A 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.
In Spanish
número entero no negativo (número natural con el cero)
The whole numbers are the counting numbers plus 0. They are the integers that are not negative, and they sit inside the rational numbers.
Formal definition and notation
The whole numbers are the set {0, 1, 2, 3, …}: the counting numbers together with zero. Equivalently, they are the nonnegative integers. The set has a smallest element, 0, but no largest. The sum and the product of two whole numbers are always whole numbers; a difference or quotient may not be.
Symbol
Read it as
Meaning
{0, 1, 2, 3, …}
the set 0, 1, 2, 3, and so on
the whole numbers
…
and so on
the list continues forever in the same pattern
n ≥ 0
n is greater than or equal to zero
a whole number n is never negative
Key facts
Whole numbers include 0, but they do not include negative numbers.
There is a smallest whole number, 0, and no largest one: you can always add 1.
A fraction or decimal that simplifies to a whole number is one: 124 = 3 and 7.0 = 7.
Adding or multiplying two whole numbers always gives a whole number. Subtracting (3 − 5 = −2) or dividing (7 ÷ 2 = 3.5) may not.
Every whole number is an integer and a rational number, but not every integer is a whole number.
Whole Number examples, step by step
Example 1: Pick out the whole numbers
Warm-up
Which of these are whole numbers: −4, 0, 7, 2.5, 93, 100?
−4 is negative, so it is not a whole number. 2.5 is between 2 and 3, so it is not.
0, 7 and 100 are whole numbers.
93 looks like a fraction, but 9 ÷ 3 = 3, a whole number.
Answer: 0, 7, 9/3 (which equals 3) and 100 are whole numbers; −4 and 2.5 are not.
Example 2: Count whole numbers in a range
Core
How many whole numbers are greater than −3.5 and less than 4.2?
Whole numbers start at 0, so the negative part of the range adds none.
The whole numbers below 4.2 are 0, 1, 2, 3 and 4.
Count them: 5. (Remember to count 0.)
Answer: There are 5 whole numbers: 0, 1, 2, 3 and 4.
Example 3: Which operations stay whole?
Test-style
Two whole numbers are combined with one operation. Which operations always give a whole number? Test with 7 and 9.
Addition: 7 + 9 = 16, whole. Adding two numbers that are 0 or more never gives a negative or a fraction.
Integers include the negatives: …, −2, −1, 0, 1, 2, …. Whole numbers are only the integers from 0 up. So −5 is an integer but not a whole number.
Whole Numbervs.Counting numbers (natural numbers)
Counting numbers start at 1: 1, 2, 3, …. Whole numbers are the same list with 0 added at the front.
Quick check: 3 questions
Question 1 of 3
Which number is NOT a whole number?
Show answer and explanation
Answer: D.−1 is negative, so it is not a whole number. 0 is the first whole number, 15 is whole, and 62 = 3 is whole even though it is written as a fraction.
Question 2 of 3
What is the smallest whole number?
Show answer and explanation
Answer: A. The whole numbers are 0, 1, 2, 3, …, so the smallest is 0. Choice B is the smallest counting number, C is not whole, and D describes the integers, which go on forever in the negative direction.
Question 3 of 3
How many whole numbers are less than 5?
Show answer and explanation
Answer: B. They are 0, 1, 2, 3 and 4: five numbers. Choice A forgets 0, C includes 5 itself, and D counts negative numbers, which are not whole.
Whole Number: frequently asked questions
Is 0 a whole number?
Yes. Zero is the first whole number: the whole numbers are 0, 1, 2, 3 and so on. That is exactly what separates them from the counting numbers, which start at 1. Zero is also an integer and an even number, and it is neither positive nor negative.
What is the difference between whole numbers and integers?
Integers include the negative numbers, and whole numbers do not. The integers are …, −3, −2, −1, 0, 1, 2, 3, …; the whole numbers are only 0, 1, 2, 3, …. Every whole number is an integer, but integers such as −7 are not whole numbers.
Are decimals or fractions ever whole numbers?
Only when they equal a whole number. 8.5 and 34 are not whole numbers, but 6.0 and 205 are, because they equal 6 and 4. The test is the value: if it lands exactly on 0, 1, 2, 3, … on the number line, it is a whole number.
