Properties of Integers

​After going through this lesson, you are expected to illustrate the different properties of operations on the set of integers.

​PROPERTIES OF ADDITION AND MULTIPLICATION

Closure Property

• Addition - For any integers a and b,    a + b is an integer.

This is close for addition since the addends are integers, the sum is also an integer.

(7) + (12) = 19                 (-15) + (12) = -3

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• Multiplication - For any integers a and b,    a · b is an integer.

This is close under multiplication, since the factors are integers, the product is also an integer.

(5)(9) = 45                      (-3)(8) = -24

​PROPERTIES OF ADDITION AND MULTIPLICATION

Commutative Property

• Addition - For any integers a and b,    a + b = b + a.

The order of addends will not affect the sum.

(6) + (-2)  = (-2) + (6)                (-8) + (-3) = (-3) + (-8)

4 = 4                                          -11 = -11

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• Multiplication - For any integers a and b,     a ·b = b · a.

The order of factors will not affect the product.

(7) (-5) = (-5) (7)                 (10) (6) = (6) (10)

-35 = -35                                    60 = 60

​PROPERTIES OF ADDITION AND MULTIPLICATION

Associative Property

• Addition - For any integers a, b, and c,      (a + b) + c = a + (b + c).

Different groupings of addends will not affect the sum.

(3 + 7) + (-4)  = 3 + (7 + -4)                (-2 + 8) + 5 = -2 + (8 + 5)

10 + (-4)  = 3 + 3                                       6 + 5 = -2 + 3

6 = 6                                                  11 = 11

• Multiplication - For any integers a, b, and c,       (a · b) · c = a · (b · c).

Different groupings of factors will not affect the product.

(4)(5) (7) = (4) (5)(7)                 (-2)(6) (-4) = (-2) (6)(-4)

(20)(7) = (4)(35)                          (-12)(-4) = (-2)(-24)

140 = 140                                          48 = 48

​PROPERTIES OF ADDITION AND MULTIPLICATION

Identity Property

• Addition - The sum of any number and zero will always be equal to the number.

The identity element of addition is 0.

15 + 0 = 15                 0 + (-8) = -8

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• Multiplication - For any integer a,      a · 1 = a.

The product of any number and 1 will always be equal to the number.

The identity element of multiplication is 1.

(32)(1) = 32                      (1)(-15) = -15

​PROPERTIES OF ADDITION AND MULTIPLICATION

Inverse Property

• Addition - For any integer a,       a + -a = 0.

The opposite of the given integer is called its additive inverse.

The sum of a number and its additive inverse is zero.

10 + (-10) = 0                  (-2) + (2) = 0

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• Multiplication - The reciprocal of the integer is called its multiplicative inverse.

The product of a number and its multiplicative inverse is one.

(3)( 1/3) = 3/3 = 1                         (-5)(-1/5) = -5/-5 = 1                                          

​PROPERTIES OF ADDITION AND MULTIPLICATION

Distributive Property of Multiplication over Addition

For any integers a, b, and c,      a (b + c) = ab + ac.

Adding first before multiplying or distributing the factor to each addend will

give the same result.

6(4 + 8) = 6(4) + 6(8)                    -5 (-3 + 10) = -5(-3) + -5(10)

6(12) = 24 + 48                                    -5(7) = 15 + (-50)

72 = 72                                              -35 = -35 

Zero Property of Multiplication

For any integer a, a · 0 = 0. Any number multiplied by 0 is equal to 0.

(-98) (0) = 0                                          (0)(12) = 0

Closure Property - Close

Commutative Property - Commutation or Exchange

Associative Property - Associate or Combine

Identity Property - Sameness

Inverse Property - Opposite

Distributive Property - Distribute/ Share

Zero Property - Zero

Closure Property - Close

Commutative Property - Commutation or Exchange

Associative Property - Associate or Combine

Identity Property - Sameness

Inverse Property - Opposite

Distributive Property - Distribute/ Share

Zero Property - Zero

Complete the following to make true statements, then determine the property applied.

​TEST YOUR MINDS!