An integer is a whole number, its opposite, or zero: the counting numbers, their negatives, and zero. The integers are …, −3, −2, −1, 0, 1, 2, 3, … and go on forever in both directions. Integers have no fractional or decimal part, so 4.5 and 12 are not integers.
In Spanish
número entero
Also called
whole-number positives and negatives, integers, signed whole numbers
Integers are the evenly spaced dots on the number line, on both sides of 0. Numbers between the dots, such as −0.5 and 2.5, are rational but not integers.
Formal definition and notation
The integers are the numbers of the form a or −a for some whole numbera. The set is written ℤ (a double-struck Z) in higher math; in school it is usually listed as {…, −2, −1, 0, 1, 2, …}. It contains the positive integers 1, 2, 3, …, the negative integers −1, −2, −3, …, and 0, which is neither positive nor negative. The sum, difference and product of two integers is always an integer; a quotient may not be (7 ÷ 2 = 3.5). Every integer is a rational number, since n = n1.
Symbol
Read it as
Meaning
−5
negative five
the integer 5 units to the left of 0
−(−5)
the opposite of negative five
equals 5; the opposite of the opposite is the number itself
{…, −1, 0, 1, …}
the set of integers
the dots mean the list continues forever both ways
Key facts
Zero is an integer. It is neither positive nor negative, and it is its own opposite.
On a number line, integers to the left are smaller: −7 < −3, even though 7 is greater than 3.
Every whole number is an integer, but not every integer is a whole number: −4 is an integer but not a whole number.
A fraction or decimal can name an integer: 124 = 3 and 6.0 are integers. 74 and 2.5 are not.
Integers describe opposite directions: temperatures below zero, elevation below sea level, debts and credits, gains and losses.
Try it: place integers
Interactive
Drag the two points, or select one and use the arrow keys. The readout tells which integer is greater and gives each one’s distance from 0. Try two negative integers: which one is greater?
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Integer examples, step by step
Example 1: Spot the integers
Warm-up
Which of these numbers are integers: −8, 3.2, 0, 15, −23, √9?
−8 is the opposite of the whole number 8, so it is an integer. 0 and 15 are integers too.
3.2 and −23 have a fractional part, so they are not integers.
√9 = 3, which is an integer, even though it is written with a root sign.
Answer: The integers are −8, 0, 15 and √9, which equals 3.
Example 2: Order integers
Core
Order from least to greatest: −5, 3, −1, 0, −8, 4.
Negative integers come first, and the farther from 0, the smaller: −8 < −5 < −1.
At 6 a.m. the temperature was −7degrees Fahrenheit. By noon it had risen 12 degrees, and by evening it had fallen 9 degrees from the noon reading. What was the evening temperature?
A rise adds and a fall subtracts: −7 + 12 − 9.
−7 + 12 = 5: the temperature crossed 0 and reached 5 degrees at noon.
5 − 9 = −4.
Answer: The evening temperature was −4 degrees Fahrenheit.
Common mistakes
✗ Mistake−6 is not an integer, because integers are positive.
✓ Instead−6 is an integer: the opposite of the whole number 6.
A negative number can still be an integer. The word integer does not mean positive only.
✗ Mistake−7 > −3, because 7 is bigger than 3.
✓ Instead−7 < −3: on the number line, −7 is farther left.
For negative integers, a larger distance from 0 means a smaller number. Owing 7 dollars is worse than owing 3.
✗ Mistake186 is a fraction, so it cannot be an integer.
✓ Instead186 = 3, an integer.
What matters is the value, not how it is written. Simplify first: if the result has no fractional part, it is an integer.
✗ Mistake0 is not an integer, because it is not positive or negative.
✓ Instead0 is an integer. It is the integer that is neither positive nor negative.
The integers are the whole numbers, their opposites, and zero, so 0 is included by definition.
A rational number is any number that can be written as a fraction of integers, such as 34 or −2.5. Integers are the rational numbers with no fractional part.
The absolute value of an integer is its distance from 0, which is never negative. −9 is an integer; its absolute value is 9.
Quick check: 3 questions
Question 1 of 3
Which number is NOT an integer?
Show answer and explanation
Answer: D.72 = 3.5, which has a fractional part. −11 is a negative integer, 0 is an integer, and 25 is a positive integer.
Question 2 of 3
Which list is in order from least to greatest?
Show answer and explanation
Answer: A. The most negative number is the least, so −9 comes before −4. Choice B orders the negatives by their size without the sign, choice C starts at 0, and choice D is greatest to least.
Question 3 of 3
What integer is 6 less than −2?
Show answer and explanation
Answer: C. 6 less than −2 is −2 − 6 = −8: move 6 units left from −2. −4 and 4 come from 6 − 2 with the wrong sign or order, and 8 drops the negative sign.
Integer: frequently asked questions
Is 0 an integer?
Yes. Zero is an integer, and it is the only integer that is neither positive nor negative. It sits in the middle of the number line and is its own opposite, because −0 = 0. It is also a whole number. In real situations it often marks a reference point, like sea level or 0 degrees.
Are fractions and decimals integers?
Only when they equal a number with no fractional part. 205 = 4 and 7.0 are integers in disguise. 34, 2.5 and −0.1 are not. A quick test: simplify the fraction or look at the digits after the decimal point. If nothing is left over, the number is an integer.
