Integers – Definition, Examples, and Rules

In math, the integers are numbers that do not contain fractions or decimals. The set includes zero, the natural numbers (counting numbers), and their additive inverses (the negative integers). Examples of integers include -5, 0, and 7. Here is everything you need to know about the integers, including examples of them and their rules for addition, subtraction, multiplication, and division.

What Are Integers?

Integers are numbers without decimals or fractions. But, unlike the whole numbers, integers include negative numbers. Typical symbols for the set of integers include the capital letter Z, as well as the blackboard bold or boldface versions of the letter. The set is Z = {…, -3, -2, -1, 0, 1, 2, 3, …}. It is a countably infinite set. All of the integers are both rational and real numbers.

Integers on the Number Line

Integers extend infinitely in both directions along the number line. Positive integers lie to the right of zero, while negative integers lie to the left.

← −5 −4 −3 −2 −1 0 1 2 3 4 5 →

The farther a number is from zero, the greater its absolute value. For example, −8 is farther from zero than −3, so |−8| > |−3|.

Understanding integers as positions on a number line makes addition and subtraction easier to visualize. Adding moves right, while subtracting moves left.

Examples of Integers and Numbers That Are Not Integers

Integers include:

• Negative integers
• Zero
• Positive integers

So, examples of integers are:

• -57
• -13
• -1
• 0
• 5
• 124

Examples of numbers that are not integers are:

• 1/3
• 0.45
• -22.6
• 132.4
• -1/2
• √3
• Pi
• 3i
• 7 ⅚

Quick Integer Classification Table

Why Is Zero an Integer?

Zero is an integer because it has no fractional or decimal part and fits the definition of an integer. It serves as the boundary between positive and negative integers.

Zero is unique because it is:

• Neither positive nor negative
• The additive identity (a + 0 = a)
• The only integer whose additive inverse is itself

Why Is the Symbol for Integers ℤ?

Mathematicians typically represent the set of integers using the symbol ℤ. The symbol comes from the German word Zahlen, meaning “numbers.”

Examples:

• ℤ = {…, -3, -2, -1, 0, 1, 2, 3, …}
• 5 ∈ ℤ (“5 is an element of the integers”)
• 1/2 ∉ ℤ (“1/2 is not an integer”)

Addition and Subtraction

The rules for adding and subtracting integers are essentially the rules for working with positive and negative numbers.

When adding two integers that have the same sign, the answer has the same sign. For example:

• 1 + 3 = 4
• (-4) + (-3) = -7
• 4 + 0 = 4
• 0 + (-2) = -2

Adding integers with different signs works the same as subtracting integers. The answer has the same sign as the number with the larger value. Note that when you subtract a negative integer, it is the same as adding its positive value. For example:

• (-7) + 2 = -5 which also equals 2 – 7
• 4 + (-8) = 4 – 8 = -4
• 3 – (-2) = 3 + 2 = 5
• (-5) – (-3) = (-5) + 3 = -2 which also equals 3 – 5
• 8 – 0 = 8

Multiplication and Division

The rules for multiplication and division are straightforward:

• If both integers are positive, the answer is positive.
• When both integers are negative, the answer is positive. The two negative signs cancel each other out.
• If the numbers have different signs, the answer is negative.
• When adding or dividing several numbers, add up how many are positive and how many are negative. If there are an odd number of negative signs, the answer is also negative. Otherwise, it is positive.
• Multiplying any integer by 0 equals 0.
• Division by zero is undefined in ordinary arithmetic and algebra. While some advanced mathematical systems use special concepts involving infinity, division by zero is not defined for integers, rational numbers, real numbers, or complex numbers.
• Multiplying two integers gives another integer, but this is not a rule for division.

Here are examples of multiplication and division using integers.

