Binary Multiplication: Signed & Unsigned Rules, Table & Examples

Binary multiplication works just like normal multiplication, but instead of using numbers from 0 to 9 (like in decimal multiplication), it only uses 0s and 1s, since binary is a base-2 number system. In binary multiplication, we follow the same steps as in decimal multiplication—like multiplying, shifting, and adding the results. The main difference is that the multiplication rules are much simpler. When you multiply any binary number:

• 0 × 0 = 0
• 0 × 1 = 0
• 1 × 0 = 0
• 1 × 1 = 1

So, the process involves just placing the correct values and adding the binary results. Binary multiplication is commonly used in computer systems, digital electronics, and programming, where all data is stored and processed in binary form. This makes binary operations, including multiplication, a very important part of computer science and digital communication.

What is Binary Multiplication?

Binary multiplication is one of the main operations we do with binary numbers, just like addition, subtraction, and division. It works in a similar way to how we multiply regular decimal numbers, but instead of using digits from 0 to 9, binary uses only two digits: 0 and 1. The process includes multiplying digits and then adding the results, just like in regular multiplication. It is often used in computers and digital devices, where all numbers and data are represented in binary form. Because of this, binary multiplication is a key part of how machines process information.

Binary Multiplication Table

Since binary numbers make use of only two digits that is 0 and 1, we get to multiply only these binary numbers while performing multiplication. The multiplication table for binary numbers is as follows:

Binary Numbers	Multiplication Value
	0
	0
	0
	1

Binary Multiplication Rules

In binary multiplication, we have a multiplier and a multiplicand. The basic rules for the multiplication of binary numbers are:

Multiplicand	Multiplier	Product
0	0	=0
0	1	=0
1	0	=0
1	1	=1

Understanding Binary Operations: How Multiplication is Different

Binary numbers use only two digits: 0 and 1, and just like in decimal math, we can add, subtract, multiply, and divide them. But the rules for each operation are a little different. Here's a simple table to show how binary addition, subtraction, and division work, so you can see how multiplication stands out from the rest.

Addition	Subtraction	Division
0 + 0 = 0	0 – 0 = 0	0 ÷ 0 = 0*
0 + 1 = 1	0 – 1 = 1 (borrowed 1)	0 ÷ 1 = 0
1 + 0 = 1	1 – 0 = 1	1 ÷ 1 = 1
1 + 1 = 0 (carry 1)	1 – 1 = 0

Note: In binary division, 0 ÷ 0 is usually considered undefined, but in some cases may be treated as 0 for simplicity.

While multiplication follows its own set of easy rules (0 × anything = 0 and 1 × 1 = 1), it's important to understand how these operations differ to avoid confusion when solving binary math problems.

Multiplication of Binary Numbers

As binary numbers comprise of only two values i.e. 0 and 1, the process of multiplication of these numbers becomes easier as compared to decimal numbers. The steps involved in multiplying binary numbers are given below:

Example: Multiply 11101 by 1001.

Step 1: Write the multiplicand 11101 and the multiplier 1001 one below the other in proper columns.

Step 2: Start the multiplication process from the extreme right digit of the multiplier which is 1 in this case, with all the digits of the multiplicand.

Step 3: Add the placeholder ‘X’ before starting the multiplication with the next digit of the multiplier in the next row.

Step 4: Repeat the same procedure till the leftmost digit in the multiplier is multiplied by all the digits in the multiplicand.

Step 5: The product obtained in each row is called the partial product. Finally, all the partial products are added using the rules for binary addition.

(Rules for binary addition are: 0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, 1 + 1 = 0, 1 carry).

Let us look at the actual multiplication:

1	1	1	0	1
x	1	0	0	1
1	1	1	0	1
0	0	0	0	0	x
0	0	0	0	0	x	x
+	1	1	1	0	1	x	x	x
1	0	0	0	0	0	1	0	1

Therefore, we can say that the product of 11101 and 1001 is 100000101. We can also check our result by changing binary to decimal numbers. The decimal equivalent of 11101 is 29 and that of 1001 is 9. And the product of 29 and 9 is 261 which is written as 100000101 in binary notation.

