What Is an Integer? | The Whole-Number Rulebook

An integer is a whole number that can be negative, positive, or zero, with no decimal or fractional part.

Integers show up everywhere in math class, from basic counting to algebra, graphs, and coding. If you’ve ever seen -3, 0, or 14 and wondered what group they belong to, you’re already looking at integers.

This page gives you a clean definition, how integers behave, how they differ from other number types, and the sign rules that trip people up. You’ll also get quick checks you can use to spot integers in the wild.

What Is an Integer? A Clear Definition

An integer is any whole number with no fractional part. That includes negative whole numbers, zero, and positive whole numbers. If a number has digits after a decimal point, it is not an integer.

You’ll often see integers written with a special symbol:

• Z stands for the set of integers.
• It’s usually written as ℤ (a stylized Z).
• So, ℤ = {…, -3, -2, -1, 0, 1, 2, 3, …}

That “…” matters. It means the list continues without end in both directions.

Integer Meaning With Real-World Uses

Integers feel less abstract once you tie them to everyday situations that naturally go above and below a middle point.

Common Places You See Integers

• Temperature: -5° and 12° are integers.
• Elevators: Basement levels use negative numbers, ground level can be 0.
• Money Balances: A debt of 20 can be modeled as -20.
• Sports Stats: Goal difference can be negative, zero, or positive.
• Coordinates: Points on a grid often land on integer values.

The pattern is the same: integers are built for counting steps and measuring changes that can move forward or backward.

How Integers Fit Into The Number Family

Integers are one stop in a bigger “family tree” of numbers. Seeing where they sit helps you stop mixing them up with similar sets.

Integers Vs. Whole Numbers

Whole numbers usually mean 0 and the positive counting numbers: 0, 1, 2, 3, … Whole numbers leave out the negatives. Integers include them.

Integers Vs. Rational Numbers

Rational numbers can be written as a fraction of two integers, like 3/4 or -11/2. Every integer is rational (since 5 = 5/1), yet many rational numbers are not integers (like 5/2).

Integers Vs. Decimals

Some decimals are rational, some are not, and most are not integers. A quick test: if there’s a decimal point with digits after it, it’s not an integer. Even 2.0 is often treated as equal to 2 in value, yet the notation “2.0” signals a decimal form and can matter in science or computing.

Integers Vs. Natural Numbers

Natural numbers are commonly 1, 2, 3, … Some textbooks include 0, some don’t. Integers always include 0 and all negative whole numbers, so they cover every natural-number convention plus more.

If you want a formal, widely cited definition, Encyclopaedia Britannica’s entry is a solid reference point. Encyclopaedia Britannica’s “integer” definition matches the standard classroom meaning.

Seeing Integers On A Number Line

A number line is the simplest way to “feel” integers. Every integer lands on a tick mark with equal spacing. Zero sits in the center, positives move right, negatives move left.

Ordering Integers

On the number line, numbers farther right are greater. That makes comparisons easy once your brain accepts one rule: among negative numbers, the one with the larger absolute value is actually smaller.

• -2 > -5 (because -2 is to the right of -5)
• -10 < -3
• 0 is greater than any negative integer

Absolute Value And Distance

The absolute value of an integer is its distance from 0, written with bars: | -7 | = 7. Absolute value never comes out negative.

This is handy for “how far apart” questions. The distance between -4 and 3 is 7 because you move 7 steps from -4 to 3.

Integer Operations That Always Work

Integers have a nice feature: when you add, subtract, or multiply integers, you always get an integer.

Addition

Addition combines moves on the number line. Adding a positive number moves right. Adding a negative number moves left.

• 6 + 4 = 10
• 6 + (-4) = 2
• -6 + (-4) = -10

Subtraction

Subtraction is often easier if you rewrite it as “add the opposite.”

• 8 – 3 = 8 + (-3) = 5
• 8 – (-3) = 8 + 3 = 11
• -5 – 7 = -5 + (-7) = -12

Multiplication

Multiplication follows sign rules you can memorize and then trust.

• (+) × (+) = (+)
• (-) × (-) = (+)
• (+) × (-) = (-)
• (-) × (+) = (-)

Division

Division is where integers stop being “closed.” Sometimes you get an integer, sometimes you don’t.

• 12 ÷ 3 = 4 (integer result)
• 12 ÷ 5 = 2.4 (not an integer)
• -9 ÷ 2 = -4.5 (not an integer)

If you want a math reference that states properties and notation clearly, Wolfram MathWorld is a strong source. Wolfram MathWorld’s “Integer” entry lays out the standard set definition and common facts.