IntegerAn 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.Number System · Grade 6, Grade 7, Grade 8Counting NumbersCounting numbers are the numbers you say when you count objects one by one: 1, 2, 3, 4, 5 and so on, without end. They start at 1, not 0, and include no fractions, decimals or negative numbers. Adding 0 to the counting numbers gives the whole numbers.Number System · Kindergarten, Grade 1, Grade 2ZeroZero (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 5NonnegativeNonnegative means greater than or equal to zero: such a number is either 0 or positive, so x ≥ 0. The nonnegative integers are 0, 1, 2, 3 and so on. It is not the same as positive, because 0 is nonnegative but not positive. Distances, absolute values and principal square roots are always nonnegative.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 5Negative 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 7Rational 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 1AdditionAddition is the operation that puts amounts together to find the total. The numbers you add are called addends, the answer is the sum, and the sign is the plus sign, +. For example, 7 + 5 = 12. On a number line, adding a positive number means moving to the right.Operations · Kindergarten, Grade 1, Grade 2NumberA 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 2MultiplicationMultiplication is the operation that combines equal groups to find a total. The fact 4 × 6 = 24 means 4 groups of 6, or 6 + 6 + 6 + 6. The numbers you multiply are called factors, and the result is the product. Order does not matter: 6 × 4 is also 24.Operations · Grade 2, Grade 3, Grade 4SubtractionSubtraction is the operation that finds how much is left when you take one amount away from another, or how far apart two numbers are. It is written with a minus sign: 9 − 4 = 5. The answer is called the difference. Subtraction undoes addition, so you can check 9 − 4 = 5 by adding 5 + 4 = 9.Operations · Grade 1, Grade 2, Grade 3DivisionDivision is the operation of splitting a number into equal groups. In 12 ÷ 3 = 4, the dividend 12 is shared into 3 equal groups (the divisor), and each group gets 4 (the quotient). Division undoes multiplication, because 3 × 4 = 12. If the groups do not come out even, the amount left over is the remainder.Operations · Grade 3, Grade 4, Grade 5UndefinedUndefined describes an expression that has no value, most often because it divides by zero. 70 is undefined, since no number times 0 gives 7. A vertical line has an undefined slope, and 3x − 2 is undefined at x = 2. Undefined does not mean zero: 07 = 0, but 70 has no value.Problem Solving · Grade 6, Grade 8, Algebra 1SimplifySimplify 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 6Place ValuePlace value is the value of a digit based on its position in a number. In 3,507 the 5 is worth 500, because it sits in the hundreds place. Each place is worth 10 times the place to its right: ones, tens, hundreds, thousands, and after the decimal point, tenths and hundredths.Number System · Grade 3, Grade 4, Grade 5DigitA digit is one of the ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 that we use to write every number. The number 507 has three digits: 5, 0 and 7. What a digit is worth depends on its place: in 507, the 5 stands for 5 hundreds, or 500.Number System · Grade 1, Grade 2, Grade 3DecimalA decimal is a number written with a decimal point to show parts of a whole. Decimals use place value to represent tenths, hundredths, thousandths and more: in 3.14, the 1 means 1 tenth and the 4 means 4 hundredths. So 0.75 = 75100, and 0.5 is one half.Number System · Grade 4, Grade 5, Grade 6Improper FractionAn improper fraction has a numerator that is greater than or equal to its denominator, so its value is 1 or more. Examples are 74, 92 and 55. Every improper fraction can be rewritten as a whole number or a mixed number: 74 = 134. Improper does not mean wrong.Number System · Grade 4, Grade 5, Grade 6Irrational 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 1Even NumberAn even number is an integer that can be divided by 2 with no remainder, so it can be split into two equal groups or into pairs. Even numbers end in 0, 2, 4, 6 or 8, such as 14, 236 and −10. Zero is even. Every even number can be written as 2n for some integer n.Number System · Grade 2, Grade 3, Grade 4Odd NumberAn odd number is a whole number that cannot be split into two equal groups: dividing it by 2 always leaves a remainder of 1. Odd numbers end in 1, 3, 5, 7 or 9, such as 7, 23 and 409. Every odd number can be written as 2k + 1, one more than a multiple of 2.Number System · Grade 2, Grade 3, Grade 4Natural NumberA natural number is one of the numbers 1, 2, 3, 4 and so on, used for counting and ordering. Natural numbers are positive and whole, so they include no fractions, decimals, negatives or zero in most US school math. Some books also include 0, so check the convention. The set is written ℕ.Number System · Grade 5, Grade 6, Grade 7