Binary Calculator

Calculated using standard mathematical and industry formulas.

Free online Binary Calculator calculator for academic, engineering, and statistical math.

How to use this calculator?

1. For conversion: enter a binary string of 0s and 1s to get decimal, hexadecimal, and octal equivalents.
2. For arithmetic: enter two binary numbers and select the operation.
3. The result is shown in binary, decimal, and hexadecimal simultaneously.

Formula & calculation method

Binary to decimal positional notation

Decimal = Σ bit_i × 2^i (from right, i = 0) · Binary add: treat as base-2, carry when sum ≥ 2

Right-to-left positional values: 1, 2, 4, 8, 16, 32, 64, 128, …

Step-by-step calculation example

10110₂ = 0×1 + 1×2 + 1×4 + 0×8 + 1×16 = 22₁₀ = 0x16 = 0o26.

Key insights to remember

• Binary (base 2) uses only 0 and 1 — each digit is one bit. 8 bits = 1 byte.
• 10110 in binary = 1×16 + 0×8 + 1×4 + 1×2 + 0×1 = 22 in decimal.
• Computers store all data in binary — text, images, and programs are ultimately sequences of bits.
• Binary addition: 1 + 1 = 10 (carry 1). 111 + 1 = 1000.

How to interpret your results?

Binary to decimal

Each bit position represents a power of 2, from right to left: 2⁰=1, 2¹=2, 2²=4, 2³=8, etc.

Bit length

The number of digits in the binary representation. An n-bit number can represent values from 0 to 2ⁿ − 1.

Frequently asked questions

How does the Binary Calculator calculator work?

It uses standard algebraic, geometric, or statistical theorems to return exact values.

How accurate are the results?

All calculations use double-precision floating-point arithmetic for exact accuracy.

Can I copy the step-by-step work?

Yes, step-by-step solutions are formatted clearly for homework or documentation.

How are undefined operations handled?

The calculator flags mathematical errors such as division by zero or negative square roots.

Is this tool suitable for students and engineers?

Yes, it is designed for students, teachers, engineers, and scientists.

Does it support negative numbers and decimals?

Yes, all real numbers (positive, negative, fractions, and decimals) are supported.

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