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Binary Multiplication

Q: What happens when you multiply a binary number by 10 (which is 2 in decimal)?

Multiplying a binary number by $10_2$ is just like multiplying a decimal number by 10. You simply add a $0$ to the end of the number. For example, $1101_2 \times 10_2 = 11010_2$.

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Ever wonder how computers do math so fast? It all starts with binary! This lesson will teach you binary multiplication, a simple but powerful skill. It's just like regular multiplication, but you only have to worry about two numbers: $0$ and $1$ . Let's get started!

Binary Multiplication — an original Algebra911 reference diagram defining binary multiplication and a worked example. — Binary Multiplication: A Step-by-Step Guide for Students

What Is Binary Multiplication?

Binary multiplication is the method used to multiply two numbers that are in the binary, or base-2, number system. This system is the fundamental language of computers, using only two digits: $0$ (off) and $1$ (on) to represent all information. While it might sound complicated, the process of multiplying binary numbers is surprisingly similar to the long multiplication you already know with decimal numbers (base-10).

Think about how you multiply $12 \times 3$ . You multiply $2 \times 3$ , then $10 \times 3$ , and add the results. Binary multiplication follows the same core idea of multiplying, shifting positions, and then adding everything up. The big difference, and what makes it much simpler in some ways, is that you only ever multiply by $0$ or $1$ . You don't need to memorize a large times table!

Understanding this process is like getting a peek under the hood of a computer. Every time a program performs a calculation, from a simple calculator app to a complex video game, it's breaking the problem down into operations like binary multiplication. Mastering this skill helps build a strong foundation for understanding how digital technology works.

What Are the Four Basic Rules of Binary Multiplication?

The best part about binary multiplication is its simplicity. In the decimal system, you have to remember $100$ different multiplication facts (from $0 \times 0$ to $9 \times 9$ ). In binary, there are only four rules to learn, and they are incredibly intuitive. These four rules are the entire foundation for every binary multiplication problem you will ever solve.

Here are the only rules you need to know:

$0 \times 0 = 0$
$0 \times 1 = 0$
$1 \times 0 = 0$
$1 \times 1 = 1$

That's it! Notice a pattern? The rules are identical to the rules for multiplying by $0$ and $1$ in our familiar decimal system. If a $0$ is involved in the multiplication, the answer is $0$ . The only way to get a $1$ is by multiplying $1 \times 1$ . This simplicity removes the need for complex memorization and lets you focus on the process of arranging and adding the numbers correctly.

How Do You Perform Binary Multiplication? A Step-by-Step Guide

Performing binary multiplication is a systematic process that combines the four simple rules of multiplication with the rules of binary addition. If you can do long multiplication with regular numbers, you can definitely do it with binary. Let's break it down into four clear steps.

Set Up the Problem: Write the two binary numbers one on top of the other, just like you would for a decimal multiplication problem. It's helpful to align them to the right. The top number is called the multiplicand, and the bottom number is the multiplier.
Multiply to Find Partial Products: Starting with the rightmost digit of the multiplier (the bottom number), multiply it by the entire multiplicand (the top number). Write this result, called a partial product, below the line. Since you are only multiplying by $0$ or $1$ , this step is easy:
- If the digit is $1$ , the partial product is just a copy of the multiplicand.
- If the digit is $0$ , the partial product is a row of zeros.
Shift and Repeat: Move one digit to the left in the multiplier. Multiply this digit by the multiplicand to get your second partial product. Write this new partial product on a new line, but shifted one position to the left. You can add a placeholder $0$ on the right to help with alignment. Repeat this for every digit in the multiplier, shifting one additional space to the left each time.
Add the Partial Products: The final step is to add all of your partial products together using the rules of binary addition. Remember these key addition facts:
- $0 + 0 = 0$
- $0 + 1 = 1$
- $1 + 0 = 1$
- $1 + 1 = 10$ (This means you write down $0$ and carry over a $1$ to the next column).
- $1 + 1 + 1 = 11$ (This means you write down $1$ and carry over a $1$ to the next column).
The sum you get is the final answer to your multiplication problem.

