Number basics

What Are Integers?

Integers are the whole numbers, their negatives and zero: …, −3, −2, −1, 0, 1, 2, 3, … and so on in both directions. They have no fractional or decimal part. This guide explains how to tell whether a number is an integer, how integers are ordered on the number line, the rules for adding, subtracting, multiplying and dividing them, and the mistakes that trip people up.

Integers defined

An integer is any number you can reach from 0 by counting whole steps forward or backward. Positive integers are 1, 2, 3 and up; negative integers are −1, −2, −3 and down; and 0 sits between them. Nothing in between counts — 2.5, 3/4 and −0.1 are not integers.

The word comes from Latin integer, meaning “whole” or “untouched” — a number that has not been broken into pieces. That is the test to apply: if a number has a fractional part that is not zero, it is not an integer.

Integers extend the whole numbers (0, 1, 2, 3, …) by adding their negatives. They are in turn part of the rational numbers, because any integer can be written as a fraction over 1: −7 = −7/1.

Is it an integer? Examples

The test is about the value, not the way the number is written. A fraction or a square root that works out to a whole number is an integer.

A quick way to check: an integer has nothing to the right of the decimal point except zeros. If a calculator shows any other digit after the point, the number is not an integer.

Positive, negative and zero

Every integer falls into exactly one of three groups.

• Positive integers: 1, 2, 3, … These are the counting numbers, also written ℤ⁺.
• Negative integers: −1, −2, −3, … Each is the opposite of a positive integer.
• Zero: neither positive nor negative. It is its own opposite.

Two related phrases come up often. Non-negative integers are 0, 1, 2, 3, … — the same set as the whole numbers. Non-positive integers are 0, −1, −2, −3, …

Even and odd integers

Even and odd extend to negative integers too. An integer is even if it divides by 2 with no remainder, so −4, −2, 0, 2 and 4 are all even. An integer is odd if it leaves a remainder of 1, so −3, −1, 1 and 3 are odd. Zero is even, because 0 ÷ 2 = 0 exactly.

Consecutive integers

Consecutive integers follow one another with a gap of 1, such as 15, 16, 17 or −2, −1, 0. Word problems often use them: three consecutive integers that add to 48 are 15, 16 and 17, because the middle one is 48 ÷ 3 = 16.

Integers on the number line

On a number line, the integers are evenly spaced points: 0 in the middle, positives to the right and negatives to the left. Two ideas follow directly from the picture.

Ordering integers

A number further right is always greater. That makes ordering negatives easy to get right: −2 is to the right of −7, so −2 > −7, even though 7 is bigger than 2. Think of temperatures — −2°F is warmer than −7°F.

Opposites and absolute value

Every integer has an opposite the same distance from 0 on the other side: the opposite of 5 is −5, and the opposite of −5 is 5. That distance is the absolute value, written with bars: |−8| = 8 and |8| = 8. Absolute value is never negative, because a distance cannot be. The absolute value calculator handles full expressions as well as single numbers.

Adding, subtracting, multiplying and dividing integers

Integer arithmetic uses the same four operations as ordinary arithmetic, with rules for the signs. The complete set of sign rules, with a number-line explanation of each case, is in negative number rules. The short version:

• Adding: same signs, add the sizes and keep the sign. Different signs, subtract the smaller size from the larger and take the sign of the larger.
• Subtracting: change it to adding the opposite. 5 − (−3) becomes 5 + 3.
• Multiplying and dividing: same signs give a positive answer; different signs give a negative one.

Integers are closed under +, − and ×, but not ÷

Add, subtract or multiply any two integers and the answer is always an integer. Division breaks that: 12 ÷ 4 = 3 is an integer, but 7 ÷ 2 = 3.5 is not. That is why division with integers is often stated with a remainder instead — 17 ÷ 5 = 3 remainder 2 — and why fractions had to be invented. The remainder calculator gives the quotient and remainder, including for negative numbers, where conventions differ: −17 ÷ 5 is −4 remainder 3 if the quotient is rounded down, or −3 remainder −2 if it is rounded toward zero. Both satisfy quotient × 5 + remainder = −17.

Properties of integers

The familiar laws of arithmetic all hold for integers. These are what let you rearrange and simplify expressions safely.

Subtraction and division are neither commutative nor associative: 3 − 8 = −5 but 8 − 3 = 5. The guides to the commutative and associative properties and the distributive property cover each law and where it breaks down.

One thing integers lack is a multiplicative inverse. The number that multiplies 4 to give 1 is 1/4, which is not an integer. Only 1 and −1 have integer inverses.

Integers in everyday use

Negative integers show up wherever a quantity can go below a zero point.

• Temperature: a morning low of −8°F rising to an afternoon high of 15°F is a rise of 15 − (−8) = 23 degrees.
• Elevation: sea level is 0. A diver at −40 feet and a gull at 25 feet are 25 − (−40) = 65 feet apart.
• Money: a balance of $120 followed by a $150 purchase leaves 120 − 150 = −30, an overdraft of $30.
• Sports: a golf score of −3 means three strokes under par; yardage lost on a football play is a negative gain.
• Buildings: some elevators label basement floors −1 and −2, with the ground floor as 0.

In each case the integer records both a size and a direction, which is the reason negatives exist at all.

Common mistakes

• Thinking −9 is bigger than −2. On the number line −9 is further left, so it is smaller: −9 < −2.
• Leaving 0 out. Zero is an integer, a whole number and an even number. It is just not positive or negative.
• Treating “integer” as “positive whole number”. That describes the counting numbers. Integers include the negatives.
• Judging by appearance. 12/4 and √16 are integers once evaluated; 0.999 is not, however close it is to 1.
• Squaring a negative without brackets. −3² means −(3²) = −9, because exponents come before the minus sign under PEMDAS. (−3)² = 9. The order of operations calculator shows which one an expression really means, and exponent rules covers powers of negative numbers.
• Assuming integer division always works out. 7 ÷ 2 leaves the integers. Say whether you want 3.5, or 3 remainder 1.

Frequently asked questions

Calculators for this

Every one of these shows the working, so you can check the method as well as the answer.

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