Number basics

Whole Numbers and Natural Numbers

Whole numbers are 0, 1, 2, 3 and so on, with no fractions, decimals or negatives. Natural numbers are the counting numbers 1, 2, 3 and so on — though some textbooks and fields also count 0 as natural. This guide sets out both definitions, explains where the conventions on 0 differ, compares them with integers and counting numbers, and works through examples of what does and does not qualify.

Whole numbers vs natural numbers: the short answer

The two sets are almost identical. The only number in question is 0.

So under the usual school convention, the whole numbers are the natural numbers plus 0. Under the other convention, the two sets are the same thing. Neither convention is wrong; they are different definitions, and a textbook, course or paper should tell you which it is using. When it matters and nobody has said, write it out: “positive integers” for 1, 2, 3, … and “non-negative integers” for 0, 1, 2, 3, … leave no room for doubt.

What are whole numbers?

Whole numbers are the numbers you get by starting at 0 and adding 1 again and again: 0, 1, 2, 3, 4, 5, and on forever. There is no largest whole number, because you can always add 1 more.

Three kinds of number are left out:

• Negative numbers, such as −1 or −40.
• Fractions, such as 1/2 or 7/3, unless they simplify to a whole number.
• Decimals, such as 2.5 or 0.01, unless every digit after the point is zero.

The smallest whole number is 0. On a number line, the whole numbers are the evenly spaced points starting at 0 and running to the right — individual dots, not the stretches of line between them.

In everyday speech, “a whole number of” means a count with nothing left over. Eighteen cookies shared among three people is a whole number each, because 18 ÷ 3 = 6. Shared among four, it is not: 18 ÷ 4 = 4.5. The remainder calculator shows how much is left over when a division does not come out whole.

What are natural numbers?

Natural numbers are the numbers used for counting objects: 1, 2, 3, 4, and so on. They are written with the symbol ℕ. If you count the chairs in a room, you say “one, two, three”, never “zero” first, which is why many definitions start at 1.

Is 0 a natural number?

It depends on the convention, and both are in wide use.

• 0 is not natural in most US elementary, middle and high school courses, and in much of number theory. There, ℕ = {1, 2, 3, …} and the natural numbers are the same as the counting numbers and the positive integers.
• 0 is natural in set theory, mathematical logic and most of computer science, where counting often starts at 0 (the first item in a list is item 0). The international standard for mathematical notation, ISO 80000-2, also includes 0.

Why two conventions exist

Each starting point suits a different job. If natural numbers are for counting objects, starting at 1 matches how people count. If they are for answering “how many?”, 0 is a valid answer — an empty box holds 0 items — and starting at 0 makes the definitions in set theory and programming tidier, since 0 is where counting up from nothing begins. Both uses are old and well established, so neither has displaced the other.

Writers who need to be exact use marked symbols. ℕ₀ or ℕ ∪ {0} means the natural numbers including 0. ℕ⁺, ℕ*, ℕ₁ or ℤ⁺ means the positive integers, starting at 1. Symbols vary between books, so check the definition near the front.

Counting numbers and positive integers

Three more names describe nearly the same sets, which is where most of the confusion comes from.

• Counting numbers are 1, 2, 3, … — always starting at 1, because you never count “zero” as the first object.
• Positive integers are 1, 2, 3, … — the integers greater than 0.
• Non-negative integers are 0, 1, 2, 3, … — the integers that are not negative, which is exactly the whole numbers.

Unlike “natural numbers”, none of these has a competing definition, which is why careful writers prefer them.

Comparing the number sets

The sets nest: each one contains the one above it in this table and adds something new.

Every natural number is a whole number, every whole number is an integer, and every integer is a rational number. The reverse statements are all false: −3 is an integer but not a whole number, and 1/2 is rational but not an integer.

Examples: whole, natural, or neither?

Judge each number by its value after simplifying, not by how it is written.

An improper fraction is a whole number only when the top divides evenly by the bottom. 18/3 is 6, but 19/3 is 6 1/3, which is not whole — the improper fraction calculator shows the whole part and the remainder.

What you can and cannot do with whole numbers

A set is closed under an operation if doing that operation on any two members always gives another member. Whole numbers are closed under two of the four basic operations.

Those two gaps are the reason the larger number systems exist: negative numbers fill the gap left by subtraction, and fractions fill the gap left by division. The rules for working past zero are in negative number rules.

Addition and multiplication of whole numbers are commutative and associative — order and grouping do not change the answer, as the guide to commutative and associative properties explains. Two numbers play special roles: 0 is the additive identity (n + 0 = n) and 1 is the multiplicative identity (n × 1 = n). Multiplying by 0 always gives 0, and dividing by 0 is undefined.

Primes, composites and the two exceptions

Every whole number greater than 1 is either prime or composite. The two exceptions are 0 and 1, which are neither — see prime and composite numbers for why.

Writing and reading whole numbers

Whole numbers are written with the ten digits 0 through 9, and the position of each digit sets its value. In 4,082 the 4 is worth 4,000, the 0 holds the hundreds place empty, the 8 is worth 80 and the 2 is worth 2. The place value guide has a full chart from ones to billions and down into decimals.

Two counting facts follow from where the sets start. There are 10 whole numbers below 10 (0 through 9), but only 9 natural numbers below 10 under the school convention (1 through 9). The sum of the first n natural numbers is n(n + 1)/2, so 1 + 2 + … + 10 = 10 × 11 ÷ 2 = 55; adding 0 to the start does not change the total.

Common mistakes

• Including negatives. −5 has no fractional part, but whole numbers stop at 0. Negative numbers like −5 are integers, not whole numbers.
• Leaving 0 out of the whole numbers. Zero is always a whole number. The debate is only about natural numbers.
• Assuming one convention for natural numbers. Check the definition in use before answering whether 0 is natural.
• Rejecting numbers by how they look. 18/3, √25 and 4.0 are whole numbers once simplified.
• Calling 0.5 “half a whole number”. It is not a whole number at all. Rounding it gives a whole number — the rounding calculator shows the nearest whole number for any decimal — but the value itself is not one.

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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