Binary, Decimal and Hex Conversion: A Step-by-Step Guide

Programmers move between number systems constantly: decimal for people, binary for bits, and hexadecimal as a compact way to write binary. This guide shows how to convert between all three by hand, using the number 46 as a running example, plus the shortcuts that make it fast.

Positional Number Systems in One Paragraph

In any positional system, each digit is multiplied by the base raised to its position, counting from 0 on the right. Decimal 46 is 4×10¹ + 6×10⁰. Binary uses base 2 and hexadecimal uses base 16, but the rule is identical. Only the base changes.

System	Base	Digits	46 is written
Binary	2	0–1	101110
Octal	8	0–7	56
Decimal	10	0–9	46
Hexadecimal	16	0–9, A–F	2E

Binary to Decimal

Write the place values (powers of two) above each bit and add the ones that have a 1:

place:  32  16   8   4   2   1
bit:     1   0   1   1   1   0
sum:    32 + 8 + 4 + 2 = 46

Memorising the powers of two up to 2¹⁶ (65,536) makes this very quick: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536.

Decimal to Binary

Method 1, repeated division by 2. Divide, write down the remainder, and repeat with the quotient. Read the remainders from bottom to top.

46 ÷ 2 = 23  r 0
23 ÷ 2 = 11  r 1
11 ÷ 2 =  5  r 1
 5 ÷ 2 =  2  r 1
 2 ÷ 2 =  1  r 0
 1 ÷ 2 =  0  r 1
reading upward: 101110

Method 2, subtract powers of two. Take the largest power of two that fits, subtract it, and repeat. 46 − 32 = 14, 14 − 8 = 6, 6 − 4 = 2, 2 − 2 = 0. The powers used are 32, 8, 4, and 2, so bits 5, 3, 2, and 1 are set: 101110.

The second method is faster in your head once you know the powers of two.

Binary to Hex and Back

This is the easiest conversion, because one hex digit is exactly four bits. Group the binary number into fours from the right, pad the leftmost group with zeros, and replace each group:

101110  →  0010 1110  →  2 E  →  0x2E

Going the other way, expand each hex digit to four bits: 0x4A → 0100 1010.

Hex	Binary	Hex	Binary
0	0000	8	1000
1	0001	9	1001
2	0010	A	1010
3	0011	B	1011
4	0100	C	1100
5	0101	D	1101
6	0110	E	1110
7	0111	F	1111

Octal works the same way with groups of three: 101 110 → 5 6 → 0o56. That is why Unix permissions, which come in groups of three bits, use octal. See Unix File Permissions in Binary.

Hex to Decimal and Decimal to Hex

Hex to decimal: multiply each digit by its power of 16. 0x2E = 2×16 + 14 = 46. For a longer example, 0x1F4 = 1×256 + 15×16 + 4 = 500.

Decimal to hex: divide repeatedly by 16 and read the remainders upward, or convert to binary first and then group by four. 500 ÷ 16 = 31 r 4, 31 ÷ 16 = 1 r 15 (F), 1 ÷ 16 = 0 r 1, giving 0x1F4.

Quick Reference: Values Worth Memorising

Decimal	Binary	Hex	Why it matters
15	1111	0xF	one nibble, all ones
127	0111 1111	0x7F	largest signed byte
128	1000 0000	0x80	sign bit of a byte
255	1111 1111	0xFF	largest unsigned byte
1023	ten ones	0x3FF	10-bit maximum (common ADC range)
65535	sixteen ones	0xFFFF	largest unsigned 16-bit
2147483647	0 + 31 ones	0x7FFFFFFF	largest signed 32-bit

Negative Numbers

All the methods above are for non-negative values. Negative integers are stored in two’s complement, so −46 in 8 bits is 11010010 (0xD2), not −101110. Converting those is covered in Two’s Complement Explained.

Converting in Code

bin(46), oct(46), hex(46)       # ('0b101110', '0o56', '0x2e')
format(46, "08b")               # '00101110'
int("101110", 2), int("2E", 16), int("56", 8)   # (46, 46, 46)

(46).toString(2);               // "101110"
(46).toString(16).toUpperCase(); // "2E"
parseInt("101110", 2);          // 46
Number("0x2E");                 // 46

#include <stdio.h>
#include <stdlib.h>
printf("%X %o\n", 46, 46);      /* 2E 56 */
long n = strtol("101110", NULL, 2);   /* 46 */

Practice

The fastest way to build fluency is to predict an answer and then check it. Type a value into the Bitwise Calculator in one base and read the result in the other two, or use the Bitwise Hex Calculator to see each hex digit expanded into its four bits.