) ∙ (𝟑 + −𝟖 )
𝟔 + (−𝟏𝟔)
= −𝟏𝟎

18. The Closure Property of Multiplication
states that the product of any two Real
numbers is also a real number .
If a and b represent Real Numbers ,
then a ∙ b = c where c ∈ 𝑹.

19. The Identity Property of Multiplication.
illustrates that whenever we multiply a number
by 1, the product is the same number .
If a represents a number, then
𝒂 ∙ 𝟏 = 𝟏 ∙ 𝒂 = 𝒂.
Example:
𝟕 ∙ 𝟏 = 𝟕 𝟏 ∙ 𝟕 = 𝟕

20. The Zero Property of Multiplication.
tells that whenever we multiply a number by
zero, its product is zero .
If a represents a number, then
𝒂 ∙ 𝟎 = 𝟎 ∙ 𝒂 = 𝟎.
Example:
𝟕 ∙ 𝟎 = 𝟎 𝟎 ∙ 𝟕 = 𝟎

21. The Inverse Property of Addition.
Explains that any number except zero has its
reciprocal, and whenever this number is multiplied to
its reciprocal, the product is equal to 1. We also call
the two numbers as reciprocal of each other.
If a represents a non-zero number,
then 𝒂
𝟏
𝒂
=
𝟏
𝒂
𝒂 = 𝟏
Example:
𝟒
𝟏
𝟒
= 𝟏
𝟏
𝟒
𝟒 = 𝟏

22. Fundamental Operations
with Real Numbers

23. Fundamental Operations with Real Numbers
The concept of the absolute value of a real number is important to signed numbers.
Signed numbers are numbers which are preceded by plus or minus sign. However, a
number that has no sign is understood to be positive.
The absolute value of a real number x denoted by |x| is defined as
x, if x > 0 (i.e. is x positive)
-x, if x < 0 (i.e. is x negative)
0, if x = 0
According to the definition, the absolute value of any nonzero number is always
positive.
For example, |4| = 4 ; |-4| = - (-4) = 4 ; |0| = 0

24. Rules Governing the Operations on Signed Numbers
The operation on the set of real numbers is governed by the following rules:
Rule 1: To add two real numbers with like signs, add their absolute values and prefix
the common sign. For example,
a. 2 + 5 = 7 b. – 2 + (-5) = -7 c. -23 + (-8) = -31
Rule 2: To add two real numbers with unlike signs, subtract the smaller absolute value
from the bigger absolute value, and prefix the sign as that of the bigger absolute value.
For example,
a. 8 + (-11) = -3 b. -12 + 17 = 5 c. -25 + 43 = 18
d. -8 + 11 = 3 e. -25 – (-17) = -8 f. 60 + (-80) = -20

25. Rule 3: To subtract two real numbers with like signs, change the sign of the subtrahend and
proceed to algebraic addition (Rules 1 or 2). For example,
a. 8 – 15 = 8 + (-15) = -7 d. 40 – 58 = 40 + (-58) = -18
b. -23 – 15 = -23 + (-15) = - 38 e. -8 – 15 = -8 + (-15) = -23
c. -8 – (-15) = -8 + 15 = 7 f. -95 – (-80) = -95 + 80 = -15
Rule 4: To multiply (or divide) two numbers having like signs, multiply
(or divide) their absolute values and prefix a plus sign. For example,
a. 8 (2) = 16 b. (-8) (-2) = 16
c. (-5) (-3) = 15 d. 8 ÷ 2 = 4 e. -20 ÷ -4 = 5

26. Rule 5: To multiply (or divide) two numbers having unlike signs,
multiply (or divide) their absolute value and prefix a minus sign. For
example,
a. (-10) (2) = -20 b. (-5) (3) = -15
c. -10 ÷ 2 = -5 d. -24 ÷ 6 = -4

27. Performing Operations on Series of Numbers
In a series of numbers involving the basic
operations in Arithmetic, the following give the
order of performing the operations:

28. First, perform powers and extract roots.
Illustration: −24
−
3
−8 = −16 − −2 = −16 + 2 = −14
Second, perform multiplication or division in order of occurrence.
Illustration: −62
÷ 2 2 = −36 ÷ 4 = −9 ;
−42
÷ 2 ∙ 4 = −16 ÷ 8 = −2

29. Third, perform addition or subtraction in order.
Illustration: (−5)2
−
3
−64 ÷ 4 + 2 3 = 25 −
−4 ÷ 4 + 6 = 25 − −1 + 6 = 32
In the presence of parentheses, quantities within these symbols are to be
performed first.
Illustration: a. −42
÷ −2 4 = −16 ÷ −8 = 2

30. b. −62
÷ 2 3 ÷
3
−64 ÷ 2 2 =
−36 ÷ 6 ÷ −4 ÷ −4 = −6 ÷ 1 = −6
c.
3
−64 −
3
−1
3
−8 −
3
−27 =
−4 − −1 −2 − −3 = −4 − 2 + 3 = −3

31. TRY THIS!
1. −92
÷ −3 ∙ 9 − 58 ÷ 2
= −81 ÷ −3 ∙ 3 − 29
= 27 3 − 29
= 81 − 29
= 52

32. TRY THIS!
2. 52
∙ 9 − 3 ÷ (7 + 8 ÷ 4)
= 25 ∙ 3 − 3 ÷ (7 + 2)
= 75 − 3 ÷ 9
= 72 ÷ 9
= 8