What is the difference between whole numbers and integers?
Whole numbers are 0, 1, 2, 3 and so on. Integers include all of those plus their negatives: −1, −2, −3 and so on. So every whole number is an integer, but negative integers like −12 are not whole numbers. Both sets have no fractions or decimals.
NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2Whole NumberA whole number is 0 or a positive counting number: 0, 1, 2, 3, and so on. Whole numbers do not include fractions, decimals, or negative numbers, so 12 is a whole number but 8.5, 34 and −3 are not. Every whole number is also an integer.Number System · Grade 3, Grade 4, Grade 5Positive IntegerA positive integer is a whole number greater than zero: 1, 2, 3, 4 and so on. Positive integers have no fraction or decimal part and are never 0 or negative. They are also called counting numbers, and the set is written ℤ+. When a test says n is a positive integer, n cannot be 0, −3 or 2.5.Problem Solving · Grade 6, Grade 7, Algebra 1DifferenceA difference is the result of subtracting one number from another. In 12 − 7 = 5, the difference is 5. It tells how much bigger one number is than another, or how far apart they are on a number line. “The difference of a and b” means a − b, in that order.Operations · Grade 1, Grade 2, Grade 3QuotientA quotient is the answer you get when you divide one number by another. In 20 ÷ 4 = 5, the quotient is 5. The number being divided is the dividend and the number you divide by is the divisor. When the division is not exact, the quotient comes with a remainder, or as a decimal or fraction.Operations · Grade 3, Grade 4, Grade 5Rational NumberA rational number is any number that can be written as a fraction of two integers, ab, where b is not 0. Fractions, integers, terminating decimals, and repeating decimals are rational numbers: 34, −5 = −51, 1.6 and 0.333… all qualify. Numbers such as √2 and π are irrational.Number System · Grade 7, Grade 8, Algebra 1OppositeThe opposite of a number is the number the same distance from 0 on the other side of the number line. The opposite of 7 is −7, and the opposite of −7 is 7. Opposites add to 0, and 0 is its own opposite. The opposite of a negative number is positive.Number System · Grade 6, Grade 7, Grade 8Number LineA number line is a straight line with numbers placed at equal distances along it, so every point stands for a number. Zero sits in the middle, positive numbers go to the right and negative numbers to the left. Numbers increase from left to right, so a number line shows order, distance and adding as moving.Number System · Grade 2, Grade 3, Grade 6Negative NumberA negative number is a number less than zero, written with a minus sign, such as −3, −0.5 or −23. On a number line, negative numbers sit to the left of 0. They describe amounts below a starting point: temperatures below zero, depths below sea level, or money owed. Zero itself is neither negative nor positive.Number System · Grade 6, Grade 7DegreeA degree is the standard unit for measuring an angle, written with the symbol °. One degree is 1360 of a full turn, so a full turn is 360°, a straight angle is 180° and a right angle is 90°. An angle that turns through 25 one-degree angles measures 25°.Geometry · Grade 4, Grade 5, Grade 7SimplifySimplify means to rewrite a number or expression in a shorter, easier form without changing its value. To simplify a fraction, divide the numerator and denominator by a common factor until none is left: 1824 = 34. To simplify an expression, combine like terms and do the operations you can: 3x + 2x = 5x.Number System · Grade 4, Grade 5, Grade 6Absolute ValueThe absolute value of a number is its distance from zero on the number line, so it is never negative. It is written with two vertical bars: |−6| = 6 and |6| = 6. A positive number and 0 keep their value; a negative number loses its minus sign.Number System · Grade 6, Grade 7, Grade 8Decimal PointA decimal point is the dot that separates the whole-number part of a number from its fractional part. Digits to its left are ones, tens and hundreds; digits to its right are tenths, hundredths and thousandths. In 46.25 the point sits between 6 ones and 2 tenths, and you read it aloud as “and”.Number System · Grade 4, Grade 5, Grade 6ZeroZero (0) is the number that represents none or no quantity. It is neither positive nor negative and sits in the middle of the number line, between the negative and positive numbers. Adding zero to a number does not change it, as in 9 + 0 = 9, so zero is called the additive identity.Number System · Grade 3, Grade 4, Grade 5Irrational NumberAn irrational number cannot be written as a fraction of two integers. Its decimal form does not end and does not repeat in a pattern. Examples are √2 = 1.41421… and π = 3.14159…. The square root of any whole number that is not a perfect square is irrational.Number System · Grade 8, Algebra 1Consecutive IntegersConsecutive integers are integers that follow each other in order, each 1 more than the one before, such as 7, 8, 9 or −2, −1, 0. In algebra, write them as n, n + 1, n + 2. Consecutive even or consecutive odd integers are 2 apart: n, n + 2, n + 4.Algebra · Grade 7, Grade 8, Algebra 1Real NumberA real number is any number that can be placed on the number line. The real numbers are all the rational numbers (fractions, integers, terminating and repeating decimals) together with all the irrational numbers (such as √2 and π). Square roots of negative numbers, like √−4, are not real.Number System · Grade 8, Algebra 1Additive InverseThe additive inverse of a number is the value that adds to it to give zero. The additive inverse of 9 is −9, because 9 + (−9) = 0, so it is the same thing as its opposite. To find it, change the sign: the additive inverse of −14 is 14, and 0 is its own.Number System · Grade 6, Grade 7, Grade 8