conversion

Binary to Decimal Converter

Convert binary numbers to decimal instantly. See the positional breakdown showing each bit\'s power-of-2 contribution.

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Number Base Conversion Tips

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Pick a scenario to see how the calculator works, then adjust the values

ASCII Letter A

Convert the ASCII code for uppercase A (65) across all bases.

Key values: Decimal 65 · Binary 1000001 · Hex 41

Byte Maximum

The largest value a single byte can hold (255).

Key values: Decimal 255 · Binary 11111111 · Hex FF

Hex Color Code

Convert a common hex color code (deep sky blue) to other bases.

Key values: Hex 00BFFF · Decimal 49151 · Binary nibbles

Convert a Number

Enter a number in any base to see it converted to binary, octal, decimal, hexadecimal, and any custom base 2-36.

Documentation

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The Binary Number System

Binary (base-2) uses only two digits: 0 and 1. Each position represents a power of 2, just as each position in decimal represents a power of 10:

\underbrace{1}_{2^3}\;\underbrace{0}_{2^2}\;\underbrace{1}_{2^1}\;\underbrace{1}_{2^0} = 8 + 0 + 2 + 1 = 11

Conversion Method

Multiply each binary digit by its positional power of 2, then sum:

(b_n b_{n-1} \ldots b_1 b_0)_2 = \sum_{i=0}^{n} b_i \times 2^i

Worked Example

Convert $11010110_2$ to decimal:

1 \!\times\! 128 + 1 \!\times\! 64 + 0 \!\times\! 32 + 1 \!\times\! 16 + 0 \!\times\! 8 + 1 \!\times\! 4 + 1 \!\times\! 2 + 0 \!\times\! 1 = 214

Powers of 2 Reference

Position	$2^0$	$2^1$	$2^2$	$2^3$	$2^4$	$2^5$	$2^6$	$2^7$
Value	1	2	4	8	16	32	64	128

An 8-bit byte can represent $2^8 = 256$ values (0 to 255). A 16-bit value reaches 65,535. A 32-bit value reaches approximately 4.3 billion.

Why Computers Use Binary

Electronic circuits have two stable states: on (high voltage) and off (low voltage). Binary maps directly to these states, making it naturally resistant to electrical noise. A circuit doesn't need to distinguish between 10 voltage levels (as base-10 would require) — just two. This simplicity enables reliable computation at billions of operations per second.

Common Binary Patterns

Binary	Decimal	Significance
`11111111`	255	Maximum 8-bit unsigned value; max RGB channel
`10000000`	128	Highest bit set (sign bit in signed integers)
`01111111`	127	Maximum signed 8-bit value
`00000001`	1	Least significant bit (LSB)

Frequently Asked Questions

How do you convert binary to decimal?

Multiply each binary digit by its positional power of 2 and sum the results. For example, 1011 in binary = $1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 8 + 0 + 2 + 1 = 11$ in decimal.

Why do computers use binary instead of decimal?

Electronic circuits have two stable states: on (high voltage) and off (low voltage). Binary maps directly to these states, making it naturally resistant to electrical noise. A circuit only needs to distinguish between two voltage levels, not ten.

What is the largest number an 8-bit byte can represent?

An unsigned 8-bit byte can represent values from 0 to 255. With all bits set to 1 (11111111), the value is 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = 255.

What are common binary patterns I should memorize?

11111111 = 255 (max byte), 10000000 = 128 (sign bit), 01111111 = 127 (max signed byte), 00000001 = 1 (LSB). Knowing these makes working with binary data much faster.

How many binary digits are needed for a given decimal number?

A decimal number N requires approximately $\lceil \log_2(N + 1) \rceil$ binary digits. For example, 255 needs 8 bits, 256 needs 9 bits, and 1024 needs 11 bits.

Binary to Decimal Converter

Convert binary numbers to decimal instantly. See the positional breakdown showing each bit\'s power-of-2 contribution.

Try an Example

ASCII Letter A

Byte Maximum

Hex Color Code

Convert a Number

Related conversion Variants

Decimal to Binary

Hex to Decimal

Decimal to Hex

Binary to Hex

Octal to Decimal

More Math Calculators

Calculator Search

Try an Example

ASCII Letter A

Byte Maximum

Hex Color Code

Convert a Number

The Binary Number System

Conversion Method

Worked Example

Powers of 2 Reference

Why Computers Use Binary

Common Binary Patterns

Frequently Asked Questions

How do you convert binary to decimal?

Why do computers use binary instead of decimal?

What is the largest number an 8-bit byte can represent?

What are common binary patterns I should memorize?

How many binary digits are needed for a given decimal number?