Binary Converter: Convert Between Binary, Decimal, Octal and Hexadecimal

Every number system is positional: a digit's value depends on its position. In base b, the digit string d_nd_n-1…d₁d₀ means d_n×bⁿ + d_n-1×b^n-1 + … + d₁×b¹ + d₀×b⁰. The only thing that changes between binary, octal, decimal and hexadecimal is b and the allowed digits.

Binary to decimal

Take 10110100. Number the bits from the right, starting at 2⁰: 1×2⁷ + 0×2⁶ + 1×2⁵ + 1×2⁴ + 0×2³ + 1×2² + 0×2¹ + 0×2⁰ = 128 + 32 + 16 + 4 = 180.

Decimal to binary

Divide by 2 repeatedly and read the remainders bottom-up. 180 ÷ 2 = 90 remainder 0; 90 ÷ 2 = 45 remainder 0; 45 ÷ 2 = 22 remainder 1; 22 ÷ 2 = 11 remainder 0; 11 ÷ 2 = 5 remainder 1; 5 ÷ 2 = 2 remainder 1; 2 ÷ 2 = 1 remainder 0; 1 ÷ 2 = 0 remainder 1 — reading the remainders from last to first gives 10110100.

Binary, octal and hex shortcuts

Because 8 = 2³ and 16 = 2⁴, conversions between these three are digit-by-digit: group binary digits in threes for octal and in fours for hexadecimal. 1111 1111 becomes F F in hex and 377 in octal — the same value, three skins.

A base converter takes one number and shows it in every base at once, so you never convert twice by hand. This converter handles binary, decimal, octal and hexadecimal in a single pass.

Step by step

1. Type the value — letters A–F are allowed for hexadecimal input.
2. Choose the base it is written in from the "This value is in" list.
3. Press Convert. The results show the value in all four bases immediately.
4. Read the extra lines under the results: the minimum bit length and the signed/unsigned note for the decimal value.

What you get

• Binary — shown in groups of 4 bits for readability, e.g. 1111 1111
• Decimal — plain base-10, with thousands separators
• Octal — base 8, handy for old-school file permissions
• Hexadecimal — uppercase base 16, the compact form programmers read
• Bit length — how many bits the value needs, and the standard width it fits in
• Signed/unsigned note — how the same decimal value reads as a signed 32-bit two's complement integer

Invalid digits

Each base only accepts its own digits. If you type 102 while Binary is selected, the converter refuses and tells you exactly which digit is wrong — 2 is not a binary digit. Switch the base selector to match the value, or fix the digit, and convert again.

The four bases are four ways of writing the same numbers. Binary mirrors the on/off states of a circuit, decimal is what people count in, octal survived in Unix file permissions, and hexadecimal is the compact shorthand that programmers read every day.

Why hex is the favorite shorthand

One hexadecimal digit covers exactly 4 bits, so a byte is always two hex digits. 1111 1111 is hard to scan; FF is not. That is why error codes, memory addresses and color codes are printed in hex — it is a faithful, compressed picture of the underlying bits. For heavier math on top of these numbers, our scientific calculator evaluates trig, logarithms, powers and roots.

After each conversion the tool reports how many bits the value needs. The rule is simple: the bit length is the position of the highest set bit. 255 is 1111 1111, so it needs 8 bits. 256 is 1 0000 0000, so it needs 9 bits — one more bit than the number it just beat.

Rounding up to a standard width

Real hardware stores bits in fixed widths: 8, 16, 32 or 64 bits. A value that needs 9 bits cannot live in a byte, so it is stored in 16 bits. The converter shows the smallest standard width that fits your value — 8 bits (1 byte), 16 bits (2 bytes), 32 bits (4 bytes), 64 bits (8 bytes), and beyond.

Unsigned versus signed decimal

An 8-bit register holding 1111 1111 is 255 if the bits are read as unsigned, but −1 if they are read as signed two's complement, where the leftmost bit means negative. Nothing in the bits changes — only the interpretation. The converter shows the signed 32-bit reading whenever the value fits, so you can see both readings side by side. If you want to explore what bits mean for storage sizes, try the data storage converter to move between bits, bytes, kilobytes and gigabytes.

The two's complement trick

In two's complement, a 32-bit pattern's signed value equals the unsigned value minus 2³² when the top bit is set. So 11111111 11111111 11111111 11111111 is 4,294,967,295 unsigned and 4,294,967,295 − 4,294,967,296 = −1 signed. This is the standard the two's complement article on Wikipedia describes, and it is how virtually all modern processors store negative integers.

Some conversions come up constantly. The table below covers the byte boundaries — the values where the bit length jumps, which is exactly when storage sizes change.

When the input is rejected

• A digit that does not belong: 2–9 in binary, 8–9 in octal, G–Z in hex. The error names the offending digit.
• Wrong base selected: make sure the selector matches the notation you typed.
• Negative values: a leading minus is accepted and converts the magnitude; the outputs are shown for the positive value with the sign preserved on decimal.
• Very large values: the converter uses arbitrary-precision integers, so numbers far beyond 2⁵³ convert exactly — output strings can simply get long.