Multiplication 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.
Multiplication counts equal groups quickly. An array shows why the order of the factors does not matter, and the area model shows how to split a big product into easy ones.
Formal definition and notation
For whole numbers, a × b is the total number of objects in a equal groups of b, which equals adding b to itself a times. It is also the number of unit squares in an a-by-b rectangle (the area model). Multiplication extends to fractions, decimals and negative numbers and keeps the same properties: it is commutative, associative and distributive over addition, 1 is its identity (a × 1 = a), and any number times 0 is 0.
Symbol
Read it as
Meaning
3 × 8
3 times 8
the usual sign in arithmetic
3 · 8
3 times 8
a raised dot, used in algebra so the sign is not confused with the letter x
3(8), 3x
3 times 8, 3 times x
parentheses or a number written next to a variable also mean multiply
Associative property: (2 × 5) × 13 = 2 × (5 × 13) = 130. Grouping a 2 with a 5 makes 10 and speeds up mental math.
Identity and zero: a × 1 = a and a × 0 = 0, so 358 × 0 = 0.
Distributive property: break a factor apart, 7 × 14 = 7 × 10 + 7 × 4 = 98. Long multiplication is built on this.
Signs: a positive times a negative is negative, and a negative times a negative is positive: (−4)(−5) = 20.
Multiplying a whole number by 10 puts a 0 on the end: 46 × 10 = 460, because each digit moves one place left.
Parts of a multiplication fact
Parts of a multiplication fact
factor × factor = product
factor
a number being multiplied (the number of groups, or the size of each group)
product
the result, the total
Check a product with division: product ÷ one factor = the other factor. Since 4 × 6 = 24, 24 ÷ 4 = 6.
Try it: rows times columns
Interactive
Drag the corner or use the arrow keys to change the rows and columns. The number of unit squares is the product of the two side lengths. Try 6 by 4, then 4 by 6.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Multiplication examples, step by step
Example 1: Use a fact you know
Warm-up
Find 8 × 7 by breaking 7 into 5 + 2.
8 × 5 = 40.
8 × 2 = 16.
Add the parts: 40 + 16 = 56.
Answer:8 × 7 = 56.
Example 2: Two-digit by two-digit
Core
Multiply 36 × 24.
Split 24 into tens and ones: 36 × 20 and 36 × 4.
36 × 20 = 720.
36 × 4 = 144.
Add the partial products: 720 + 144 = 864.
Answer:36 × 24 = 864.
Example 3: A multi-step word problem
Test-style
A theater has 28 rows with 15 seats in each row. Every seat is sold for 12 dollars. How much money does a sold-out show bring in?
Seats: 28 × 15 = 420.
Money: 420 × 12.
420 × 10 = 4200 and 420 × 2 = 840.
4200 + 840 = 5040.
Answer: A sold-out show brings in 5,040 dollars.
Common mistakes
✗ MistakeLeaving out the placeholder zero: 36 × 24 as 144 + 72 = 216.
✓ InsteadThe second row is 36 × 20 = 720, so 144 + 720 = 864.
The 2 in 24 stands for 2 tens, so its partial product must be shifted one place left, which the zero does.
✗ Mistake9 × 0 = 9.
✓ Instead9 × 0 = 0.
Nine groups of zero objects contain nothing. Adding 0 leaves a number the same; multiplying by 0 always gives 0.
✗ Mistake(−4) × (−5) = −20.
✓ Instead(−4) × (−5) = 20.
Two negative factors give a positive product. Only one negative factor gives a negative product.
✗ Mistake7 × 14 = 7 × 10 + 4 = 74.
✓ Instead7 × 14 = 7 × 10 + 7 × 4 = 70 + 28 = 98.
When you split a factor, the other factor must multiply every part.
Multiplication is the operation; the product is its answer. In 5 × 9 = 45, the multiplication gives the product 45.
Quick check: 3 questions
Question 1 of 3
Which sum means the same as 6 × 4?