Integer Types, Notation, And Quick Checks

Use this table when you’re sorting numbers into groups or trying to name the set a number belongs to. It’s built to answer the “Is this an integer?” question fast, while also showing where integers sit among related sets.

Number Set Or Idea	What It Includes	Quick Check
Natural Numbers (N)	Counting numbers (often 1, 2, 3, …)	No negatives; usually no 0
Whole Numbers	0 and counting numbers	Starts at 0; still no negatives
Integers (ℤ)	…, -3, -2, -1, 0, 1, 2, 3, …	No decimal point; negatives allowed
Rational Numbers (Q)	Any number written as a/b with integers a, b (b ≠ 0)	Fractions and repeating decimals fit here
Irrational Numbers	Decimals that never end and never repeat (like √2)	Decimal pattern never repeats
Real Numbers (R)	All rationals and irrationals	Every point on the number line
Opposites	Pairs like 7 and -7	Same distance from 0, different signs
Absolute Value	Distance from 0, written |x|	Result is never negative
Parity (Even/Odd)	Even integers end in 0, 2, 4, 6, 8; odd end in 1, 3, 5, 7, 9	Look at the last digit

Sign Rules People Mix Up

If integers feel tricky, it’s usually the signs. The math is steady. The confusion comes from how our brains read minus signs in different roles.

Minus Sign Vs. Negative Sign

These look the same, yet they can mean different things.

• Negative sign: -8 is a single integer.
• Minus sign: 10 – 8 is an operation between two numbers.

When you see two negatives in a row, stop and label them. That split-second habit prevents a lot of mistakes.

Adding A Negative

Adding a negative is the same as subtracting the positive number. The clean mental move is “step left on the number line.”

• 9 + (-2) = 7
• -1 + (-6) = -7

Subtracting A Negative

Subtracting a negative flips it to addition. This is the “double negative” idea that shows up all over algebra.

• 5 – (-3) = 8
• -4 – (-2) = -2

Operation Rules Cheat Sheet

This table is built for homework speed. Use it when you need the rule and a clean sample in one glance. It focuses on the operations where sign mistakes happen most.

Operation Pattern	Rule	Sample
(+) + (-)	Subtract absolute values; keep sign of the larger absolute value	7 + (-10) = -3
(-) + (-)	Add absolute values; result stays negative	-4 + (-9) = -13
(+) – (-)	Change to addition	6 – (-2) = 8
(-) – (+)	Change to adding a negative	-3 – 5 = -8
(-) × (-)	Result is positive	(-6) × (-2) = 12
(-) × (+)	Result is negative	(-6) × 2 = -12
Integer ÷ Integer	Result can be non-integer; check divisibility	15 ÷ 3 = 5, yet 15 ÷ 4 = 3.75

How To Tell If A Value Is An Integer

When you’re scanning a worksheet or a word problem, these quick checks save time.

Quick Tests

• Look for a decimal point: If you see digits after it, it’s not an integer.
• Look for a fraction bar: A fraction can be an integer only if it simplifies to a whole number (like 8/4 = 2).
• Look at the context: “Three and a half” is not an integer. “Three steps down” often points to -3.

Integer Traps That Show Up In Class

• Negative zero: In standard arithmetic, -0 is the same as 0.
• Mixed numbers: 2 1/3 is not an integer.
• Repeating decimals: 0.333… is rational (1/3), not an integer.
• Square roots: √9 is an integer (3). √8 is not.

Mini Practice: Build Integer Confidence

Try these without rushing. If you get stuck, use the number line idea and the “add the opposite” rewrite for subtraction.

Sort Each Value As Integer Or Not

• -12
• 0
• 5.6
• 18/3
• -7/2
• √16

Compute Each One

• -8 + 13
• 6 – (-9)
• -4 – 11
• (-3) × (-7)
• 20 ÷ 5

Check Your Thinking With Two Habits

• Say the operation out loud: “Subtract a negative” should trigger an addition rewrite.
• Track distance from 0: Absolute value helps you judge which side wins in mixed-sign addition.

Why Integers Matter In Later Math

Integers are the first number set where subtraction always makes sense without leaving the set. That single fact feeds a lot of later skills, like solving equations, working with slopes, and reading graphs.

Once you’re steady with integer rules, algebra stops feeling like a new language. Signs become predictable. Steps become repeatable. That’s when speed starts to show up, even without memorizing piles of tricks.

One last check: an integer never has a fractional piece. If it’s a clean whole number in either direction from zero, it belongs in ℤ.