PROPERTIES OF INTEGERS
The number line:
Representing integers on a number line:
Pic credits: onlinemathforall.com
CONSECUTIVE INTEGERS
The numbers -3,-2, -1, 0, 1, 2…. are consecutive integers. Consecutive integers differ by 1.
They can be represented by n, n+1, n+2,….. or as n-1, n, n+1….., where n is an integer.
Consecutive even integers:
-8, -6, -4 …… are consecutive even integers, as are 2, 4, 6 ……
They can be represented by 2n, 2n+2, 2n+4… where n is an integer.
Consecutive odd integers:
-7, -5, -3 …… are consecutive odd integers, as are 3, 5, 7……
They can be represented by 2n+1, 2n+3, 2n+5… where n is an integer.
The integer 0 (zero) is neither positive nor negative. Division by zero is not defined.
If n is any number, then n + 0 = n (Additive Identity is zero)
Properties of the integer 1
•
If n is any number, then 1 × n = n (Multiplicative Identity is 1)
•
For any number n ≠ 0, n × (1/n) = 1 (
•
Multiplying or dividing an expression by 1 does not change the value of that expression
•
The number 1 is not a prime number since it has only one positive divisor
Represent the following numbers on the same number line:
-4
7
2
-6
0
Write the integers represented by the letters:
l=
v=
s=
p=
q=
Properties of Integers with Examples
Property
Closure under addition
If a, b ∈ Z, then a + b ∈ Z
Closure under multiplication
If a, b ∈ Z, then a × b ∈ Z
Commutativity of addition
If a, b ∈ Z, then
a+b=b+a
Commutativity of multiplication
If a, b ∈ Z, then
a × b= b × a
Associativity of Addition
If a, b, c ∈ Z, then
a + ( b + c) = (a + b) + c
Associativity of multiplication
If a, b, c ∈ Z, then
a × ( b × c) = (a × b) × c
Existence of Additive Identity
There exists an element
0 ∈ Z such that for all a ∈ Z
a+0=a
Existence of Multiplicative Identity
There exists an element
0 ∈ Z such that for all a ∈ Z
a×0=0
Example
If 2 and 3 are two integers, then their sum 2+3 is
also an integer.
If 2 and 3 are two integers, then their product 2×3 is
also an integer.
If 2 and 3 are two integers, then the sum 2+3 is equal
to the sum 3+2.
The order of integers does not affect the addition we can add numbers in any order.
If 2 and 3 are two integers, then the product 2×3 is
equal to the product 3×2.
2.(-4) = (-4).2
The order of integers does not affect the product we can multiply numbers in any order.
If 2, 3 and 4 are three integers, then the sum
2 + (3 + 4) is equal to the sum (2 + 3) + 4.
The grouping of numbers in a sum does not affect the
result.
If 2, 3 and 4 are three integers, then the product
2 × (3 × 4) is equal to the product (2 × 3) × 4.
The grouping of numbers in a product does not affect
the result.
If 2 is any integer, then 2 + 0 = 2
If 2 is any integer, 2 × 0 = 0
Additive Inverse
For every a ∈ Z, there is a solution
x ∈ Z to a + x = 0 (therefore x = -a)
Law of Distribution over Addition
If a, b, and c ∈ Z, then
a (b + c) = a × b + a × c
For the integer 2, the additive inverse is the negative
of 2.
2 + (-2) = 0
If 2, 3, and 4 are three integers, then
2 (3 + 4) = 2 × 3 + 2 × 4
(-2)(3 × 4) = (-2) × 3 + (-2) × 4
The multiplication is "distributed" over all the terms
inside the parentheses.
Subtraction is neither commutative nor associative.
If a, b ∈ Z, then a – b ≠ b – a
2–3≠3-2
Division is neither commutative nor associative.
If a, b ∈ Z, then a ÷ b ≠ b ÷ a
2÷3≠3÷2
EXTENSION
Ordering Properties
The ordering properties are those concerning the > operator.
For two integers a and b,
If a, b > 0, then a + b > 0
(closure of “> 0” under addition)
If a, b > 0, then a × b > 0
(closure of “> 0” under multiplication)
Exactly one of either a > b, or a = b, or a < b is correct.
Choose five of the words from the box to complete these statements correctly:
Zero
multiply
divide product
commutative associative
quotient
one
multiple
factor
1. You multiply to find the ___________ of two or more factors.
2. Any number times ___________ equals zero.
3. The ___________ property helps us to know that 4 × 5 = 5 × 4.
4. When any number except zero is divided by itself, the ___________ is one.
5. You cannot ___________ a number by zero.
Now, choose one of the unused words and use it to write a true mathematical statement.
COMMUTATIVE, ASSOCIATIVE AND DISTRIBUTIVE PROPERTIES
Commutative property of addition
The order in which two numbers are added does not change the sum/result. Pick up two
integers of your choice and test the property below.
A+B=B+A
_____ + ______ = ______ + ______
______ = ______
Commutative property of multiplication
The order in which two numbers are multiplied does not change the product/result. Pick up two
integers of your choice and test the property below.
A×B=B×A
_____ × ______ = ______ × ______
______ = ______
Does the commutative property work for subtraction or division? TRY IT!!!
Is 4 – 2 = 2 – 4?
Is 4 ÷ 2 = 2 ÷ 4?
What can you conclude?
Associative property of addition
Grouping numbers (with parentheses or brackets) differently does not change the sum. Pick up
three integers of your choice and test the property below.
(A + B) + C = A + (B + C)
(_____ + _____) + _____ = (_____ + _____) + _____
_____ + _____ = _____ + _____
_____ = _____
Associative property of multiplication
Grouping numbers (with parentheses or brackets) differently does not change the product.
Pick up three integers of your choice and test the property below.
(A × B) × C = A × (B × C)
(_____× _____) × _____ = (_____× _____) × _____
_____ × _____ = _____× _____
_____ = _____
Does the associative property work for subtraction or division? TRY IT!!!
Is (3 – 2) – 1 = 3 – (2 – 1)?
Is (8 ÷ 4) ÷ 2 = 8 ÷ (4 ÷ 2)?
What do you observe?
Distributive property of multiplication over addition and subtraction
A × (B + C) = A × B + A × C
A × (B - C) = A × B - A × C
3 × (2+ 4) = 3 × 2 + 3 × 4
3 × (4 - 2) = 3 × 4 - 3 × 2
_____ × _____ = _____ + _____
_____ × _____ = _____ - _____
_____ = _____
_____ = _____
Identity property of addition (additive identity)
Zero added to any number equal that number itself. Addition of zero does not change its
value.
A+0=A
_____ + 0 = ______
Identity property of multiplication
Any number multiplied by 1 always equals that number. Multiplication by one does not change
its value.
A×1=A
_____ × 1 = ______
Inverse property of addition (additive inverse)
A number added to its inverse (or opposite) will always equal zero.
A + (-A) = 0
_____ + _____ = 0
Inverse property of multiplication
Any number multiplied by its reciprocal or inverse will always equal 1.
A×
1
𝐴
=1
_____ × ______ = 1
Multiplication property of zero
A number multiplied by zero always equals zero.
A×0=0
_____ × 0 = 0
DIVISION BY ZERO IS NOT DEFINED!!
Symmetric property
If A = B, then B = A
If (A × B) = C, then C = (A × B)
𝑨
𝟎
= ??