Binary and Bitwise Math Made Easy: A Practical Guide

A binary calculator lets you perform arithmetic and bitwise operations — add, subtract, multiply, divide, AND, OR, XOR and shifts — on numbers entered in binary, decimal, hex or octal, showing the answer in every base at once. That combination saves you from juggling a separate converter for each step. The Binary Calculator validates each input against its base and uses exact big-integer math, so large values never suffer rounding errors.

Who needs binary and bitwise math

This kind of math shows up wherever computers meet numbers. Computer science students learning Boolean logic need to see how AND, OR and XOR behave bit by bit. Embedded developers manipulate hardware registers and bit masks, where a single flipped bit changes behaviour. Network engineers compute subnet masks and address ranges. In all of these, the ability to enter a value in one base and instantly read it in the other three removes a whole layer of manual conversion and the mistakes that come with it.

How to run binary math in your browser

1. Choose the base — binary, decimal, hex or octal — for each operand and type the two numbers.
2. Select the operation: add, subtract, multiply, divide, AND, OR, XOR, left shift or right shift.
3. Read the result immediately in binary, decimal, hexadecimal and octal.
4. Copy whichever representation you need with one click.

AND, OR, XOR and shifts compared

The bitwise operators are easiest to grasp side by side. Each compares two numbers bit by bit, while shifts move bits left or right.

Operation	Rule	Example	Result
AND	1 only if both bits are 1	1100 & 1010	1000
OR	1 if either bit is 1	1100 | 1010	1110
XOR	1 if exactly one bit is 1	1100 ^ 1010	0110
Left shift	× 2ⁿ	5 << 2	20
Right shift	÷ 2ⁿ (remainder dropped)	20 >> 2	5

Shifts are a favourite trick for fast multiplication and for building bit masks, while XOR is the workhorse of checksums and simple toggling.

Key features

• Independent base selection for each operand
• Arithmetic and bitwise operations, including left and right shifts
• Result shown in binary, decimal, hex and octal simultaneously
• Per-base input validation with clear error messages
• Exact big-integer math — no floating-point rounding
• 100% private, free and works offline

Try the Binary Calculator now — it's free and runs entirely in your browser.

Frequently asked questions

How do you add two binary numbers?

Add column by column like decimal, but carry at 2 instead of 10: 1 + 1 = 10 (write 0, carry 1). So 1011 + 0110 = 10001, which is 11 + 6 = 17 in decimal. The calculator shows the sum in every base.

What's the difference between AND, OR and XOR?

They compare bits. AND gives 1 only when both bits are 1, OR when at least one is 1, and XOR when exactly one is 1. For 1100 and 1010 that's 1000, 1110 and 0110 respectively.

What does a bit shift do?

Shifting left by n multiplies by 2ⁿ; shifting right divides by 2ⁿ and discards the remainder. So 5 << 2 = 20 and 20 >> 2 = 5. Shifts power fast multiplication and mask construction.

How does binary division handle remainders?

It performs integer division: the quotient is truncated toward zero and the remainder is shown separately. Dividing 1010 (10) by 11 (3) gives a quotient of 11 (3) and a remainder of 1.

How do I convert between binary, decimal, hex and octal?

Each hex digit equals exactly four binary bits and each octal digit three bits, which is why programmers use them as binary shorthand. Enter a number in any base and the other three appear automatically.

Related free tools

• Scientific Calculator — trig, logs and powers
• Unit Converter — including data storage units
• Base64 Encoder — encode and decode text and data
• Percentage Calculator — everyday percentage sums

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