Number Base Converter

Continue in the tool

More pages for this tool

binary to decimal

Binary to Decimal Converter

Each binary digit carries an increasing power of 2 from right to left: the rightmost bit is worth 2⁰, the next 2¹, and so on up to the most significant bit, which alone accounts for the largest share of the final value. The tool adds up these weights in BigInt, accepting up to 256 digits in the source base and an optional minus sign, without the 32- or 64-bit ceiling that trips up ordinary calculators.

Positional weight and the most significant bit

The decimal value of a binary number comes from summing 2^k for each position k, counting from the right (k=0) to the left, whenever the bit there is 1. In 1011₂, the set bits sit at positions 0, 1 and 3: 2⁰+2¹+2³ = 1+2+8 = 11₁₀, the same result as multiplying each digit by its weight and adding. No conversion table is needed, just the sum of the powers of 2 matching the bits that are on.
The most significant bit alone accounts for nearly half of the total value: in an 8-bit byte, 10000000₂ is worth 128, exactly half of the largest representable value, 255 (11111111₂). Flipping only that bit to 0 and every other bit to 1, 01111111₂, results in 127, a drop of just 1 from 128 despite seven bits changing state. Adding one more position, 100000000₂ with 9 bits, already crosses the byte ceiling and reaches 256, the first value that no longer fits in 8 bits.
The tool computes in BigInt, not in Number or in the 32- or 64-bit integers most calculators use: it accepts up to 256 digits in the source base, enough for numbers far beyond what an ordinary processor represents natively. The negative sign is treated as a plain minus prefix over the absolute value, not as two's complement: -1010₂ converts to -10₁₀ by flipping just the sign, with no bit inversion or adding 1, the technique processors use internally to represent negative numbers.

Examples

Max byte

Input: 11111111₂
Expected output: 255₁₀

The whole byte turned on adds 128+64+32+16+8+4+2+1, every power of 2 from 2⁰ to 2⁷ at once.

IPv4 octet

Input: 11000000₂
Expected output: 192₁₀

The two highest bits, 128 and 64, already add up to 192 alone: it is the first octet in addresses like 192.168.x.x.

Crossing the byte boundary

Input: 100000000₂ (9 bits)
Expected output: 256₁₀

A ninth bit to the left no longer fits in a single byte: 256 is the first decimal value that 8 bits alone cannot represent, since the largest is 255.

Full tool FAQ

How do I convert binary to decimal by hand?

Multiply each bit by its weight, which is a power of 2 counted from the right starting at 2 to the power of 0, and add the results. For example, binary 1011 is 1x8 + 0x4 + 1x2 + 1x1 = 11. A quick check: 11111111 is always 255, because it is eight set bits, 2^8 - 1. To see the weight table laid out, use the Step by step tab.

How do I convert hexadecimal to decimal by hand?

How do I convert decimal to binary, fractions included?

Why is 0.1 not exact in binary?

What is two's complement?

What is IEEE 754?

Why do octal and hexadecimal exist?

Which bases are used in computing? Is base64 a numeric base?

What does the tool accept as input and what is the limit?

Are the numbers I type sent to a server?

Frequently asked questions

Do negative numbers work?

Yes: just prefix the binary value with a minus sign, like -1010₂, which the tool converts to -10₁₀. That differs from the two's complement processors use internally, where the highest bit marks the sign without a separate character.

Why does the most significant bit matter so much?

Because it alone accounts for nearly half of the total value: in an 8-bit byte, the MSB is worth 128, more than the sum of the other seven bits combined (127). Getting that one bit wrong changes the result more than getting any combination of the rest wrong.

Up to how many binary digits does the tool accept?

Up to 256 digits in the source base, computed entirely in BigInt: no floating-point rounding enters the calculation, unlike what would happen if it used JavaScript's Number type past 53 bits.

Next steps

Continue in the tool

Open full tool

binary to decimal

Binary to Decimal Converter

Positional weight and the most significant bit

The decimal value of a binary number comes from summing 2^k for each position k, counting from the right (k=0) to the left, whenever the bit there is 1. In 1011₂, the set bits sit at positions 0, 1 and 3: 2⁰+2¹+2³ = 1+2+8 = 11₁₀, the same result as multiplying each digit by its weight and adding. No conversion table is needed, just the sum of the powers of 2 matching the bits that are on.

The most significant bit alone accounts for nearly half of the total value: in an 8-bit byte, 10000000₂ is worth 128, exactly half of the largest representable value, 255 (11111111₂). Flipping only that bit to 0 and every other bit to 1, 01111111₂, results in 127, a drop of just 1 from 128 despite seven bits changing state. Adding one more position, 100000000₂ with 9 bits, already crosses the byte ceiling and reaches 256, the first value that no longer fits in 8 bits.

The tool computes in BigInt, not in Number or in the 32- or 64-bit integers most calculators use: it accepts up to 256 digits in the source base, enough for numbers far beyond what an ordinary processor represents natively. The negative sign is treated as a plain minus prefix over the absolute value, not as two's complement: -1010₂ converts to -10₁₀ by flipping just the sign, with no bit inversion or adding 1, the technique processors use internally to represent negative numbers.

Examples

Max byte

Input: 11111111₂
Expected output: 255₁₀

The whole byte turned on adds 128+64+32+16+8+4+2+1, every power of 2 from 2⁰ to 2⁷ at once.

IPv4 octet

Input: 11000000₂
Expected output: 192₁₀

The two highest bits, 128 and 64, already add up to 192 alone: it is the first octet in addresses like 192.168.x.x.

Crossing the byte boundary

Input: 100000000₂ (9 bits)
Expected output: 256₁₀

A ninth bit to the left no longer fits in a single byte: 256 is the first decimal value that 8 bits alone cannot represent, since the largest is 255.

Full tool FAQ

How do I convert binary to decimal by hand?

How do I convert hexadecimal to decimal by hand?

How do I convert decimal to binary, fractions included?

Why is 0.1 not exact in binary?

What is two's complement?

What is IEEE 754?

Why do octal and hexadecimal exist?

Which bases are used in computing? Is base64 a numeric base?

What does the tool accept as input and what is the limit?

Are the numbers I type sent to a server?

Frequently asked questions

Do negative numbers work?

Why does the most significant bit matter so much?

Up to how many binary digits does the tool accept?

Up to 256 digits in the source base, computed entirely in BigInt: no floating-point rounding enters the calculation, unlike what would happen if it used JavaScript's Number type past 53 bits.