Integers

As a fundamental part of mathematics, integers extend whole numbers to include negatives. This extension allows representation of situations involving loss, debt, temperature changes, or any scenario where values can go below zero, providing a complete framework for arithmetic operations and problem-solving with both positive and negative values.

Table Of Contents

What are integers?
What is the difference between integers and whole numbers?
Are all whole numbers integers?
Is zero an integer?
Do integers include fractions or decimals?
What are the properties of integers?
What are the rules for multiplying and dividing integers?
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What are integers?

The set of Integers ( $\mathbb{Z}$ ) includes positive and negative whole numbers, as well as zero, without fractional or decimal parts.

Example: ( $\mathbb{Z}$ ) = {…, −3, −2, −1, 0, 1, 2, 3, …}

What is the difference between integers and whole numbers?

Whole Numbers ( $\mathbb{N}_\mathbf{0}$ ): Include all non-negative numbers without fractions or decimals.
Example: ( $\mathbb{N}_\mathbf{0}$ ) = {0, 1, 2, 3, 4, …}.

Integers ( $\mathbb{Z}$ ): Extend whole numbers to include negative numbers:
Examples: ( $\mathbb{Z}$ ) = {…, -3, -2, -1, 0, 1, 2, 3, …}

Are all whole numbers integers?

Yes, all whole numbers are part of the integer set, which includes zero, positive numbers, and their negative counterparts.

Is zero an integer?

Yes, zero is part of the integers, acting as the neutral element that separates positive and negative numbers.

Do integers include fractions or decimals?

No, integers do not include fractions or decimals. They consist of whole numbers, their negatives, and zero.

What are the properties of integers?

Here are the properties explained with examples:

Closure: The sum, difference, or product of any two integers is always an ( $\mathbb{Z}$ ).
Example: ${3}+{5}={8}$ ; ${3}-{5}={-2}$ and ${-1} \times {3}={-3}$
However, division of integers is not always closed.
Example: ${4}\div{3}={1,33...}$ , which is not an ( $\mathbb{Z}$ ).
Commutative Property: The order does not affect the result for addition or multiplication.
Example: ${a} + {b} = {b} + {a}$
Example: ${a} \times {-b} = {-b} \times {a}$
Associative Property: Grouping does not affect the result for addition or multiplication.
Example: $({a} + {b})+{c} = {a} + ({b}+{c})$
Example: $({a} \times {-b}) \times {c} = {a} \times ({-b} \times {c})$
Distributive Property: Multiplication distributes over addition or subtraction.
Example: ${a} \times ({b}+{c})$ = ( ${a}\times {b}) + ({a} \times {c}$ )
Example: ${-a} \times ({b}-{c})$ = ( ${-a}\times {b}) - ({-a} \times {c})$ .
Additive Inverses: Every integer has an opposite that sums to zero.
The additive inverse of 5 is -5 because 5+(-5)=0
The additive inverse of -8 is 8 because -8+8=0

What are the rules for multiplying and dividing integers?

Multiplication:
Positive $\times$ Positive = Positive ( ${3}\times{4}={12}$ )
Negative $\times$ Negative = Positive ( ${-3}\times{-4}={12}$ )
Positive $\times$ Negative = Negative ( ${3}\times{-4}={-12}$ )

Division:
Positive $\div$ Positive = Positive ( ${12}\div{3}={4}$ )
Negative $\div$ Negative = Positive ( ${-12}\times{-4}={3}$ )
Positive $\div$ Negative = Negative ( ${12}\times{-4}={-3}$ )