Multiplication of Binary Numbers with Decimal Points

Multiplying binary numbers with decimal points is an easy procedure. It is similar to multiplying two binary numbers without decimals. The only difference is, after performing the entire multiplication we need to place the decimal point by counting the decimal places in the multiplier and the multiplicand.

Let us understand this with an example:

Example: Multiply: 1011.01 and 110.1

Solution: We will perform simple binary multiplication and insert a decimal point in the final answer:

1	0	1	1	0	1
x	1	1	0	1
1	0	1	1	0	1
0	0	0	0	0	0	x
1	0	1	1	0	1	x	x
+	1	0	1	1	0	1	x	x	x
1	0	0	1	0	0	1	0	0	1

The answer obtained by multiplying 101101 and 1101 is 1001001001. Now as we have to multiply 1011.01 and 110.1, the final answer is 1001001.001.

Signed Binary Multiplication

Signed Binary Multiplication is also known as 2’s complement multiplication. We can perform this multiplication by simply multiplying the magnitudes of the two numbers and then extending it to the original sign bit of the number.

It is to be noted that unlike addition when we multiply an n-bit number with an m-bit number, it results in an n+m-bit number.

Let us understand this signed multiplication using an example:

Example: Multiply -5 and 7 in signed binary multiplication.

Solution: We know that in binary numbers -5 is written as 1011 and 7 is written as 0111.

In order to perform signed multiplication, we simply need to perform binary multiplication using simple rules, that is , , and .

After final multiplication, we have to extend each row to the number of sign bits, in this case, 8-bit. Once all the rows are extended we can add the rows together using rules of addition, and give the result in an 8-bit representation.

1	0	1	1
x	0	1	1	1
1	1	1	1	1	0	1	1
1	1	1	1	0	1	1	x
1	1	1	0	1	1	x	x
+	0	0	0	0	0	x	x	x
1	0	1	1	0	1	1	1	0	1

The final result is represented in 8-bit ignoring the extra two digits in the front.

So, the signed multiplication result for 1011 and 0111 is 11011101.

Solving unsigned binary multiplication is an easy process. This multiplication can be solved like any other decimal multiplication.

Let us check a solved example for better understanding:

Example: Multiply 13 and 9 in binary digits.

Solution: 13 in binary can be written as 1101 and 9 in binary is denoted as 1001.

Performing unsigned multiplication:

So, unsigned multiplication for 1101 and 1001 is 1110101. That is 13 multiplied by 9 gives 117.

Solved Examples of Binary Multiplication

Example 1: Solve 1001 × 110

Step-by-step solution:

We are multiplying 1001 (which is 9 in decimal) with 110 (which is 6 in decimal).

Binary Multiplication:

1001

× 110

--------

0000 ← 1001 × 0 (rightmost digit)

+ 10010 ← 1001 × 1 (next digit, shifted one place to the left)

+100100 ← 1001 × 1 (next digit, shifted two places to the left)

---------

 110110

Answer: 1001 × 110 = 110110 (Binary)

Check in Decimal:
1001 (binary) = 9
110 (binary) = 6
9 × 6 = 54
110110 (binary) = 54 

Example 2: Solve 111 × 101

We are multiplying 111 (which is 7 in decimal) by 101 (which is 5 in decimal).

Step-by-step solution:

111

× 101

-------

111 ← 111 × 1 (rightmost digit)

+ 0000 ← 111 × 0 (next digit, shifted one place)

+11100 ← 111 × 1 (next digit, shifted two places)

---------

 100011

Answer: 111 × 101 = 100011 (Binary)

Check in Decimal:
111 (binary) = 7
101 (binary) = 5
7 × 5 = 35
100011 (binary) = 35 

We hope that the above article is helpful for your understanding and exam preparations. Stay tuned to the Testbook App for more updates on related topics from Mathematics, and various such subjects. Also, reach out to the test series available to examine your knowledge regarding several exams.

If you are checking Binary Multiplication article, also check related maths articles:
Convert Octal to Binary	Binary Number System
Octal Number System	Types of Number System
Octal Number System MCQs	Decimal to Octal Conversion

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