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Learn What is Binary Multiplication? A fundamental operation in Mathematics, computer science and digital electronics. Discover how binary numbers are multiplied using a systematic approach, exploring concepts like bitwise operations and carrying.
Contents
What is Binary Multiplication?
Binary multiplication is the process of multiplying two binary numbers together. Binary numbers are numbers expressed in the base-2 system, which uses only two digits: 0 and 1. In binary multiplication, each digit of one binary number is multiplied by each digit of the other binary number, similar to the process of multiplication in the decimal system.
The basic rules of binary multiplication are as follows:
- Multiplying 0 by any number gives 0.
- Multiplying 1 by any number gives the same number.
- When multiplying two 1s, the result is 1, and a carry of 1 is generated.
- 1 * 1 = 1 with a carry of 1
To perform binary multiplication, you start by multiplying the least significant bit (LSB) of the second binary number by each bit of the first binary number. The results are then added together, taking into account any carry generated. This process is repeated for each bit of the second binary number, shifting the partial products to the left by one position for each iteration.
Here’s an example of binary multiplication:
1101 <- Binary number A
0000 <- Partial product (LSB of B multiplied by A)
- 1101 <- Partial product (Second bit of B multiplied by A, shifted left by 1) +1101 <- Partial product (Third bit of B multiplied by A, shifted left by 2) +1101 <- Partial product (MSB of B multiplied by A, shifted left by 3)
100111 <- Binary product of A and B
In this example, the binary number A is 1101, and the binary number B is 101. After performing the binary multiplication, the product is 100111.
Binary multiplication is an essential operation in digital systems and computer arithmetic, as it forms the basis for various operations like addition, subtraction, division, and more complex arithmetic operations.
What is an Example of a Binary Multiplication?
Let’s multiply two binary numbers: 101 (5 in decimal) and 110 (6 in decimal).
101
x 110
101 (Partial product: 101)
1010 (Partial product: 101 shifted one place to the left)
+10100 (Partial product: 101 shifted two places to the left)
111100 (Final product: 30 in decimal)
In this example, we multiply each digit of the second number (110) with each digit of the first number (101). We start with the rightmost digit of the second number (0), multiply it with each digit of the first number (101), and write down the partial product. Then we shift the partial product one place to the left and add it to the next partial product obtained by multiplying the next digit of the second number (1) with the first number (101). We repeat this process for the last digit of the second number (1) and add it to the previous partial products. Finally, we obtain the final product (111100) in binary, which is equal to 30 in decimal.
What are the Rules for Binary Multiplication?
Binary multiplication follows the same principles as multiplication in decimal (base-10) system, but with only two possible digits, 0 and 1. Here are the rules for binary multiplication:
Multiplying 0 by any number results in 0: When you multiply any binary number by 0, the result is always 0. This is similar to multiplying any decimal number by 0, which also gives 0.
Example:
0 * 1 = 0
0 * 0 = 0
Multiplying 1 by any number results in the same number: When you multiply any binary number by 1, the result is always the same number. This is similar to multiplying any decimal number by 1, which also gives the same number.
Example:
1 * 1 = 1
1 * 0 = 0
Multiplying any number by itself results in the number squared: When you multiply a binary number by itself, the result is the square of that number.
Example:
1 * 1 = 1 (1 squared)
0 * 0 = 0 (0 squared)
When multiplying two binary digits, the result is determined by their product:
0 multiplied by 0 gives 0.
0 multiplied by 1 gives 0.
1 multiplied by 0 gives 0.
1 multiplied by 1 gives 1.
Example:
1 * 1 = 1
1 * 0 = 0
0 * 1 = 0
0 * 0 = 0
Multiplying multi-digit binary numbers follows the same principles as decimal multiplication. You multiply each digit in the first number by each digit in the second number, starting from the rightmost digit and moving left. The results are then added together.
Example:
101 (5 in decimal)
x 110 (6 in decimal)
1010 (10 in decimal)
+ 1010 (10 in decimal)
11110 (30 in decimal)
These rules are fundamental to binary multiplication and can be extended to perform multiplication on larger binary numbers.
|
Multiplicand |
Multiplier |
Product |
|
0 |
0 |
0 × 0 = 0 |
|
0 |
1 |
0 × 1 = 0 |
|
1 |
0 |
1 × 0 = 0 |
|
1 |
1 |
1 × 1 = 1 |
How do you Multiply Binary Numbers?
To multiply binary numbers, you can use a method similar to long multiplication. Here’s a step-by-step guide on how to do it:
Write down the two binary numbers you want to multiply. Align them so that the rightmost digits are lined up.
Start with the rightmost digit of the second (multiplicand) binary number and multiply it with each digit of the first (multiplier) binary number. Write the intermediate results below each digit of the multiplier, shifted to the right according to their position.
If the digit of the multiplier is 0, the intermediate result will be all 0s. If the digit is 1, the intermediate result will be the same as the multiplicand.
Add up all the intermediate results, aligning them vertically, and perform binary addition. Start from the rightmost column and work towards the left, carrying over any carry values as needed.
The final result will be the sum obtained from the previous step. It will be a binary number that represents the product of the two original binary numbers.
Here’s an example to illustrate the process. Let’s multiply the binary numbers 1101 and 101:
1101 (multiplier)
× 101 (multiplicand)
0000 (intermediate result, digit 1 of the multiplier is 0)
+ 1101 (intermediate result, digit 2 of the multiplier is 1)
1101 (product)
Therefore, the product of 1101 and 101 is 1101 in binary.
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Categories: Math
