Why do computers use binary?

Electronic circuits have two stable states — on (1) and off (0). Binary maps perfectly to this physical reality, making it reliable and simple to implement in hardware.

Why is hexadecimal used instead of binary?

Binary is verbose — 156 requires 8 digits in binary vs only 2 in hex. Since each hex digit maps exactly to 4 binary bits, hex is a compact human-readable way to represent binary data.

What is the 0x prefix?

0x is a notation prefix indicating that the number that follows is hexadecimal, not decimal. 0x9C = 156 decimal. Similarly, 0b before a number indicates binary.

How are Unix file permissions related to octal?

Unix permissions (rwx) for owner/group/others map to 3-bit binary for each group. chmod 755 means 111 101 101 in binary (7=111, 5=101) — read/write/execute for owner, read/execute for others.

Formula Guide

How to Convert Between Binary, Decimal, Octal & Hex

Number systems differ by their base — how many unique digits they use before rolling over to the next position. Decimal (base 10) uses 0–9. Binary (base 2) uses only 0 and 1 and is the native language of computers. Hexadecimal (base 16) uses 0–9 then A–F and is widely used in colour codes, memory addresses, and debugging. Octal (base 8) uses 0–7 and appears in Unix file permissions.

Last updated: March 31, 2026

The Formula

Decimal to Binary: divide by 2, record remainders bottom-to-top
Binary to Decimal: Σ(bit × 2^position) reading right to left from position 0
Decimal to Hex: divide by 16, record remainders (10=A, 11=B … 15=F)
Hex to Decimal: Σ(digit × 16^position)

Quick shortcut: each hex digit equals exactly 4 binary bits. F = 1111, A = 1010, etc. Convert binary → hex by grouping bits in sets of 4 from the right.

Variable Definitions

Symbol	Name	Description
Base	Radix	The number of unique digits in the number system (2 for binary, 8 for octal, 10 for decimal, 16 for hex)
bit	Binary Digit	A single 0 or 1 — the smallest unit of information in computing

Step-by-Step Example

Convert decimal 156 to binary and hexadecimal.

Given

Decimal value:156

Solution

1
Divide by 2 repeatedly: 156÷2=78 r0 | 78÷2=39 r0 | 39÷2=19 r1 | 19÷2=9 r1 | 9÷2=4 r1 | 4÷2=2 r0 | 2÷2=1 r0 | 1÷2=0 r1
2
Read remainders bottom to top: Binary = 10011100
3
Verify: 128+16+8+4 = 156: 10011100 → 128+0+0+16+8+4+0+0 = 156 ✓
4
Divide 156 by 16: 156 ÷ 16 = 9 remainder 12 (12 = C)
5
9 ÷ 16 = 0 remainder 9: Hex digits bottom to top: 9C

156 in decimal = 10011100 in binary = 0x9C in hexadecimal.

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Common Mistakes to Avoid

Reading remainders in the wrong order — for decimal to binary, read remainders from bottom to top (last remainder is the most significant bit).

Forgetting hex letters — 10=A, 11=B, 12=C, 13=D, 14=E, 15=F. A remainder of 12 is written as C.

Mixing up the 0x prefix — 0x9C means hex, not decimal. The prefix is just a notation convention.

Wrong bit position weighting — binary positions from right are 2⁰=1, 2¹=2, 2²=4, 2³=8, 2⁴=16, etc.

Frequently Asked Questions

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