• 4 x 5 = 20
• (-2) x (-3) = 6
• (-6) x 3 = -18
• 7 x (-2) = -14
• 2 x (-3) x 4 = -24
• (-2) x 2 x (-3) = 12
• 12 / 2 = 6
• (-2) / 6 = -1/3
• (-10) / 5 = -2
• 14 / (-7) = -2
• (-6) / (-2) = 3

Real-World Examples of Integers

Integers appear whenever quantities can be positive, negative, or zero.

Examples include:

• Temperatures above and below 0°C
• Elevations above and below sea level
• Bank account gains and losses
• Sports scores and point differentials
• Building floors above and below ground level
• Profit and loss calculations

For example, a temperature of −10°C is represented by the integer −10, while a temperature of 15°C is represented by the integer 15.

Rules and Properties of Integers

The algebraic properties of integers include:

• Closure
• The identity element
• Associative property
• Commutative property
• Distributive property
• Inverse elements
• No division by zero

Note that the integers are closed under subtraction, but are not closed under division. In other words, if you divide one integer by another, the answer is not necessarily another integer. For example, 2 divided by 4 is 1/2 or 0.5 (not an integer). Division of an integer by zero is undefined.

Here is a summary of these properties.

Absolute Value and Integers

The absolute value of an integer is its distance from zero on the number line, regardless of direction.

Examples:

Absolute value is written using vertical bars:

|−7| = 7

|4| = 4

Because distance cannot be negative, absolute value is always zero or positive.

Integers vs Whole Numbers, Natural Numbers, and Rational Numbers

Integers belong to a larger hierarchy of number sets.

Common Mistakes When Working With Integers

Mistake: Assuming all negative numbers are integers.

Example:

−2.5 is negative but not an integer.

Mistake: Forgetting that subtracting a negative becomes addition.

Example:

5 − (−3) = 8

Mistake: Assuming division of integers always produces an integer.

Example:

5 ÷ 2 = 2.5

Mistake: Thinking zero is positive.

Zero is neither positive nor negative.

Frequently Asked Questions

Is 0 an integer?

Yes. Zero is an integer and separates positive and negative integers on the number line.

Is a negative number always an integer?

No. Negative fractions and decimals such as −1/2 and −2.7 are not integers.

Is 3.0 an integer?

Mathematically, yes. The value 3.0 is equal to 3, which is an integer.

Is √4 an integer?

Yes. Since √4 = 2, the result is an integer.

Is √3 an integer?

No. √3 is irrational and is not an integer.

Are integers rational numbers?

Yes. Every integer can be written as a fraction with denominator 1.

For example:

5 = 5/1

−8 = −8/1

History of Integers

Ancient civilizations used positive whole numbers for counting, but negative numbers were not widely accepted for centuries. Chinese mathematicians were using positive and negative numbers as early as the 2nd century BCE, often representing them with different colored counting rods.

European mathematicians were initially skeptical of negative numbers because quantities less than zero seemed to have no physical meaning. By the 17th century, negative numbers became accepted as useful mathematical tools, particularly in algebra and accounting.

The modern concept of integers as a formal number set developed during the 19th century as mathematicians established rigorous foundations for arithmetic and number theory.

References

• Clapham, Christopher; Nicholson, James (2014). The Concise Oxford Dictionary of Mathematics (5th ed.). Oxford University Press. ISBN 978-0-19-967959-1.
• Frobisher, Len (1999). Learning to Teach Number: A Handbook for Students and Teachers in the Primary School. The Stanley Thornes Teaching Primary Maths Series. Nelson Thornes. ISBN 978-0-7487-3515-0.
• Goldrei, Derek (1998). Classic Set Theory: A Guided Independent Study (1st ed.). Boca Raton, FL: Chapman & Hall/CRC. ISBN 978-0-412-60610-6.
• Mac Lane, Saunders; Birkhoff, Garrett (1999). Algebra (3rd ed.). American Mathematical Society. ISBN 0-8218-1646-2.
• Mendelson, Elliott (2008). Number Systems and the Foundations of Analysis. Dover Books on Mathematics. Courier Dover Publications. ISBN 978-0-486-45792-5.