What Is an Integer?

An integer is any number that has no fractional or decimal part and can be positive, negative, or zero. In other words, integers are the counting numbers, their negatives, and zero, all placed together on a single number line. So \(-7\), \(0\), \(4\), and \(152\) are all integers, but \(\frac{1}{2}\), \(2.75\), and \(\sqrt{2}\) are not, because they include a fractional or non-terminating decimal part.

A quick way to check if a number is an integer is to ask: "Can I write this number without a fraction, decimal, or square root sign?" If yes, and it does not include an imaginary or irrational value, it is an integer.

Integers are easiest to picture on a number line, which stretches in both directions from zero. Positive integers sit to the right of zero, negative integers sit to the left, and zero sits exactly in the middle. If you have not worked with negative values before, it helps to first review negative numbers and opposite numbers, since every positive integer has a matching negative integer the same distance from zero.

Since integers only fall on these evenly spaced marks, a number like \(1.5\) would land between two ticks and would not count as an integer. Once you can plot integers this way, it becomes much easier to work through comparing and ordering numbers, since the number line directly shows which integer is greater or smaller.

Integers belong to a larger family of number types, and it is easy to mix them up. The table below shows how they differ, based mainly on whether negative numbers and zero are included. For a fuller picture of where integers fit among fractions, decimals, and irrational numbers, see understanding the number systems.

Every whole number is an integer, and every natural number is also an integer, but the reverse is not true. A number like \(-6\) is an integer, yet it is neither a whole number nor a natural number.

Here are some quick examples to build intuition:

• Integers: \(-100\), \(-12\), \(-1\), \(0\), \(3\), \(58\), \(2024\)
• Not integers: \(\frac{3}{4}\), \(-2.5\), \(\sqrt{5}\), \(1.333...\), \(\pi\)

Notice that \(4.0\) is still an integer, because the decimal part is zero, but \(4.5\) is not, since it has a nonzero fractional part. Also, integers can be as large or as small (in the negative direction) as you like; there is no smallest or largest integer.

Consecutive integers are integers that follow each other in order, each one exactly \(1\) more than the one before it, such as \(4, 5, 6\) or \(-3, -2, -1\). If \(n\) is an integer, the next consecutive integer is \(n + 1\), and the one after that is \(n + 2\). This idea shows up often in word problems that ask you to find several unknown integers in a row.

Integers appear everywhere in everyday math: temperatures above and below zero, money gained or owed, elevations above and below sea level, and yardage gained or lost in a game. Understanding integers is also the foundation for later topics such as place value, absolute value, and operations with positive and negative numbers, so getting comfortable with the definition and the number line now will make those topics much easier.

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