Binary to Decimal Converter — Free, Both Ways

How to Use the Binary Converter

1. Type a binary number (only 0s and 1s) in the Binary box — the decimal value appears instantly.
2. Or type a decimal number in the Decimal box — the binary equivalent appears instantly.
3. Hex, octal and bit-count update automatically with every keystroke.
4. Invalid input (like the digit 2 in binary) shows a clear error instead of a wrong answer.

How to Convert Binary to Decimal by Hand

Short answer: write each bit over its power of 2 (1, 2, 4, 8, 16… from the right), multiply, and add. Four steps:

1. Write the binary number and number each position from the right, starting at 0. The rightmost bit is position 0.
2. Find each bit’s place value: the bit in position n is worth 2ⁿ — so the positions are worth 1, 2, 4, 8, 16, 32…
3. Multiply each bit (0 or 1) by its place value.
4. Add everything up. Zeros contribute nothing, so in practice you only add the values sitting under the 1s.

This is the method exams test, and it is exactly what the converter above automates — type the same number into the Binary box to check your hand calculation.

Binary Place-Value Table

Every binary digit sits in a position worth a power of 2. Bit 0 (the rightmost) is worth 2⁰ = 1, and each step left doubles the value:

Memorize the landmarks: 2¹⁰ = 1,024 (≈ 1 thousand), 2²⁰ = 1,048,576 (≈ 1 million), 2³⁰ = 1,073,741,824 (≈ 1 billion). Computer memory sizes — KB, MB, GB — are built on these.

Worked Example: What Is 101101 in Decimal?

Convert 101101₂ using the hand method. Line the bits up with their place values:

One more, the classic textbook example: 111001₂ = 32 + 16 + 8 + 1 = 57₁₀. Going the other way — decimal to binary — divide by 2 repeatedly and read the remainders backwards: 25 → 25÷2=12 r1, 12÷2=6 r0, 6÷2=3 r0, 3÷2=1 r1, 1÷2=0 r1, so 25₁₀ = 11001₂.

What Is the Binary-to-Decimal Formula?

Formally, an n-bit binary number dₙ₋₁…d₁d₀ converts as:

Two terms worth knowing: the rightmost bit (d₀) is the LSB (least significant bit) and the leftmost is the MSB (most significant bit). The formula is the same idea as decimal notation: in 507, the 5 means 5×100 because it sits in the hundreds place — binary just uses powers of 2 instead of powers of 10.

Shortcut — the doubling method: scan the bits left to right. Start at 0; for each bit, double your running total and add the bit. For 1011: 0 → 1 → 2 → 5 → 11. No powers to memorize, and it works for numbers of any length.

Binary to Decimal Conversion Table

The first 21 values, with hex equivalents — the range homework questions live in:

Common Binary Values Worth Memorizing

A handful of values show up constantly in computing — recognizing them by sight saves real time:

Pattern to remember: n ones always equal 2ⁿ − 1 — that is why 11111111 is 255 and not 256.

How Does Two’s Complement Work? (Signed Binary)

Short answer: in two’s complement, the leftmost bit carries a negative weight — so 8-bit 11111111 reads as −1, not 255.

Computers store negative integers with two’s complement: to negate a number, invert every bit and add 1. Reading a signed value works in reverse — for 11111111: invert → 00000000, add 1 → 1, so the value is −1. The tradeoff: one bit is spent on the sign, so a signed 8-bit number runs from −128 to +127 (the max, 01111111, is 127).

Note: the converter above shows the unsigned (whole-number) value — type 11111111 and you get 255. Two’s complement is the programmer’s signed interpretation of the same bits.

Binary to Hex and Octal: The Grouping Trick

Short answer: chop the bits into groups of 4 (from the right) for hex, or groups of 3 for octal, then convert each small group.

Because 16 = 2⁴, one hex digit replaces exactly 4 bits. Pad 101101 to 0010 1101: 0010 = 2, 1101 = 13 = D, so 101101₂ = 2D₁₆ (check: 2×16 + 13 = 45 ✓).

Octal works the same with groups of 3 (8 = 2³): 101 101 → 5, 5 → 55₈ (check: 5×8 + 5 = 45 ✓). This is exactly what the Hex and Octal chips above the tool compute on every keystroke — try typing any binary number and watch them update.

Can Binary Numbers Have Decimal Points?

Short answer: yes — binary fractions work like decimal fractions, but with halves. Bits after the point are worth ½, ¼, ⅛… So 0.101₂ = 1×½ + 0×¼ + 1×⅛ = 0.625₁₀.

The catch: just as ⅓ has no finite decimal form (0.333…), some decimals have no finite binary form — 0.1₁₀ repeats forever in binary. That is the real reason computers sometimes show 0.1 + 0.2 as 0.30000000000000004.

Common Mistakes to Avoid

1. Reading left to right. Binary place values start at the right. In 1011, the leftmost 1 is worth 8, not 1.
2. Forgetting the LSB. The rightmost bit is always worth 2⁰ = 1. Every odd binary number ends in 1.
3. Using digits other than 0 and 1. A “2” in a binary string is invalid — the converter flags it instead of guessing.
4. Reading binary digits as a decimal number. 101 is “one-zero-one”, worth 5 — not “one hundred one”.
5. Thinking leading zeros change the value. 00101₂ = 101₂ = 5. Padding is cosmetic — except when it marks the bit-width for two’s complement.

Where Binary-to-Decimal Matters

IP addresses and subnet masks are four 8-bit numbers in disguise: 255.255.255.0 is 11111111.11111111.11111111.00000000 — try decoding one in our IP Subnet Calculator. Text encoding: the letter “A” is ASCII 65 (1000001₂). Color codes like #FF0000 are hex — each pair of digits is one binary byte (FF = 255 = full red). File permissions (chmod 755) are octal: 7 = 111, 5 = 101. And debugging: reading memory dumps, error flags, and hardware registers means converting binary to decimal by sight.

Binary Converter FAQs