Show answer and explanation
Answer: B. 6 groups of 4 is 4 added 6 times, which is 24. 6 + 4 adds the factors, 6 + 6 + 6 uses only 3 groups, and 64 just writes the digits side by side.
Question 2 of 3
What is 25 × 12?
Show answer and explanation
Answer: A.25 × 12 = 25 × 10 + 25 × 2 = 250 + 50 = 300. 37 adds the factors, 270 makes a regrouping error, and 3,000 adds an extra zero.
Question 3 of 3
What is (−3) × 7 × (−2)?
Show answer and explanation
Answer: D.(−3) × 7 = −21, and −21 × (−2) = 42. Two negative factors make a positive product. −42 ignores one sign, −21 stops early, and 2 adds the numbers.
Multiplication: frequently asked questions
Why is multiplication called repeated addition?
For whole numbers, multiplying is a shortcut for adding the same number again and again. 5 × 3 means five 3s: 3 + 3 + 3 + 3 + 3 = 15. The idea stops being literal for fractions and decimals, such as 12 × 8 = 4, where it means taking a part of a group instead.
What are the properties of multiplication?
There are five main ones. Commutative: order does not matter. Associative: grouping does not matter. Identity: multiplying by 1 changes nothing. Zero: multiplying by 0 gives 0. Distributive: a(b + c) = ab + ac, so 6 × 23 = 6 × 20 + 6 × 3 = 138. Together they make mental math and algebra work.
How do you write multiplication in algebra?
Algebra avoids the × sign because it looks like the letter x. Instead you write a raised dot, 4 · 5, parentheses, 4(5) or (4)(5), or put a number right next to a variable: 4x means 4 times x, and ab means a times b.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 3.OA.A.1, CCSS 3.OA.B.5, CCSS 3.OA.C.7
Cite this entry
Effortless Math Team. “Multiplication.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/multiplication/
Flashcard
TermMultiplication4 × 6
DefinitionCombining equal groups to find a total. 4 × 6 is 4 groups of 6: 6 + 6 + 6 + 6 = 24.
FactorA 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 6Whole 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 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 2SquareA square is a quadrilateral with four equal sides and four right angles. It is a special rectangle and a special rhombus at the same time. For a square with side s, the perimeter is 4s, the area is s2, and each diagonal is s√2.Geometry · Kindergarten, Grade 2, Grade 3AreaArea is the amount of surface a flat shape covers, measured in square units. It counts how many unit squares fit inside the shape, which is why the units are squared. A rectangle’s area is length times width, so a 5 by 3 rectangle has area 15 square units.Geometry · 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 7AdditionAddition 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 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 6Distributive PropertyThe distributive property says that multiplying a number by a sum gives the same result as multiplying it by each term and then adding: a(b + c) = ab + ac. It multiplies a factor by each term inside parentheses, so 3(x + 4) = 3x + 12. It works with subtraction too, and using it backward is factoring.Algebra · Grade 6, Grade 7, Grade 8ExponentAn exponent tells how many times to use a base as a factor. It is the small raised number to the upper right of the base: in 53, the base is 5 and the exponent is 3, so 53 = 5 × 5 × 5 = 125. Any nonzero number with an exponent of 0 equals 1.Algebra · Grade 6, Grade 7, Grade 8Repeated 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 3ProductThe product is the answer you get when you multiply two or more numbers. In 4 × 6 = 24, the product is 24, and 4 and 6 are the factors. In word problems, the words “product of” always mean multiply, so “the product of 3 and 7” is 21.Operations · Grade 3, Grade 4, Grade 5Skip CountingSkip counting is counting forward or backward by the same number each time instead of by 1, such as 2, 4, 6, 8 or 5, 10, 15, 20. Each jump adds the same amount, so skip counting by 5 six times reaches 6 × 5 = 30. It is a quick way to count groups and learn multiplication facts.Number System · Kindergarten, Grade 1, Grade 2Order 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 7MultipleA 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 6DivisionDivision 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 5