The zero property of multiplication says that any number multiplied by 0 equals 0: a × 0 = 0 and 0 × a = 0. For example, 58 × 0 = 0. If even one factor in a product is 0, the whole product is 0. It explains why 5 ÷ 0 has no answer.
In Spanish
propiedad del cero en la multiplicación
Also called
multiplication property of zero, zero property, multiplicative property of zero
However you picture it, groups of nothing or no groups at all, a product with a factor of 0 is 0.
Formal definition and notation
For every real numbera, a · 0 = 0 · a = 0. It follows from the distributive and identity properties: a · 0 = a(0 + 0) = a · 0 + a · 0, and subtracting a · 0 from both sides gives a · 0 = 0. The converse is the zero product property: if ab = 0, then a = 0 or b = 0.
Identity: a + 0 = a and a × 1 = a, the number stays. Zero property: a × 0 = 0, the number is lost.
Zero Property of Multiplicationvs.Zero product property
The zero product property goes the other way: if ab = 0, then a = 0 or b = 0. It solves equations like x(x − 4) = 0.
Quick check: 3 questions
Question 1 of 3
What is 0 × 813?
Show answer and explanation
Answer: B. Any number times 0 is 0. 813 confuses this with adding 0, 1 is a guess from the identity property, and 8130 appends a zero digit.
Question 2 of 3
Evaluate 5 × (6 − 6) + 2.
Show answer and explanation
Answer: D.6 − 6 = 0, so 5 × 0 = 0 and 0 + 2 = 2. 32 multiplies 5 × 6 and adds 2, 7 adds 5 and 2, and 0 forgets the final + 2.
Question 3 of 3
Which statement is true?
Show answer and explanation
Answer: D.0 ÷ 9 = 0 because 9 × 0 = 0. 9 ÷ 0 is undefined, 9 × 0 is 0, and 0 ÷ 9 is not 9.
Zero Property of Multiplication: frequently asked questions
Why is anything times zero equal to zero?
Multiplication counts equal groups. 7 × 0 means 7 groups with nothing in each, which is nothing. 0 × 7 means zero groups of 7, which is also nothing. As repeated addition, 7 × 0 = 0 + 0 + 0 + 0 + 0 + 0 + 0 = 0. The same holds for fractions, decimals and negatives.
Is the zero property the same as the zero product property?
They are related but not the same. The zero property of multiplication says a × 0 = 0. The zero product property says the reverse: if a product equals 0, at least one factor is 0. Algebra students use the second to solve equations such as (x − 2)(x + 7) = 0, giving x = 2 or x = −7.
What about zero divided by a number?
Zero divided by any nonzero number is 0: 0 ÷ 4 = 0, because 4 × 0 = 0. A number divided by zero is undefined: 4 ÷ 0 would need a number that gives 4 when multiplied by 0, and the zero property says every such product is 0.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 3.OA.B.5, CCSS 3.OA.D.9
Cite this entry
Effortless Math Team. “Zero Property of Multiplication.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/zero-property-of-multiplication/
Flashcard
TermZero Property of Multiplicationa × 0 = ?
DefinitionAny number times 0 is 0: 58 × 0 = 0. One zero factor makes the whole product 0. But 5 ÷ 0 is undefined.
Real 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 1SideA 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 2UndefinedUndefined 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 1NumberA 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 2Repeated AdditionRepeated addition is adding the same number again and again. It is one way to understand multiplication: 3 + 3 + 3 + 3 = 4 × 3 = 12. The number of times you add is one factor, and the number being added is the other. It only works when all the addends are equal.Operations · Grade 2, Grade 3Identity PropertyThe identity property says a number keeps its value when you add 0 or multiply by 1: a + 0 = a and a × 1 = a. So 0 is the additive identity and 1 is the multiplicative identity. For example, 15 + 0 = 15 and 15 × 1 = 15. It is how equivalent fractions work.Operations · Grade 1, Grade 3, Grade 6EvaluateEvaluate means to find the single number value of an expression. Replace each variable with the number you are given, then calculate using the order of operations. For example, 2x + 7 evaluated at x = 3 is 2(3) + 7 = 13. The result is one number, not a new expression.Algebra · Grade 6, Grade 7, Algebra 1DivisionDivision 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 5MultiplicationMultiplication 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 4ZeroZero (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 5Order 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 7FactoringFactoring rewrites a number or expression as a product of factors. It is the reverse of multiplying or distributing: 12 = 3 × 4, 6x + 9 = 3(2x + 3) and x2 + 5x + 6 = (x + 2)(x + 3). Factoring changes the form, not the value, so multiplying the factors back gives the original.Algebra · Grade 6, Grade 7, Grade 8Quadratic EquationA quadratic equation is an equation that can be written in the form ax2 + bx + c = 0, where a is not 0. Its graph is a parabola, and it may have zero, one, or two real solutions. You can solve it by factoring, by taking square roots, by completing the square, or with the quadratic formula.Algebra · Algebra 1, Algebra 2Commutative PropertyThe commutative property says that changing the order of the numbers does not change the result of adding or multiplying them: a + b = b + a and a × b = b × a. For example, 4 + 9 = 9 + 4 and 3 × 5 = 5 × 3. Subtraction and division are not commutative.Operations · Grade 1, Grade 3, Grade 6Associative PropertyThe associative property says that when you add or multiply three or more numbers, the way you group them does not change the result: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c). It lets you pick the easiest pair first. It does not work for subtraction or division.Operations · Grade 1, Grade 3, Grade 6