Binary Calculator
Add, subtract, multiply and divide binary numbers, and see every result in binary, decimal, hexadecimal and octal.
Results
| Item | Value |
|---|
Steps
| Step | Working |
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Quick Answer
Binary uses only 0 and 1, and each place is worth twice the one to its right: 1, 2, 4, 8, 16 and so on. 1011 is 8 + 0 + 2 + 1 = 11, and 110 is 4 + 2 = 6. Adding them gives 17, which is 10001 in binary.
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Table of Contents
- Binary Calculator
- Quick Answer
- Related Calculators
- How This Binary Calculator Works
- How to Use the Binary Calculator
- Binary Rules
- Key Terms
- Worked Example
- Real-Life Examples
- Binary vs Decimal vs Hexadecimal
- Binary, Decimal and Hex Chart
- Where Binary Is Used
- Common Mistakes to Avoid
- Frequently Asked Questions
- References & Sources
How This Binary Calculator Works
Enter two binary numbers and choose an operation. The calculator reads each number in base 2, does the arithmetic, and shows the result in binary, decimal, hexadecimal (base 16) and octal (base 8). The steps show the decimal values so you can check the working.
Quick explanation: each binary digit (bit) is worth a power of 2, starting from 2⁰ = 1 on the right.
- Operations: add, subtract, multiply and divide.
- Size: up to 64 digits per number.
- Division: shows the quotient and the remainder.
How to Use the Binary Calculator
- Enter the first binary number, using only 0 and 1.
- Choose add, subtract, multiply or divide.
- Enter the second binary number.
- Press Calculate.
- Read the result in binary, decimal, hex and octal, and check the steps.
Binary Rules
- Place values: 1, 2, 4, 8, 16, 32, 64, 128 … from right to left
- Addition: 0 + 0 = 0, 0 + 1 = 1, 1 + 1 = 10 (write 0, carry 1)
- Subtraction: 10 − 1 = 1 (borrow from the next place)
- Multiplication: like decimal, using only 0 × and 1 × rows
- Binary to decimal: add the place values where there is a 1
Key Terms
- Bit: a single binary digit, 0 or 1.
- Byte: 8 bits, which can hold values from 0 to 255.
- Base: how many digits a number system uses, such as 2 for binary.
- Hexadecimal: base 16, using 0 to 9 and A to F.
- Carry: a 1 moved to the next place when a column adds up to 2 or more.
Worked Example
Add 1101 and 111.
- 1101 = 8 + 4 + 1 = 13
- 111 = 4 + 2 + 1 = 7
- 13 + 7 = 20
- 20 = 16 + 4 = 10100 in binary
Real-Life Examples
- Homework: 1010 × 11 = 11110, which is 10 × 3 = 30 in decimal.
- Subnet math: a mask of 11111111 (255) has 8 bits on, a common part of network settings.
- Byte values: 11111111 is 255 in decimal and FF in hex, the largest value one byte can hold.
- Checking code: confirm that a bit pattern such as 1000000 equals 64 before using it as a flag.
Binary vs Decimal vs Hexadecimal
- Binary (base 2): digits 0 and 1. What computers use inside.
- Decimal (base 10): digits 0 to 9. What people use every day.
- Hexadecimal (base 16): digits 0 to 9 and A to F. Each hex digit stands for exactly 4 binary digits.
- Example: 11111111₂ = 255₁₀ = FF₁₆.
Binary, Decimal and Hex Chart
| Decimal | Binary | Hex |
|---|---|---|
| 0 | 0 | 0 |
| 1 | 1 | 1 |
| 2 | 10 | 2 |
| 3 | 11 | 3 |
| 4 | 100 | 4 |
| 5 | 101 | 5 |
| 8 | 1000 | 8 |
| 10 | 1010 | A |
| 15 | 1111 | F |
| 16 | 10000 | 10 |
| 32 | 100000 | 20 |
| 64 | 1000000 | 40 |
| 100 | 1100100 | 64 |
| 128 | 10000000 | 80 |
| 255 | 11111111 | FF |
Where Binary Is Used
- Computers: all data is stored as bits, 0s and 1s.
- Networking: IP addresses and subnet masks are built from binary.
- Colors: web colors use hexadecimal, a short way of writing binary.
- Permissions: file settings in some systems use binary flags.
Common Mistakes to Avoid
- Using digits other than 0 and 1. Binary has only two digits.
- Forgetting to carry. 1 + 1 = 10, so carry the 1 to the next place.
- Reading place values left to right. The smallest place, 1, is on the right.
- Dropping leading zeros in fixed-size values. 00001010 and 1010 are the same number, but some systems need all 8 bits.
- Mixing up hex letters. A to F stand for 10 to 15.
Frequently Asked Questions
Add each column from the right. 1 + 1 = 10, so write 0 and carry 1. 1 + 1 + 1 = 11, so write 1 and carry 1.
Add the place values (1, 2, 4, 8 …) where the digit is 1. 1011 = 8 + 2 + 1 = 11.
Divide by 2 repeatedly and write down the remainders from bottom to top.
11111111.
10.
Yes. Computers often store them using two's complement. This calculator shows negatives with a minus sign.
Electronic circuits are simplest and most reliable with two states, on and off.
Subtract each column from the right. When you need to take 1 from 0, borrow from the next place, which turns the 0 into 10 (2 in decimal).
Split the binary number into groups of 4 from the right, then change each group to one hex digit. 1111 0101 = F5.
A way computers store negative numbers: flip every bit and add 1. In 8 bits, −1 is 11111111.
11111111, which is 255 in decimal.
References & Sources
| Source | Used For | Link |
|---|---|---|
| Wikipedia, Binary number | The base-2 number system. | https://en.wikipedia.org/wiki/Binary_number |
| Wikipedia, Hexadecimal | Base 16 and its link to binary. | https://en.wikipedia.org/wiki/Hexadecimal |