Which of these are integers: −8, 2.5, 0, 3/4?
Answer: −8 and 0
2.5 and 3/4 have a part left over after the whole, so they are not integers.
📖 Math Dictionary
Integer
/IN-tuh-jer/
Definition
An integer is any whole number (positive, negative, or zero) without fractions or decimals. The set of integers is: {..., −3, −2, −1, 0, 1, 2, 3, ...}
⚡Integer at a glance
- Symbol
- ℤ, taken from the German word Zahlen ("numbers")
- What it includes
- Positive whole numbers, negative numbers and zero
- What it leaves out
- Fractions, decimals, π and √2
- Smallest and largest
- Neither. The set runs forever in both directions
- Usually taught in
- Grades 6-8 (Key Stage 3)
💡Example
Examples of integers: −5, −1, 0, 7, 100. Non-integers: 3.5, ½, π, √2 (these have decimal or fractional parts).
🎓Used in Grades
Grades 6-8Key Stage 3
Types of integer
There are three types of integer. Positive whole numbers are the counting numbers: 1, 2, 3 and so on. Negative numbers are their opposites: −1, −2, −3. Zero is the third type.
Zero is an integer included in the set, even though it is neither positive nor negative. It is the dividing point between the positive and negative integers.
Whole numbers are 0, 1, 2, 3 and so on. Every whole number is an integer. The reverse is not true, because whole numbers leave out the negatives.
Integers on a number line
Number lines make integers easy to see. Put zero in the middle. Positive and negative integers sit on either side, one step apart.
Each integer has a partner the same distance from zero. These pairs are called additive inverses. For example, 6 and −6 are additive inverses, because 6 + (−6) = 0.
Number lines also show order. Any number further to the right is larger. That is why −2 is greater than −9.
Integers in the real world
Real world examples make negative numbers feel normal. A temperature of −8 °C is eight degrees below freezing. A bank balance of −$25 means you owe $25.
Elevators use the same idea. Floor 3 is above ground and floor −1 is the basement. Height works this way too: a mountain top is +2,400 m and a submarine is −300 m, with sea level at 0.
Working with positive and negative integers
Adding a negative is the same as subtracting: 5 + (−3) = 2.
Subtracting a negative is the same as adding: 5 − (−3) = 8.
Two negatives multiplied give a positive: (−4) × (−3) = 12. A positive times a negative gives a negative: (−4) × 3 = −12.
Division follows the same sign rules as multiplication: (−12) ÷ (−4) = 3, and (−12) ÷ 4 = −3.
How integers fit with other number types
Every integer is a rational number, because it can be written as a fraction over 1. For example, 4 = 4/1.
Every integer is also one of the real numbers. Real numbers fill in every gap, including fractions, decimals and values like π. So the real numbers form a much larger set than the integers.
The word integer comes from Latin and means "whole" or "untouched". The symbol ℤ comes from a German word, Zahlen, which simply means "numbers".
✏️Practice Checks
What is the additive inverse of −12?
Answer: 12
It sits the same distance from zero on a number line, with the opposite sign. −12 + 12 = 0.
A diver is 15 m below sea level. Write this as an integer.
Answer: −15
Sea level is 0, so distances below it are written as negative numbers.
Work out −6 + 10.
Answer: 4
Start at −6 on the number line and move 10 steps right.
❓Integer FAQ
Is zero an integer?
Yes. Zero is an integer. It is the only one that is neither positive nor negative.
Is every whole number an integer?
Yes. Whole numbers are 0, 1, 2, 3 and so on, and all of them are integers. The reverse is not true, because negative numbers are integers but not whole numbers.
Are fractions and decimals integers?
No. Values like 3.5, ½ and 0.25 have a part left over after the whole. A decimal such as 4.0 is an exception, because it is really the integer 4.
What is the difference between integers and real numbers?
Integers are spaced one step apart on a number line with nothing in between. Real numbers fill every gap, including fractions, decimals and values like π.
What are additive inverses?
Two integers the same distance from zero with opposite signs. They always add to zero, so −7 + 7 = 0.
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