At the end of the lesson, the learner should be able to: recall the different properties of real numbers write equivalent statements involving variables using the properties of real numbers

1. Properties of Real
Numbers

2. Objectives
At the end of the lesson, you should be able
to:
•recall the different properties of real
numbers
•write equivalent statements involving
variables using the properties of real
numbers

3. Introduction
•In solving different operations on real
numbers, recognizing some patterns can
help you solve faster.
•In this lesson, you will study some of these
patterns known as the properties of real
numbers.

4. Learn about It!
The following are the properties of
real numbers. There are five such
properties, most of which have variants
for addition and multiplication.

5. Commutative Property
•This property has variants for both addition
and multiplication.
•Commutative Property of Addition
•a + b = b + a
•Example: 1 + 2 = 2 + 1
•The order in which you add numbers does
not matter. You will always get the same
sum.

6. Commutative Property
•Commutative Property of Multiplication
a x b = b x a
For example: 1 x 2 = 2 x 1
•The order in which you multiply numbers
does not matter. You will always get the
same product.

7. More Examples
•Is 3 + 7 equal to 7 + 3?Yes! 3 + 7 = 10 and 7
+ 3 = 10.
✓Therefore, 3 + 7 = 7 + 3.

8. •What is another way to write 10 + n? 10 + n
may be written as n + 10.

9. •Is 3 x 7 equal to 7 x 3? Yes! 3 x 7 = 21 and 7 x
3 = 21.
✓Therefore, 3 x 7 = 7 x 3.

10. •What is another way to write n ∙ 8? may be
written as 8n.

11. Associative Property
•This property also has variants for both
addition and multiplication.
•Associative Property of Addition
(a + b) + c = a + (b + c)
•The order in which you add a group of
numbers does not matter. You will always
get the same sum.

12. Associative Property
•Associative Property of Multiplication
(a x b) x c = a x (b + c)
•The order in which you multiply a group of
numbers does not matter. You will always
get the same product.

13. Examples
•In adding three numbers, will (3 + 4) + 5 be
equal to 3 + (4 + 5)?
✓Yes!
(3 + 4) + 5 = 12
3 + (4 + 5) = 12
✓Therefore, (3 + 4) + 5 = 3 + (4 + 5).

14. •What is another way to write n + (2n + 3)?
• n + (2n + 3) may be written as (n + 2n) + 3.

15. •In multiplying three numbers, will (3 x 4) x
5 be equal to 3 x (4 x 5)?
✓Yes!
(3 x 4) x 5 = 12 x 5 = 60
3 x (4 x 5) = 3 x 20 = 60
✓Therefore, (3 x 4) x 5 = 3 x (4 x 5).

16. •What is another way to write 2 ∙ (3 ∙ n) ?
•2 ∙ (3 ∙ n) may be written as
(2 ∙ 3) ∙ n.

17. Distributive Property of
Multiplication over Addition
•The Distributive Property of Multiplication
over Addition states that
a x (b + c) = a x b + b x c or
a (b + c) = ab + bc.
•Multiplying a number a by a sum b + c will
give you the same value as multiplying a by
both b and c and then adding the products.

18. Examples
•Evaluate 5(6 + 9).
•According to the Distributive Property, there are
two ways to evaluate this problem:
5(6 + 9) = 5(15) = 75
•and
5(6 + 9) = 5 x 6 + 5 x 9
= 30 + 45
= 75

19. •What is another way to write 2 (x + 3)?
2 (x + 3) may be written as 2x + 2(3).
•This expression can be simplified into
2x + 6.

20. Identity Property
•Identity Property of Addition
a + 0 = a
•The sum of any real number and 0 is equal
to the same number.

21. Identity Property
•Identity Property of Multiplication
a x 1 = a
•The product of any real number and 1 is
always the same number.

22. Inverse Property
•Inverse Property of Addition
a + (-a) = 0
•The additive inverse of a number is the
number that has the same absolute value but
with the opposite sign. The sum of a real
number and its additive inverse is always
equal to 0.

23. •Inverse Property of Multiplication
a x 1
𝑎
= 1
•The multiplicative inverse of a nonzero real
number is its reciprocal. The product of a
nonzero real number and its multiplicative
inverse is always equal to 1.

24. •The Identity Property of Addition can be
observed in the equations 6 + 0 = 6 and 2x +
0 = 2x.
•The Identity Property of Multiplication can
be observed in the equations 8 x 1 = 8 and
3x x 1 = 3x.

25. •The Inverse Property of Addition is shown
by the equations
42 + (-42) = 0
and
5x + (-5x) = 0.

26. •The Inverse Property of Multiplication is
shown by the equations
•3 x 1
3
= 1 and
•2
5
x 5
2
= 1.

27. Try It!
Find the value of that will satisfy each
equation.
1. 20 x 32 = a + 20
2. -6x + a = 0
3. a(x + 6) = 4 x x + 4 x 6

28. Solution
1. a = 32; this is an application of the
Commutative Property of Multiplication.
2. 6x; this is an application of the Inverse
Property of Addition.
3. a = 4; this is an application of the
Distributive Property.

29. Tips
•The Inverse Property of Multiplication does
not apply to the real number 0. This is
because the reciprocal of 0 is
1
0
, which is
undefined.
•Subtraction and division are simply
considered as inverse operations of addition
and multiplication, respectively. You can
change any subtraction or division problem
into addition or multiplication.

30. Key Points
•The properties of real numbers can be used to
help you solve operations on real numbers
easier and rewrite real numbers with variables.
•There are 5 major properties, most of which are
broken into addition and multiplication. These
are the Commutative Property, the Associative
Property, the Distributive Property, the Identity
Property and the Inverse Property.