Binary Calculator
Use the following calculators to perform the addition, subtraction, multiplication, or division of two binary values, as well as convert binary values to decimal values, and vice versa.
Modify the values and click the Calculate button to use
Binary Calculation—Add, Subtract, Multiply, or Divide
Convert Binary Value to Decimal Value
Convert Decimal Value to Binary Value
Understanding the Binary System
The binary system is a numerical system that functions virtually identically to the decimal number system that people are likely more familiar with. While the decimal number system uses the number 10 as its base, the binary system uses 2. Furthermore, although the decimal system uses the digits 0 through 9, the binary system uses only 0 and 1, and each digit is referred to as a bit. Apart from these differences, operations such as addition, subtraction, multiplication, and division are all computed following the same rules as the decimal system.
Almost all modern technology and computers use the binary system due to its ease of implementation in digital circuitry using logic gates. It is much simpler to design hardware that only needs to detect two states, on and off (or true/false, present/absent, etc.). Using a decimal system would require hardware that can detect 10 states for the digits 0 through 9, and is more complicated.
Binary/Decimal Conversion
| Decimal | Binary |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 2 | 10 |
| 3 | 11 |
| 4 | 100 |
| 7 | 111 |
| 8 | 1000 |
| 10 | 1010 |
| 16 | 10000 |
| 20 | 10100 |
While working with binary may initially seem confusing, understanding that each binary place value represents 2n, just as each decimal place represents 10n, should help clarify.
Example: Decimal 8
8 × 1 = 8
Example: Decimal 18
(1 × 10) + (8 × 1) = 10 + 8 = 18
Binary 10010
10010 = (1 × 24) + (0 × 23) + (0 × 22) + (1 × 21) + (0 × 20) = 18
Steps to Convert Decimal to Binary
- Find the largest power of 2 within the given number.
- Subtract that value from the given number.
- Find the largest power of 2 within the remainder.
- Repeat until there is no remainder.
- Enter 1 for each binary place value that was found and 0 for the rest.
Converting Binary to Decimal
Determine all of the place values where 1 occurs, and find the sum of the values.
Example: 10111 = (1 × 24) + (0 × 23) + (1 × 22) + (1 × 21) + (1 × 20) = 23
Binary Addition
Binary addition follows the same rules as decimal addition except carrying occurs when the sum equals 2.
- 0 + 0 = 0
- 0 + 1 = 1
- 1 + 0 = 1
- 1 + 1 = 0, carry 1
Example:
10110
+ 10111
--------
101100
Binary Subtraction
Borrowing occurs when 1 is subtracted from 0. The 0 becomes 2 and the borrowing column is reduced by 1.
- 0 – 0 = 0
- 0 – 1 = 1, borrow 1
- 1 – 0 = 1
- 1 – 1 = 0
Example 1:
1111
- 01101
--------
01010
Example 2:
110
- 011
------
001
Binary Multiplication
Binary multiplication is simpler because results are either the first term or 0.
- 0 × 0 = 0
- 0 × 1 = 0
- 1 × 0 = 0
- 1 × 1 = 1
Example:
10111
× 11
--------
10111
+101110
--------
1000101
Binary Division
The process is similar to long division in the decimal system, but uses binary subtraction instead.
Related Calculators:
Sample Size Calculator, Number Sequence CalculatorExternal Resources:
Binary Calculator on Calculator.net