Worked Example: Multiplying Two Simple Binary Numbers

Example 1

Let's multiply $11_{2}$ by $10_{2}$ .

Step 0: Decimal Check (Optional)
First, let's convert these numbers to decimal to predict our answer. $11_{2}$ is $2^{1} + 2^{0} = 2 + 1 = 3$ . The number $10_{2}$ is $2^{1} + 0 = 2$ . So, we are calculating $3 \times 2$ , and our answer should be $6$ .

Step 1: Set Up
Write the problem vertically.

\begin{array}{ccc} 1 & 1_{2} \\ \times & 1 & 0_{2} \end{array}

Step 2: Multiply by the First Digit
Multiply the top number ( $11_{2}$ ) by the rightmost digit of the bottom number, which is $0$ .
$11_{2} \times 0 = 00_{2}$ . We write this as our first partial product.

\begin{array}{ccc} 1 & 1_{2} \\ \times & 1 & 0_{2} \\ 0 & 0 \end{array}

Step 3: Shift and Multiply by the Second Digit
Now, multiply the top number ( $11_{2}$ ) by the next digit of the bottom number, which is $1$ .
$11_{2} \times 1 = 11_{2}$ . We write this second partial product on a new line, shifted one place to the left.

\begin{array}{ccc} 1 & 1_{2} \\ \times & 1 & 0_{2} \\ 0 & 0 \\ 1 & 1 \end{array}

Step 4: Add the Partial Products
Finally, add the two partial products together using binary addition.

\begin{array}{cccc} 1 & 1_{2} \\ \times & 1 & 0_{2} \\ 0 & 0 \\ + & 1 & 1 & 0 \\ 1 & 1 & 0_{2} \end{array}

Our final answer is $110_{2}$ . Let's check this against our decimal prediction. $110_{2}$ is $2^{2} + 2^{1} + 0 = 4 + 2 + 0 = 6$ . It matches! Our calculation is correct.

Worked Example: A Problem with Binary Addition and Carrying

Example 2

Let's solve a slightly harder problem: $101_{2} \times 11_{2}$ .

Step 0: Decimal Check (Optional)
Let's find the decimal equivalents. $101_{2}$ is $2^{2} + 0 + 2^{0} = 4 + 0 + 1 = 5$ . The number $11_{2}$ is $2^{1} + 2^{0} = 2 + 1 = 3$ . We are calculating $5 \times 3$ , so our answer should be $15$ .

Step 1: Set Up
Write the numbers vertically.

\begin{array}{cccc} 1 & 0 & 1_{2} \\ \times & 1 & 1_{2} \end{array}

Step 2: First Partial Product
Multiply the top number ( $101_{2}$ ) by the rightmost digit of the bottom number, which is $1$ .
$101_{2} \times 1 = 101_{2}$ .

\begin{array}{cccc} 1 & 0 & 1_{2} \\ \times & 1 & 1_{2} \\ 1 & 0 & 1 \end{array}

Step 3: Second Partial Product (Shifted)
Multiply the top number ( $101_{2}$ ) by the next digit of the bottom number, which is also $1$ .
$101_{2} \times 1 = 101_{2}$ . Write this result on a new line, shifted one place to the left.

\begin{array}{cccc} 1 & 0 & 1_{2} \\ \times & 1 & 1_{2} \\ 1 & 0 & 1 \\ 1 & 0 & 1 \end{array}

Step 4: Add the Partial Products
Now we add the partial products. This is where carrying comes in.

\begin{array}{ccccc} 1 \\ 1 & 0 & 1_{2} \\ \times & 1 & 1_{2} \\ 1 & 0 & 1 \\ + & 1 & 0 & 1 & 0 \\ 1 & 1 & 1 & 1_{2} \end{array}

Let's go column by column from right to left:

Column 1 (ones place): $1 + 0 = 1$ .
Column 2 (twos place): $0 + 1 = 1$ .
Column 3 (fours place): $1 + 0 = 1$ .
Column 4 (eights place): $1$ (from the bottom partial product) = $1$ .

