Number Systems — Binary, Decimal, Octal, and Hexadecimal

We grow up counting in base 10, so other number systems feel alien at first. But they all work the same way — just with a different number of digits available. Understanding binary and hexadecimal is genuinely useful if you do anything with computers, programming, or networking. And even if you don't, it's one of those things that changes how you see numbers.

What Is a "Base"?

The base of a number system tells you how many unique digits it uses before rolling over to the next place value. Our everyday system is base 10 (decimal) — it has 10 digits: 0 through 9. When you run out of single digits, you move to the next column (ones → tens → hundreds...).

Binary is base 2 — only two digits: 0 and 1. Hex is base 16 — sixteen symbols: 0-9 plus A-F. Same logic, different column sizes.

The Four Main Number Systems

Name	Base	Digits Used	Common Use
Decimal	10	0–9	Everyday math
Binary	2	0, 1	Computer hardware, logic circuits
Octal	8	0–7	Unix file permissions, some legacy systems
Hexadecimal	16	0–9, A–F	Colors in web design, memory addresses, debugging

Binary — Base 2

Computers use binary because transistors (the basic building blocks of all digital hardware) have two states: on (1) and off (0). Every piece of data your computer processes is ultimately represented as a sequence of 0s and 1s.

Each digit in a binary number is called a bit. The columns represent powers of 2 (not powers of 10 like in decimal).

Decimal place values: ...1000 100 10 1
Binary place values: ... 8 4 2 1

Binary 1011 = (1×8) + (0×4) + (1×2) + (1×1)
= 8 + 0 + 2 + 1 = 11 in decimal

Decimal to Binary Conversion

The easiest method: repeatedly divide by 2, writing down the remainder each time. Read the remainders bottom to top.

Convert 25 to binary:
25 ÷ 2 = 12 remainder 1
12 ÷ 2 = 6 remainder 0
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1

Read bottom to top: 11001
Check: 16 + 8 + 0 + 0 + 1 = 25 ✓

Hexadecimal — Base 16

Hex needs 16 symbols but we only have 10 digits (0-9). So letters A-F fill in for values 10-15:

Decimal	10	11	12	13	14	15
Hex	A	B	C	D	E	F

In hex, each column is a power of 16 (1, 16, 256, 4096...).

Hex 2F = (2 × 16) + (15 × 1) = 32 + 15 = 47
Hex FF = (15 × 16) + (15 × 1) = 240 + 15 = 255

Why is FF = 255 significant? Because 255 is the maximum value that fits in one byte (8 bits), which is why RGB color values go from 0 to 255. In web design, #FF0000 means red at full intensity, green at zero, blue at zero — pure red.

Hexadecimal to Decimal

Convert 3A7 (hex) to decimal:
3 × 16² + A × 16¹ + 7 × 16⁰
= 3 × 256 + 10 × 16 + 7 × 1
= 768 + 160 + 7
= 935

Binary to Hexadecimal (The Fast Way)

One hex digit always represents exactly 4 binary digits (called a nibble). This makes converting between them extremely convenient — no arithmetic needed, just look up the 4-bit groups.

Binary	Hex	Decimal
0000	0	0
0001	1	1
0100	4	4
1000	8	8
1010	A	10
1111	F	15

Convert binary 11010110 to hex:
Split into groups of 4: 1101 0110
1101 = 13 = D
0110 = 6
Result: D6

This is exactly why programmers prefer hex over long strings of binary — D6 is a lot easier to read and type than 11010110, and they represent the same value.

Octal — Base 8

Octal uses digits 0-7. Each column is a power of 8. It was popular in older computing systems and is still used for Unix/Linux file permissions.

Octal 57 = (5 × 8) + (7 × 1) = 40 + 7 = 47 in decimal

Unix file permissions like chmod 755 use octal — 7 means full permissions (rwx = 4+2+1), 5 means read + execute (4+1). If you've ever set file permissions on a server, you've been using octal.

Quick Conversion Reference

Decimal	Binary	Octal	Hex
0	0	0	0
8	1000	10	8
10	1010	12	A
16	10000	20	10
32	100000	40	20
64	1000000	100	40
255	11111111	377	FF

Using the Base Converter

Converting between these systems by hand is fine for learning, but for practical work the SolveCalc base converter converts between decimal, binary, octal, and hex instantly. Just enter the number and the base you're converting from.

Conclusion

All number systems work on the same fundamental principle — place value and a fixed base. Once you understand that each column represents the next power of the base, you can work in any of them. Binary is the language of hardware, hex is the shorthand for binary that humans can actually read, decimal is what your brain uses naturally, and octal shows up in specific system contexts. The most useful pair to know is binary-to-hex (because of the clean 4-bit = 1 hex digit relationship), and if you do any web or systems work you'll use it regularly.