Binary to Decimal

About Binary to Decimal

Binary to Decimal turns binary numbers into ordinary decimal numbers. Paste one for homework, or a list of bytes, flags or register values. Each line is converted, and a line with a wrong digit is marked at that digit.

Bits can be written in groups, like 1010 0101, with or without 0b. Set Signed when the bits are a signed byte or word, and the first bit makes the number negative.

The same tool, set up for other bases: Hex to Decimal, Decimal to Hex, Decimal to Binary, Hex to Binary, Binary to Hex, Number Base Converter.

How to use it

1. Paste binary numbers, one per line, or load a file. Commas, tabs and semicolons also split a line, so a column from a spreadsheet works.
2. Check From and To, next to Convert. On a phone they are under More.
3. Press Convert (Alt C). Results shows every number in the To base and in decimal, hex, binary and octal. Output shows only the answers, line for line, for Copy and Download.
4. Open Steps to see the math for one number, the way you would do it on paper.
5. Swap (Alt S) puts the answers in the editor and turns From and To around, so you can check the way back.
6. In Options: Signed for two's complement, Groups for 1111 0000 or 4,294,967,295, Prefix for 0x, 0b or 0o, and how many digits a fraction gets.

How to convert binary to decimal by hand

Write the powers of 2 over the bits, from 1 on the right. Add the powers that have a 1 under them:

Place   32  16   8   4   2   1
Bit      1   0   1   1   0   1

32 + 8 + 4 + 1 = 45

A second way, good for long numbers: start at 0, and for each bit from the left, double what you have and add the bit. 1, 2, 5, 11, 22, 45.

Powers of 2

2⁰ = 1       2⁸  = 256        2¹⁶ = 65,536
2¹ = 2       2⁹  = 512        2²⁴ = 16,777,216
2² = 4       2¹⁰ = 1,024      2³² = 4,294,967,296
2³ = 8       2¹¹ = 2,048      2⁵³ = 9,007,199,254,740,992
2⁴ = 16      2¹² = 4,096      2⁶⁴ = 18,446,744,073,709,551,616
2⁵ = 32
2⁶ = 64      8 bits hold 0 to 255
2⁷ = 128     16 bits hold 0 to 65,535

Negative numbers and two's complement

In a signed byte or word, a first bit of 1 makes the number negative. Set Signed to the width: at 16-bit, 1111111111111110 is -2.

How two's complement works, with a table and the method by hand.

Exact for big numbers

Masks and register values can be 64 bits long. This tool uses BigInt, so 64 ones give 18446744073709551615. A converter built on JavaScript numbers rounds them to 18446744073709552000.

Why other converters round past 2^53.

Common errors

A digit that is not 0 or 1

Binary has only two digits. A 2 means the number is not binary, or a key was hit by mistake.

10201
2 is not a binary digit. Binary uses only 0 and 1. Remove the 2,
or set From to base 3 or more.

10001

The letter O for a zero

Typed by hand, a capital O and a zero look the same. The tool names the letter, so you know what to type.

10O1
The letter O is not a digit. Type the digit 0 (zero) in its place.

1001

Two numbers on one line

Bits are often grouped in fours, as in 1111 0000. When the groups are not the same size, the tool still reads one number, 483 in decimal here, but marks the line, because it may be a list of bytes.

1111 000 11
Read as one number, 111100011. If these are separate numbers, put each
on its own line or split them with commas.

1111, 000, 11

A number too big for its bits

With Signed set to a width, every number must fit in it. A ninth bit does not fit in a byte.

100000000 (Signed: 8-bit)
100000000 does not fit in 8 bits. 8 bits hold up to 11111111 in binary.
Pick a wider Signed setting, or set it to Off.

Binary to decimal in code and Excel

JavaScript  parseInt("101101", 2)          // 45
            BigInt("0b" + bits)            // exact at any length
Python      int("101101", 2)               # 45; "0b101101" works too
C           strtol("101101", NULL, 2)      // 45, <stdlib.h>
Java        Integer.parseInt("101101", 2)  // 45
            new BigInteger(bits, 2)        // any length
Excel       =BIN2DEC("101101")             → 45

BIN2DEC takes up to 10 bits, and the 10th bit is the sign: =BIN2DEC("1111111111") is -1. For a longer number, split it or use the tool above.