Our final answer is $1111_{2}$ . Let's check it. $1111_{2}$ is $2^{3} + 2^{2} + 2^{1} + 2^{0} = 8 + 4 + 2 + 1 = 15$ . Perfect! The answer is correct.

Worked Example: Tackling Longer Binary Numbers

Example 3

Let's try a problem with more digits to see how the process scales: $1101_{2} \times 101_{2}$ .

Step 0: Decimal Check (Optional)
$1101_{2}$ is $8 + 4 + 0 + 1 = 13$ . $101_{2}$ is $4 + 0 + 1 = 5$ . The decimal product is $13 \times 5 = 65$ . Our binary answer should equal $65$ .

Step 1: Set Up

\begin{array}{ccccc} 1 & 1 & 0 & 1_{2} \\ \times & 1 & 0 & 1_{2} \end{array}

Step 2: Generate All Partial Products
We have three digits in our multiplier ( $101_{2}$ ), so we will have three partial products.

Multiply $1101_{2}$ by the rightmost $1$ : $1101_{2}$ .
Multiply $1101_{2}$ by the middle $0$ : $0000_{2}$ . This is shifted one place to the left.
Multiply $1101_{2}$ by the leftmost $1$ : $1101_{2}$ . This is shifted two places to the left.

\begin{array}{ccccccc} 1 & 1 & 0 & 1_{2} \\ \times & 1 & 0 & 1_{2} \\ 1 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ + & 1 & 1 & 0 & 1 \end{array}

Step 3: Add the Partial Products
Now we sum the three rows. This addition will involve multiple carries.

\begin{array}{cccccccc} 1 & 1 & 1 \\ 1 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ + & 1 & 1 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 1_{2} \end{array}

Let's trace the addition from right to left:

Column 1: $1$ . Write $1$ .
Column 2: $0 + 0 = 0$ . Write $0$ .
Column 3: $1 + 0 + 1 = 10_{2}$ . Write $0$ , carry $1$ .
Column 4: $1 + 0 + 0 + carry 1 = 10_{2}$ . Write $0$ , carry $1$ .
Column 5: $1 + 1 + carry 1 = 11_{2}$ . Write $1$ , carry $1$ . Wait, I made a mistake in the diagram above, let's re-calculate carefully here.

Let's re-do the addition cleanly:

\begin{array}{cl} 1101 \\ 00000 \\ + & 110100 \end{array}

Column-by-column addition:

\begin{array}{cccccccc} 1 & 0 & 0 & 0 & (carries) \\ 1 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ + & 1 & 1 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 1_{2} \end{array}

Col 1: $1$ -> $1$
Col 2: $0 + 0$ -> $0$
Col 3: $1 + 0 + 1$ -> $10_{2}$ . Write $0$ , carry $1$ .
Col 4: $1 + 0 + 0 + carry 1$ -> $10_{2}$ . Write $0$ , carry $1$ .
Col 5: $0 + 1 + carry 1$ -> $10_{2}$ . Write $0$ , carry $1$ .
Col 6: $1 + carry 1$ -> $10_{2}$ . Write $0$ , carry $1$ .
Col 7: $carry 1$ -> $1$ .

The final answer is $1000001_{2}$ . Let's check our decimal prediction. We need to see if $1000001_{2}$ equals $65$ . The places are $64, 32, 16, 8, 4, 2, 1$ . So, $1 \times 64 + 1 \times 1 = 65$ . It matches! The process works perfectly even for larger numbers.

What Common Mistakes Should I Avoid?

Binary multiplication is straightforward, but it's easy to make small errors that can lead to the wrong answer. Being aware of these common pitfalls is the best way to avoid them.

