Binary, decimal and hex: converting by hand

To convert binary to decimal, add up the powers of two at every position holding a 1; to get hexadecimal, group the bits in fours and write one digit per group. The binary, decimal and hex converter checks each answer instantly: type a value in one base and the others update, all in your browser. Knowing the method pays off beyond exams, since it explains character codes, file permissions, colors and hashes.

What is a number base?

We count in decimal, base 10, with ten digits, and each position is worth ten times the previous one. Binary (base 2) has just 0 and 1, and each position doubles. Hexadecimal (base 16) has sixteen digits, 0 to 9 and a to f, and each position is worth sixteen times the last. Octal (base 8) uses the digits 0 to 7.

Binary to decimal, step by step

Take 10110110. Write each position's value from right to left: 1, 2, 4, 8, 16, 32, 64, 128. Add the positions that hold a 1:

1. The positions with a 1 are 128, 32, 16, 4 and 2.
2. 128 + 32 + 16 + 4 + 2 = 182.

So 10110110 in binary is 182 in decimal.

Decimal to binary

Divide by 2 repeatedly and write the remainders from bottom to top. For 203 the result is 11001011. In octal it's 313 and in hexadecimal cb. A quick check is to type 203 into the converter's decimal field and read the other three.

Binary to hex: four bits at a time

One hex digit equals exactly 4 bits, so you convert by grouping. 10110110 splits into 1011 and 0110. The first group is 11, written b, and the second is 6. The result is b6. The tool has a group-in-nibbles option so you can see those blocks.

That is why hex is everywhere: it's a compact way to write bytes. A byte is 8 bits, which is two hex digits, and 00 to ff covers 256 values, the same as 0 to 255.

Three places you'll meet it

• ASCII and text. The capital letter A has the code 65, which is 41 in hex and 01000001 in binary. Seeing character codes in hex is routine in encoding bugs and in URL encoding, where %41 stands for A.
• File permissions. On Unix-like systems 755 is octal. In binary it's 111 101 101, meaning read, write and execute for the owner, and read and execute for everyone else.
• Colors and hashes. The color #3b82f6 is three bytes in hex: 3b is 59, 82 is 130 and f6 is 246, as HEX vs RGB vs HSL shows. A SHA-256 hash is 64 hex characters, so 256 bits, as discussed in MD5 vs SHA-256.

How to use the converter

1. Open the converter and type a value into any of the four fields: binary, octal, decimal or hexadecimal. The others update.
2. If your number has a prefix (0x for hex, 0b for binary, 0o for octal), the tool recognizes it when it matches the base. It also ignores spaces and underscores used for grouping, and accepts a negative sign.
3. Switch on Group in nibbles to read binary in blocks.
4. For other bases, use the arbitrary base section, from 2 to 36, with a source and a target base. In base 36, for example, zz is 1295.
5. Check the quick reference table for 0 to 16.

One technical advantage: the tool computes with arbitrary-size integers, so a huge number such as a 64-character hash goes back and forth without losing a single digit. Ordinary floating-point arithmetic in many languages loses precision beyond a certain size.

Try three on your own

Convert 45 to binary, 2f to decimal and 11111111 to hex, then check them in the tool. The answers are 101101, 47 and ff. The last one is the largest value a single byte can hold, 255, which is why you see ff so often.

Mistakes people make by hand

• Counting positions left to right instead of right to left.
• Forgetting that in hex a is 10, b is 11 and so on up to f, which is 15.
• Confusing bases: 10 can be ten, two or sixteen depending on where you are.
• Grouping bits from the left. Group from the right, and pad the leftmost group with zeros if it comes up short.

When in doubt, write the base with a prefix or a subscript.

Frequently asked questions

Related tools

• Base Converter: binary, octal, decimal and hex, plus bases 2 to 36.
• Hash Generator: hashes are written in hexadecimal.
• Color Converter: a HEX color is three bytes in hexadecimal.
• Base64: another way to represent bytes as text.

You might also like: what is Base64? Encoding explained in plain English and Unix timestamp explained and the 2038 problem.