Binary multiplication is a basic operation in digital logic and computer arithmetic, and it is carried out using the base-2 number system.This process is similar to long multiplication in base-10 arithmetic, but it is simpler because it uses only two digits: 0 and 1.In current computing systems, any data processing task can be expressed in terms of basic binary operations, so studying Binary Multiplication is necessary for work in computer science and electronic engineering.The procedure starts by forming partial products using bitwise multiplication, then applies bit shifts and binary additions in sequence to produce the final result.In contrast to decimal multiplication, which often depends on memorizing a ten-by-ten table, binary multiplication can be described with a small truth table: the product is 1 only when both inputs are 1, and it is 0 in all other cases.This simple structure supports straightforward ALU design in CPUs and enables fast arithmetic and logic operations needed to run complex algorithms on different hardware platforms.

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 Multiplication Rules

Binary multiplication is carried out through a defined sequence of steps in which partial products are produced and then aligned before they are combined to form the final result.The multiplicand is compared with each bit of the multiplier, starting at the least significant bit (LSB) and moving step by step to the most significant bit (MSB).In binary multiplication, each 1 bit in the multiplier leads to writing the multiplicand as a partial product, while each 0 bit leads to writing a corresponding string of zeros.In binary multiplication, each later partial product is shifted one place to the left so that it matches the higher power of two represented by that bit of the multiplier.After all bits are processed, the resulting rows are added using standard binary addition, where a carry is produced whenever a column total exceeds what can be represented with a single base-2 digit.

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.

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:

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:

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 \(1\times0\ =\ 0\), \(0\times0\ =\ 0\), and \(1\times1\ =\ 1\).

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.

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 

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