A 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.
In Spanish
entero positivo
Also called
counting number, natural number, positive whole number
Positive integers are the dots at 1, 2, 3 and onward. Zero, negative numbers and in-between values such as 2.5 are all left out.
Formal definition and notation
The positive integers are the elements of ℤ+ = {1, 2, 3, …}: the integersn with n > 0. Starting from 1, each one is the previous one plus 1, and there is no largest positive integer. The set is closed under addition and multiplication, but not under subtraction or division (2 − 5 and 2 ÷ 5 are not positive integers).
Symbol
Read it as
Meaning
ℤ+
Z plus
the set of positive integers
ℕ
N, the natural numbers
often the same set; some books also include 0
n ∈ ℤ+
n is a positive integer
n is one of 1, 2, 3, …
Key facts
The smallest positive integer is 1. There is no largest one.
0 is an integer but not positive, so it is not a positive integer.
Nonnegative integers are 0, 1, 2, …; they include 0. Positive integers do not. A test that switches between the two words is changing whether 0 counts.
Quick check: 3 questions
Question 1 of 3
What is the smallest positive integer?
Show answer and explanation
Answer: B. 1 is the smallest positive integer. 0 is not positive, 0.1 is not an integer, and −1 is negative.
Question 2 of 3
How many positive integers are factors of 18?
Show answer and explanation
Answer: D. 1, 2, 3, 6, 9 and 18: six factors. 3 counts only primes and their product, 5 leaves out 1 or 18, and 12 counts negative factors too.
Question 3 of 3
If n is a positive integer and n2 < 30, how many values can n take?
Show answer and explanation
Answer: A.1, 4, 9, 16, 25 are below 30, but 62 = 36 is not, so n = 1, 2, 3, 4, 5: five values. 6 would include 0 or 6, and 4 misses 5.
Positive Integer: frequently asked questions
Is 0 a positive integer?
No. Zero is an integer, but it is neither positive nor negative, so it is not a positive integer. The positive integers start at 1. If a problem wants to include 0, it will say nonnegative integer or whole number instead. This one difference changes many counting answers on tests.
Are positive integers the same as natural numbers?
Usually. Many school books define the natural numbers, or counting numbers, as 1, 2, 3 and so on, which is exactly the positive integers. Some books include 0 in the natural numbers. Positive integer has only one meaning, so tests use it to avoid confusion.
Is 2.0 a positive integer?
Yes, in value. 2.0 equals 2, which is a positive integer; the decimal point with a zero after it does not add a fractional part. But 2.5 is not an integer, because it lies between 2 and 3. Ask whether the number has a nonzero part after the decimal point.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 6.NS.C.5
Cite this entry
Effortless Math Team. “Positive Integer.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/positive-integer/
Flashcard
TermPositive Integerℤ+
DefinitionWhole numbers greater than zero: 1, 2, 3, and so on. Not 0, not negatives, not decimals like 2.5.
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 8AdditionAddition 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 5Whole 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 5DifferenceA 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 7InequalityAn inequality compares values using symbols such as <, >, ≤ or ≥. Instead of saying two expressions are equal, it says one is greater, less, or at least/at most another, as in x > 5. An inequality with a variable usually has infinitely many solutions, shown as a ray on a number line.Algebra · Grade 6, Grade 7, Grade 8NumberA 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 2NonnegativeNonnegative 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 1Natural 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 7Counting 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 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 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 5Even 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 4Prime 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 6Consecutive 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 1