A 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.
In Spanish
número natural
Also called
natural numbers, counting numbers, positive integers, N
Grade level
K-5
Grade 5Grade 6Grade 7Algebra 1GEDSATACTACCUPLACER
Natural Number in a picture
Every natural number is also a whole number, an integer, a rational number and a real number, but not the other way around.
Formal definition and notation
The set of natural numbers is ℕ = {1, 2, 3, …}: it contains 1, and whenever it contains n it also contains n + 1. This is the same set as the positive integers. Some authors define ℕ = {0, 1, 2, …}, the nonnegative integers, so a text should say which it means. ℕ is closed under addition and multiplication but not under subtraction or division.
Symbol
Read it as
Meaning
ℕ
N, the natural numbers
the set {1, 2, 3, …} (or {0, 1, 2, …} in some books)
n ∈ ℕ
n is an element of N
n is a natural number
ℕ0
N zero
a label some books use for the natural numbers with 0 included
Key facts
The natural numbers are infinite: there is a smallest one (1, or 0 by the other convention) but no largest one.
Adding or multiplying two natural numbers always gives a natural number. Subtracting or dividing may not: 3 − 8 = −5 and 3 ÷ 8 = 38.
A number that looks like a decimal can still be natural: 6.0 and 123 both equal 6.
Every natural number greater than 1 is either prime or can be written in exactly one way as a product of primes.
Natural ⊂ whole ⊂ integer ⊂ rational ⊂ real: each set sits inside the next one.
Natural Number examples, step by step
Example 1: Is it a natural number?
Warm-up
Which of these are natural numbers: 14, −14, 0.5, 204, 0?
Counting numbers always start at 1. Natural numbers usually mean the same set, but some books add 0.
Quick check: 3 questions
Question 1 of 3
Which number is a natural number?
Show answer and explanation
Answer: D.279 = 3, a positive whole number. −5 is negative, and 13 and 0.25 are not whole.
Question 2 of 3
Which operation on two natural numbers always gives a natural number?
Show answer and explanation
Answer: B. A product of positive whole numbers is a positive whole number. Subtraction can give a negative (2 − 5), and division or averaging can give a fraction (1 ÷ 2, the average of 1 and 2 is 1.5).
Question 3 of 3
How many natural numbers are less than 7, using ℕ = {1, 2, 3, …}?
Show answer and explanation
Answer: A. They are 1, 2, 3, 4, 5, 6, so 6 of them. 7 counts 0 as well, 8 also counts 7, and “infinitely many” describes all natural numbers, not those below 7.
Natural Number: frequently asked questions
Is 0 a natural number?
It depends on the book. Most US school textbooks say no: the natural numbers are 1, 2, 3, and so on, the same as the counting numbers. Many set theory and computer science texts say yes and start at 0. Mathematicians have never fully agreed, so always check the definition your class or test uses.
What is the difference between natural numbers and whole numbers?
Under the usual school definition, only the number 0. Natural numbers are {1, 2, 3, …} and whole numbers are {0, 1, 2, 3, …}. Every natural number is a whole number, but 0 is a whole number that is not natural. Neither set includes negatives, fractions or decimals.
Are natural numbers the same as positive integers?
Yes, when the natural numbers start at 1. The positive integers are 1, 2, 3, and so on, exactly the same list. Tests such as the SAT usually say “positive integer” because that phrase leaves no doubt about 0. If a book includes 0 in the natural numbers, they match the nonnegative integers instead.
Updated November 2026·Reviewed by the Effortless Math team·Standards: n/a
Cite this entry
Effortless Math Team. “Natural Number.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/natural-number/
Flashcard
TermNatural Numberthe set N
DefinitionOne of the numbers 1, 2, 3, 4, … used for counting; some books include 0. Example: 9 and 20/4 are natural, but −2 and 0.5 are not.
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 8NonnegativeNonnegative 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 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 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 5NumberA 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 5SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2Decimal 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 6DifferenceA 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 3Positive 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 1Counting 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 2Prime NumberA prime number is a whole number greater than 1 with exactly two factors: 1 and itself. The first primes are 2, 3, 5, 7, 11 and 13. The number 1 is not prime, and 2 is the only even prime number. Whole numbers greater than 1 that are not prime are composite.Number System · Grade 4, Grade 5, Grade 6Rational 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 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 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 4