Binary Calculator

2
2

Calculation Result

—
Binary Arithmetic Tool

Perform binary arithmetic operations — add, subtract, and convert binary numbers. Enter any binary value and see results in decimal, hex, and octal with step-by-step computation breakdowns.

5Number Systems
∞Precision
0msLatency

What is Binary Calculator?

The Binary Calculator is a comprehensive arithmetic tool that performs addition, subtraction, and other operations directly on binary numbers. It accepts binary inputs and displays results across multiple number systems — binary, decimal, hexadecimal, and octal — with detailed step-by-step computation breakdowns.

This tool exists because binary arithmetic is the foundation of all computing. Every calculation your computer performs — from simple addition to complex floating-point operations — is ultimately executed as binary arithmetic in the CPU. Understanding how binary math works helps programmers debug low-level issues, optimize algorithms, and grasp how hardware processes data.

Whether you're a computer science student learning binary operations, an embedded systems developer working with register values, or a digital electronics engineer designing circuits, this calculator provides instant, verified binary arithmetic with full transparency into every step.

Interactive Demo

Binary Calculator Formula

Binary multiplication works by shift-and-add. For each 1-bit in the multiplier, the multiplicand is shifted left and added.

Shift-and-Add Engine
Multiplicand (A)
Multiplier (B)
Click Calculate to see partial products
Result: — ₂
Concept Guide

How Binary Arithmetic Simplifies Hardware

In decimal, you need to memorize the result of multiplying any two digits (10 × 10 = 100 combinations). In binary, the multiplication table has only 4 entries:

0 × 0 = 0, 0 × 1 = 0, 1 × 0 = 0, 1 × 1 = 1

This extreme simplicity is why binary circuits can be built from just two logic gates: AND (for multiplication) and XOR + AND (for addition with carry). The entire ALU in a CPU performs all four arithmetic operations using these basic building blocks.

Key Insight Multiplying by a binary digit is just a decision: copy the number (×1) or write zero (×0).
Binary vs Decimal Complexity
4
Binary multiplication rules
100
Decimal multiplication rules
Quick Reference

Binary Arithmetic Truth Tables

Addition
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 10 (carry)
Subtraction
0 − 0 = 0
1 − 0 = 1
1 − 1 = 0
10 − 1 = 1 (borrow)
Multiplication
0 × 0 = 0
0 × 1 = 0
1 × 0 = 0
1 × 1 = 1
More

Other Number System Conversions

Related numeral systems converters for number conversion between binary, decimal, hexadecimal, octal, and ASCII text.

01→10 Binary to Decimal Converter 01.1→10 Binary Fraction to Decimal ±01→10 Signed Binary to Decimal 01→FF Binary to Hexadecimal 01→07 Binary to Octal N→N Binary Base Converter 01+01 Binary Addition 01−01 Binary Subtraction → 1,2,3 Step-by-Step Solver 0…255 Binary Table Generator 01→A Binary to Text A→01 Text to Binary ±1.M×2ᴱ Floating Point Converter 8b→256 Bit Size Converter & | ^ ~ Bitwise Operations 01→? Binary Decoder
FAQ

Frequently Asked Questions

How does binary addition work?
Binary addition follows simple rules: 0+0=0, 0+1=1, 1+0=1, 1+1=10 (0 with carry 1), and 1+1+1=11 (1 with carry 1). You add column by column from right to left, carrying over just like decimal addition. Example: 1011 + 1101 = 11000₂ (11 + 13 = 24₁₀).
How does binary subtraction work?
Binary subtraction uses borrowing: 0-0=0, 1-0=1, 1-1=0, 0-1=1 (with borrow). When subtracting 1 from 0, you borrow from the next higher bit, which gives you 10₂ (2) minus 1 = 1. Alternatively, you can add the 2's complement of the subtrahend.
How does binary multiplication work?
Binary multiplication is simpler than decimal because you only multiply by 0 or 1. Multiply the binary number by each bit of the multiplier: if the bit is 1, copy the number (shifted left); if 0, write zeros. Then add all partial products. Example: 101 × 11 = 1111₂ (5 × 3 = 15₁₀).
What is binary overflow?
Binary overflow occurs when the result of an arithmetic operation exceeds the maximum value that can be represented with the given number of bits. For unsigned 8-bit numbers, overflow happens above 255 (11111111₂). For signed 8-bit, overflow occurs above 127 or below -128. Overflow can cause unexpected results in programs.
Can you divide binary numbers?
Yes, binary division works like long division in decimal but simpler since you only compare with 0 or 1. At each step, check if the divisor fits into the current portion: if yes, write 1 and subtract; if no, write 0 and bring down the next digit. The process continues until all digits are processed.
What are the basic rules of binary arithmetic?
Binary arithmetic has four basic operations: Addition (0+0=0, 0+1=1, 1+1=10), Subtraction (0-0=0, 1-0=1, 1-1=0, 10-1=1), Multiplication (0×0=0, 0×1=0, 1×1=1), and Division (0÷1=0, 1÷1=1). All follow the same principles as decimal arithmetic but with only two digits.
Copied to clipboard!