Number Base Converter

• Converting numbers between bases — you are on the right page.
• Converting text to and from character codes — use the binary to text converter.
• Converting negative numbers — use the two’s complement converter.

People search for a binary converter meaning either of the first two; this page does the number one.

A number written in base b is a sum of its digits multiplied by powers of b: value = Σ dᵢ·bⁱ, where the exponent is the digit’s distance from the radix point — counting up to the left and down to the right. Base b uses the digits 0 to b−1; hexadecimal borrows the letters A to F for the values 10 to 15, because there are no single decimal digits for them.

Write the power of two above each column and add the columns that hold a 1. For 11010110₂: 1×2⁷ + 1×2⁶ + 0×2⁵ + 1×2⁴ + 0×2³ + 1×2² + 1×2¹ + 0×2⁰ = 128 + 64 + 0 + 16 + 0 + 4 + 2 + 0 = 214₁₀. The zero terms are written out rather than skipped: the method is what you are graded on, and a dropped column is where the digits slip.

Divide by two repeatedly and keep the remainders: 156 → 78 r0 → 39 r0 → 19 r1 → 9 r1 → 4 r1 → 2 r0 → 1 r0 → 0 r1, giving 10011100₂.

The remainders are read bottom to top — from the last division back to the first. Reading them top to bottom is the single most common error in this conversion, and it produces a number that looks plausible and is wrong.

Between base 2, 8 and 16 you do not need division: 8 = 2³ and 16 = 2⁴, so three or four binary digits map to exactly one octal or hexadecimal digit. Group 110111011010₂ into fours from the radix point — 1101 1101 1010 — and read each group as a digit: D, D, A, so DDA₁₆ (3546₁₀). Backwards, 2AF₁₆ expands to 0010 1010 1111 = 1010101111₂ (687₁₀), where the two leading zeros are padding rather than data.

Octal works the same way with groups of three. 11110110111₂ is eleven bits, so the integer side is padded on the left to twelve: 011 110 110 111 → 3, 6, 6, 7 → 3667₈ (1975₁₀). Octal survives in Unix file permissions for exactly this reason: three permission bits are one octal digit.

Repeated division takes any base as its divisor. 1975 ÷ 16 gives 123 r7, then 7 r11, then 0 r7 — and a remainder of 11 is written B, so 1975₁₀ = 7B7₁₆. Dividing the same number by 8 gives 246 r7, 30 r6, 3 r6, 0 r3, so 1975₁₀ = 3667₈, which agrees with the grouping above.

A fractional part converts by repeated multiplication, and its integer parts are read top to bottom — the opposite direction to the division table. For 0.625: 0.625 × 2 = 1.25 (digit 1), 0.25 × 2 = 0.5 (digit 0), 0.5 × 2 = 1.0 (digit 1), giving .101₂. With the integer part, 13.625₁₀ is 1101.101₂.

Grouping runs outward from the radix point: pad the integer side on the left and the fraction side on the right. 1101.101₂ is D.A₁₆ — the fraction 101 pads right to 1010 = A.

Non-integers inside a computer are usually not stored this way at all: they use IEEE 754 binary floating point, whose layout is described on the two’s complement page and standardised in IEEE 754-2019 (opens in a new tab).

A fraction terminates in base b if and only if, written in lowest terms, every prime factor of the denominator is also a factor of b.

1/10 = 1/(2·5), and 5 does not divide 2, so 0.1 has no terminating binary form: it is 0.0(0011)₂, one non-repeating digit followed by a four-digit repetend. By the same rule 1/3 is 0.(01)₂ with period 2, 1/7 is 0.(001)₂ with period 3, and 1/5 in octal is 0.(1463)₈ with period 4. This is also why 0.1 + 0.2 is not 0.3 in any language that stores numbers in binary floating point.

A bit pattern on its own has no sign. 10110110is 182 read as unsigned and −74 read as 8-bit two’s complement. Only the width and the convention you declare decide which.

This page shows the unsigned reading in the four result cards and the signed reading beside them whenever the value is an integer that fits the chosen width. The two’s complement converter shows the complement chain, the range check and sign extension in full.

Every conversion here runs on arbitrary-precision integers and exact rationals, so a value above 2⁵³−1 is exact — which it is not in converters built on double-precision numbers. The demonstration is 2⁶⁴ − 1 = 18446744073709551615 = FFFFFFFFFFFFFFFF₁₆. A tool that answers 18446744073709552000 is telling you about its own arithmetic, not about your number.

• Most significant digit first, left to right; a subscript names the base.
• The prefixes 0b, 0o and 0x are accepted on input and never emitted on output.
• Digits A–F are case-insensitive on input and upper case on output.
• A repeating expansion is marked with an overline on screen and with parentheses in copied text, as 0.0(0011).
• Fractions are truncated at the digit budget by default rather than rounded. Textbooks differ; state which convention your course expects.

Positional notation and radix conversion are stated here as definitions rather than attributed to a source. The one external standard this page relies on is IEEE 754-2019, IEEE Standard for Floating-Point Arithmetic (opens in a new tab), cited in the section on fractions.