Binary Operation
Binary operations are mathematical operations that are performed over binary digits (0.1). There are various computer calculations that are done through binary operations. There are four main types of binary operations (addition, subtraction, multiplication, and division), as shown in the following diagram
Let’s explain each type of binary operation with definitions, rules, and examples
1. Binary Addition
Binary addition is an arithmetic operation used to add two binary numbers. It follows simple rules based on the digits 0 and 1. When adding 1 and 1, the sum is 0 with a carry of 1 to the next column.
Rules of Binary Addition
Binary addition follows four basic rules using the digits 0 and 1. The Key point is that when adding 1 + 1, write 0 and carry 1 to the next column. The following diagram shows the entire rules
Binary Addition Example – 1
Add 1101₂ and 0111₂ by starting from the rightmost bit and carrying 1 whenever the sum is 2.
The binary addition is 1101₂ + 0111₂ = 10100₂.
- Convert to decimal: 1101₂ = 13 and 0111₂ = 7.
- Add the decimal numbers: 13 + 7 = 20.
- Convert 20 to binary: 20 = 10100₂.
Therefore, 1101₂ + 0111₂ = 10100₂, which is equal to 13 + 7 = 20 in decimal.
Binary Addition with Fractions – Example 2
Binary addition can also be used to add binary fractions. Align the binary points and add the digits from right to left, carrying when necessary.
Example: Add 10.1₂ and 1.01₂.
10.10
+ 01.01
-------
11.11Step-by-step explanation:
- Align the binary points by writing 10.1 as 10.10.
- Add the fractional bits: 0 + 1 = 1.
- Add the next column: 1 + 0 = 1.
- Add the integer bits: 0 + 1 = 1, then 1 + 0 = 1.
Therefore, 10.1₂ + 1.01₂ = 11.11₂. In decimal, 2.5 + 1.25 = 3.75, so the result is correct.
2. Binary Subtraction
Binary subtraction is an arithmetic operation used to find the difference between two binary numbers. It follows simple rules based on the digits 0 and 1. When subtracting 1 from 0, borrow 1 from the next column to perform the subtraction.
Rules of Binary Subtraction
Binary subtraction follows four basic rules using the digits 0 and 1. The key point is that when subtracting 1 from 0, borrow 1 from the next column. The following table shows the basic rules.
| A | B | Difference | Borrow |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 1 |
| 1 | 0 | 1 | 0 |
| 1 | 1 | 0 | 0 |
Binary Subtraction Example – 1
Subtract 011₂ from 101₂ by starting from the rightmost bit and borrowing when necessary.
101
- 011
-----
010Step-by-step explanation:
- Rightmost column: 1 − 1 = 0.
- Middle column: 0 − 1 requires borrowing 1 from the leftmost column. The middle bit becomes 10₂, so 10₂ − 1₂ = 1.
- Leftmost column: After borrowing, 0 − 0 = 0.
The final result is 010₂. Therefore, 101₂ − 011₂ = 010₂, or 5 − 3 = 2 in decimal.
Binary Subtraction with Fractions – Example 2
Binary subtraction can also be used to subtract binary fractions. Align the binary points and subtract the digits from right to left, borrowing when necessary.
Example: Subtract 1.01₂ from 10.1₂.
10.10
- 01.01
-------
1.01Step-by-step explanation:
- Align the binary points by writing 10.1 as 10.10.
- Rightmost fractional column: 0 − 1 requires borrowing from the next column, giving 10₂ − 1₂ = 1.
- Next fractional column: After borrowing, 0 − 0 = 0.
- Integer part: 0 − 1 requires borrowing from the leftmost bit. The result of the integer subtraction is 1.
Therefore, 10.1₂ − 1.01₂ = 1.01₂. In decimal, 2.5 − 1.25 = 1.25, so the result is correct.
3. Binary Multiplication
Binary multiplication is an arithmetic operation used to multiply two binary numbers. It follows simple rules based on the digits 0 and 1. Multiplying any binary digit by 0 gives 0, while multiplying 1 by 1 gives 1.
Rules of Binary Multiplication
Binary multiplication follows four basic rules using the digits 0 and 1. The following table shows the rules.
| A | B | Product |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
Binary Multiplication Example – 1
Multiply 101₂ by 011₂ using the binary multiplication method.
101
× 011
-----
101
101
000
-----
01111Step-by-step explanation:
- Multiply 101 by the rightmost bit, 1. The result is 101.
- Multiply 101 by the middle bit, 1. Shift the result one position to the left, giving 1010.
- Multiply 101 by the leftmost bit, 0. The result is 00000.
- Add the partial products to get 1111₂.
Therefore, 101₂ × 011₂ = 1111₂, or 5 × 3 = 15 in decimal.
Binary Multiplication with Fractions – Example 2
Binary multiplication can also be used to multiply binary fractions. Multiply the numbers as usual, then place the binary point according to the total number of fractional bits in both numbers.
Example: Multiply 1.1₂ by 0.1₂.
11
× 01
----
11
00
----
011The numbers have two fractional bits in total: one in each factor. Place the binary point two positions from the right.
Therefore, 1.1₂ × 0.1₂ = 0.11₂. In decimal, 1.5 × 0.5 = 0.75, so the result is correct.
4. Binary Division
Binary division is an arithmetic operation used to divide one binary number by another. It follows the same basic long-division method used in decimal arithmetic, but it uses only the digits 0 and 1.
Rules of Binary Division
Binary division follows four basic rules based on the digits 0 and 1. The following table shows the basic rules.
| A | B | Quotient (A ÷ B) |
|---|---|---|
| 0 | 1 | 0 |
| 1 | 1 | 1 |
| 0 | 0 | Undefined |
| 1 | 0 | Undefined |
Key point: Division by 0 is undefined, just as in decimal arithmetic.
Binary Division Example – 1
Divide 1100₂ by 10₂ using binary long division.
110
_____
10 ) 1100
- 10
---
10
- 10
---
00Step-by-step explanation:
- Divide the first two bits, 11₂, by 10₂. The quotient is 1. Subtract 10₂ from 11₂ to get 1₂.
- Bring down the next bit, 0, to get 10₂. Divide 10₂ by 10₂ to get 1.
- Subtract 10₂ from 10₂ to get 0, then bring down the last bit, 0.
- The remaining value is 0, so the division is complete.
Therefore, 1100₂ ÷ 10₂ = 110₂, or 12 ÷ 2 = 6 in decimal.
Binary Division with Fractions – Example 2
Binary division can also be used to divide binary numbers containing fractional parts. The division follows the same basic process as whole-number division.
Example: Divide 1.1₂ by 10₂.
1.1₂ ÷ 10₂ = 0.11₂Step-by-step explanation:
- Convert the numbers to decimal to verify the result: 1.1₂ = 1.5 and 10₂ = 2.
- Divide 1.5 by 2 to get 0.75.
- Convert 0.75 to binary: 0.75 = 0.11₂.
Therefore, 1.1₂ ÷ 10₂ = 0.11₂. In decimal, 1.5 ÷ 2 = 0.75, so the result is correct.