Forgetting to Shift Partial Products: This is the most common mistake. Each new partial product must be shifted one additional place to the left. Forgetting to do this is like multiplying by $111$ instead of $123$ in decimal; you are not accounting for the place value of the digits. Always double-check your alignment before you start adding.
Errors in Binary Addition: The final step of adding the partial products is where many students slip up. It's crucial to be confident with binary addition, especially the carry rule: $1 + 1 = 10_{2}$ . When adding a column with multiple $1$ s, take your time. For example, $1 + 1 + 1 = 11_{2}$ (write $1$ , carry $1$ ). Rushing this step will almost always lead to an incorrect result.
Misaligning Columns: Write your work neatly! If your columns are crooked, it's very easy to add a number from the 'fours' column into the 'eights' column by mistake. Using grid paper can be a huge help in keeping everything perfectly aligned.
Ignoring the '0' Multiplier: When you multiply by a $0$ in the multiplier, the partial product is a row of zeros. Some people skip writing this row to save time. This is okay, but you MUST still shift the *next* partial product correctly. For example, when multiplying by $101_{2}$ , the partial product for the final $1$ must be shifted two places to the left, not just one. Writing the row of zeros helps prevent this shifting error.

Quick Reference: Rules and Steps

Here is a quick summary of the key rules and the process for binary multiplication. Use this as a handy reference while you practice.

The Process in 4 Steps

Setup: Write the numbers vertically, aligned to the right.
Multiply: Create partial products by multiplying the top number by each digit of the bottom number, from right to left.
Shift: Shift each new partial product one place further to the left.
Add: Sum all the partial products using binary addition rules.

Key Rules Tables

Use this table to remember the basic math facts for binary.

Binary Multiplication Rules

Expression	Result
$0 \times 0$	$0$
$0 \times 1$	$0$
$1 \times 0$	$0$
$1 \times 1$	$1$

Binary Addition Rules

Expression	Result	Note
$0 + 0$	$0$
$0 + 1$	$1$
$1 + 0$	$1$
$1 + 1$	$10$	Write $0$ , carry $1$
$1 + 1 + 1$	$11$	Write $1$ , carry $1$

Frequently Asked Questions

Why is binary multiplication important to learn?

Binary multiplication is a fundamental operation for all digital computers and electronics. Learning it helps you understand how computers perform calculations and process information at the most basic level, which is a key concept in computer science.

Is binary multiplication easier than decimal multiplication?

The multiplication part is much easier because the rules are simpler (you only multiply by 0 or 1). However, the final addition step can be trickier for beginners because you have to be careful with the binary carrying rules.

What happens when you multiply a binary number by 10 (which is 2 in decimal)?

Multiplying a binary number by $10_{2}$ is just like multiplying a decimal number by 10. You simply add a $0$ to the end of the number. For example, $1101_{2} \times 10_{2} = 11010_{2}$ .

How can I check my binary multiplication answer?

The best way to check your work is to convert the binary numbers into decimal numbers, perform the multiplication in decimal, and then convert the decimal result back into binary. If your binary answer matches the converted decimal answer, you know you did it correctly.

Do I have to know binary addition to do binary multiplication?

Yes, absolutely. The final and most important step of binary multiplication is adding all the partial products together. Without a solid understanding of binary addition, especially how to handle carrying, you won't be able to get the correct final answer.

What is a 'partial product'?

A partial product is the result you get when you multiply the top number (the multiplicand) by a single digit of the bottom number (the multiplier). In a problem like $110_{2} \times 11_{2}$ , you will have two partial products that you then need to add together.

Can you multiply binary numbers that aren't whole numbers?

Yes, you can multiply binary numbers with a 'binary point' (the equivalent of a decimal point). The process is very similar to multiplying decimals: you multiply the numbers as if the point isn't there, then count the total number of places after the point in the original numbers to place the point in the final answer.