Division 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.
Division splits a total into equal groups. Whatever cannot fill another full group is the remainder.
Formal definition and notation
For numbers a and b with b ≠ 0, a ÷ b is the number q such that b × q = a; so division is multiplication by the reciprocal, a ÷ b = a × 1b. For whole numbers that do not divide evenly, division with remainder finds the whole-number quotient q and remainder r with a = bq + r and 0 ≤ r < b. Division by 0 is undefined, because no number times 0 gives a nonzero dividend.
A fraction 34 is a number, and it also means 3 ÷ 4. The fraction bar is another division sign.
Divisionvs.Divisor and dividend
The dividend is divided; the divisor does the dividing. In 45 ÷ 5, 45 is the dividend and 5 is the divisor.
Quick check: 3 questions
Question 1 of 3
What is 72 ÷ 9?
Show answer and explanation
Answer: B.9 × 8 = 72, so 72 ÷ 9 = 8. 7 and 9 are neighboring facts, and 63 subtracts instead of dividing.
Question 2 of 3
In 45 ÷ 5 = 9, which number is the divisor?
Show answer and explanation
Answer: A. The divisor is the number you divide by: 5. 45 is the dividend and 9 is the quotient.
Question 3 of 3
What is 8 ÷ 0?
Show answer and explanation
Answer: D. No number times 0 gives 8, so 8 ÷ 0 is undefined. 0 is the answer to 0 ÷ 8, not 8 ÷ 0.
Division: frequently asked questions
What are the parts of a division problem called?
In 20 ÷ 4 = 5, 20 is the dividend (the amount being divided), 4 is the divisor (the number you divide by) and 5 is the quotient (the answer). If the division does not come out even, the amount left over is the remainder: in 23 ÷ 4 = 5 R 3, the remainder is 3.
How are division and multiplication related?
They are inverse operations: one undoes the other. Every division fact matches a multiplication fact, so 42 ÷ 6 = 7 because 6 × 7 = 42. That is why knowing the times tables makes division fast, and why you can check any division by multiplying the quotient by the divisor and adding the remainder.
Why can you not divide by zero?
Dividing asks what number times the divisor gives the dividend. For 10 ÷ 0, no number times 0 is 10, because anything times 0 is 0. For 0 ÷ 0, every number works, so there is no single answer. Either way, division by zero has no defined value.
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 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 4ReciprocalThe reciprocal of a number is 1 divided by that number, so the two multiply to 1. For a fraction, swap the numerator and denominator: the reciprocal of 34 is 43, and the reciprocal of 5 is 15. Zero has no reciprocal. Dividing by a fraction is the same as multiplying by its reciprocal.Number System · Grade 5, Grade 6, Grade 7Whole 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 5RemainderA remainder is the amount left over when one whole number does not divide another evenly. In 23 ÷ 5, four groups of 5 make 20 and 3 are left, so 23 ÷ 5 = 4 R 3. The remainder is always smaller than the divisor. A remainder of 0 means the division is exact.Operations · Grade 4, Grade 5, Grade 6QuotientA 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 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 1DividendThe dividend is the number being divided in a division problem. In 56 ÷ 8 = 7, the dividend is 56, the divisor is 8 and the quotient is 7. It is the total you start with before splitting it into equal groups. In a fraction such as 568, the dividend is the numerator.Operations · Grade 3, Grade 4, Grade 5Long DivisionLong division is a step-by-step written method for dividing large numbers. You work through the dividend one digit at a time from the left, repeating four steps: divide, multiply, subtract, bring down. For 872 ÷ 4, the steps give the quotient 218 with a remainder of 0.Operations · Grade 4, Grade 5, Grade 6DivisorA divisor is the number you divide by. In 56 ÷ 8 = 7, the divisor is 8: it tells how many equal groups to make, or how big each group is. The divisor can never be 0. A divisor of a whole number is also one of its factors, because it divides that number with no remainder.Operations · Grade 3, Grade 4, Grade 5FractionA fraction is a number that names equal parts of a whole, written ab. The bottom number, the denominator, tells how many equal parts make one whole; the top number, the numerator, tells how many of those parts you have. So 34 means 3 of 4 equal parts. Every fraction is also a point on the number line.Number System · Grade 3, Grade 4, Grade 5Inverse OperationsInverse operations are operations that undo each other. Addition and subtraction are inverses: 9 + 6 − 6 = 9. Multiplication and division are inverses: 7 × 4 ÷ 4 = 7. You use them to check answers and to solve equations, undoing each step in reverse order until the variable is alone.Operations · Grade 1, Grade 3, Grade 6SubtractionSubtraction 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 3Order of OperationsThe order of operations is the agreed sequence for evaluating an expression: parentheses first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right. Without it the same expression would have several answers. Many students remember it as PEMDAS.Algebra · Grade 5, Grade 6, Grade 7DecimalA 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 6FactorA factor is a number or expression that divides another number or expression evenly, with no remainder. Factors multiply together to make a product: 4 × 6 = 24, so 4 and 6 are factors of 24. Every whole number greater than 1 has at least two factors, 1 and itself.Number System · Grade 4, Grade 5, Grade 6MultipleA multiple of a number is the result of multiplying that number by any whole number. The multiples of 4 are 4, 8, 12, 16, and so on, continuing without end. Every number is a multiple of itself and of 1. To test a number, divide: if the remainder is 0, it is a multiple.Number System · Grade 4, Grade 5, Grade 6