Multiply Whole Numbers

Use multiplication notation

Suppose you were asked to count a pile of pennies arranged in $3$ rows with $8$ pennies in each row. Would you count them individually, or would you count a row and add that number $3$ times?

Multiplication is a way to represent repeated addition. Instead of adding $8$ three times, we can write a multiplication expression:

We call each number being multiplied a factor, and the result is the product. We read $3 \times 8$ as three times eight, and the result as the product of three and eight.

Several symbols represent multiplication: the times sign, a centered dot, and parentheses.

Example. Translate from math notation to words: (a) $7 \times 6$ (b) $12 \cdot 14$ (c) $6(13)$.

(a) Seven times six; the result is the product of seven and six. (b) Twelve times fourteen; the result is the product of twelve and fourteen. (c) Six times thirteen; the result is the product of six and thirteen.

Model multiplication of whole numbers

There are many ways to model multiplication. Instead of base-10 blocks, here we use counters — any objects used for counting. To model $3 \times 8$, start with a row of $8$ counters, then make $3$ rows of $8$:

Counting the counters, there are $24$ in all, so $3 \times 8 = 24$. If you look at the rows sideways, you’ll see $8$ rows of $3$ counters — the product is the same either way. We’ll return to that idea shortly.

Multiply whole numbers

To multiply without models, you need to know the one-digit multiplication facts fluently — make sure you do before proceeding, and memorize any you’re unsure of.

Multiplying by zero and by one follow simple patterns.

Multiplication Property of Zero: the product of any number and $0$ is $0$.

Identity Property of Multiplication: the product of any number and $1$ is the number; $1$ is called the multiplicative identity.

For example, $0 \cdot 11 = 0$, $(42)0 = 0$, $(11)1 = 11$, and $1 \cdot 42 = 42$.

Just as with addition, the order of the factors does not matter: $4 \cdot 7 = 28$ and $7 \cdot 4 = 28$; $8 \cdot 9 = 72$ and $9 \cdot 8 = 72$. This is the Commutative Property of Multiplication: changing the order of the factors does not change their product.

To multiply numbers with more than one digit, write the numbers vertically in columns, as with addition and subtraction. To multiply $27$ by $3$: first multiply $3 \times 7 = 21$ — write the $1$ in the ones place of the product and carry the $2$ tens by writing $2$ above the tens place. Then multiply $3 \times 2 = 6$ and add the carried $2$: $6 + 2 = 8$. The product is $81$.

To multiply two whole numbers:

1. Write the numbers so each place value lines up vertically (with different numbers of digits, put the smaller number on the bottom).
2. Multiply the digits in each place value, working right to left, starting with the ones place of the bottom number. If a product in a place value is more than $9$, carry to the next place value. Write the partial products, lining up the digits in the place values with the numbers above. Repeat for the tens place of the bottom number, the hundreds place, and so on, inserting a zero as a placeholder with each additional partial product.
3. Add the partial products.

Example. Multiply $62(87)$. Multiplying $7$ by $62$ gives the first partial product, $434$ ($7 \cdot 2 = 14$, write the $4$ and carry the $1$; $7 \cdot 6 = 42$, plus the carried $1$ is $43$). Next multiply the $8$ — which is $8$ tens — by $62$: write a $0$ placeholder in the ones place, then $8 \cdot 2 = 16$ and $8 \cdot 6 = 48$ plus the carry gives the second partial product, $4{,}960$. Adding the partial products, $434 + 4{,}960 = 5{,}394$, so $62(87) = 5{,}394$.

Example. Multiply $(354)(438)$. There are three partial products — from the $8$ ones, the $3$ tens, and the $4$ hundreds: $2{,}832 + 10{,}620 + 141{,}600 = 155{,}052$.

When a factor ends in zeros, a pattern appears: $47 \cdot 10 = 470$ and $47 \cdot 100 = 4{,}700$. Since $10$ has one zero, we put one zero after $47$; since $100$ has two zeros, we put two. If we multiplied $47$ times $10{,}000$, which has four zeros, the product would be $470{,}000$.

When there are three or more factors, multiply the first two, then multiply their product by the next factor. To find $8 \cdot 3 \cdot 2$, first multiply $8 \cdot 3 = 24$, then multiply $24 \cdot 2 = 48$.

Translate word phrases to math notation

Earlier we translated math notation into words; now we reverse the process. Some of the words that indicate multiplication:

Example. Translate and simplify: the product of $12$ and $27$. The word product tells us to multiply: $12 \cdot 27$, which is $324$.

Example. Translate and simplify: twice two hundred eleven. The word twice tells us to multiply by $2$: $2(211)$, which is $422$.

Multiply whole numbers in applications

We use the same strategy as before: determine what we are looking for, write a phrase that gives the information to find it, translate to math notation, simplify, and answer with a complete sentence.

Example. Humberto bought $4$ sheets of stamps. Each sheet had $20$ stamps. How many stamps did Humberto buy? The total is the product of $4$ and $20$, which is $4 \cdot 20 = 80$. Humberto bought $80$ stamps.

Example. When Rena cooks rice, she uses twice as much water as rice. How much water does she need for $4$ cups of rice? Twice as much as $4$ cups translates to $2 \cdot 4 = 8$. Rena needs $8$ cups of water.

Example. Van is planning a patio with $8$ rows of tiles and $14$ tiles in each row. How many tiles does he need? The product of $8$ and $14$ is $8 \cdot 14 = 112$. Van needs $112$ tiles.

If we want to know the size of a wall to be painted or a floor to be carpeted, we find its area — a measure of the amount of surface covered by a shape, measured in square units such as square inches, square feet, or square centimeters. For a rectangle, the area is the product of the length and the width. A rug $2$ feet by $3$ feet is made of $6$ squares that are each $1$ square foot, so its area is $2 \cdot 3 = 6$ square feet.

Example. Jen’s kitchen ceiling is a rectangle $9$ feet long by $12$ feet wide. Its area is the product of $9$ and $12$: $9 \cdot 12 = 108$. The area of the ceiling is $108$ square feet.

Key terms

factor — a number being multiplied. product — the result of multiplying numbers. multiplicative identity — the number $1$; the product of any number and $1$ is the number. Multiplication Property of Zero — the product of any number and $0$ is $0$. Commutative Property of Multiplication — changing the order of the factors does not change their product. partial product — the product of one digit of a factor and the other factor, written when multiplying multi-digit numbers. area — a measure of the surface covered by a shape, in square units; for a rectangle, the product of length and width.

Practice

Use multiplication notation

Model multiplication of whole numbers

Multiply whole numbers

Translate word phrases to math notation

Multiply whole numbers in applications

This section is adapted from Prealgebra 2e, Section 1.4: Multiply Whole Numbers by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: condensed prose, redrew the counters models as accessible inline graphics and summarized the multiplication-facts table and worked columns in prose, converted practice problems (“Try Its”) into interactive exercises with instant feedback, and adapted selected end-of-section exercises into the interactive Practice block, with each multipart exercise expanded into one question per part.