Number systems

Binary: Known as Base₂
- Digits: 0 & 1
Decimal: Known as Base₁₀
- Digits: 0-9
Octal: Known as Base ₈
- Digits: 0-7
Hexadecimal: Known as Base₁₆,
- Uses both numbers and characters.
- Digits: 0-9 | A-F
  - 10=A
  - 11=B
  - 12=C
  - 13=D
  - 14=E
  - 15=F
Binary Coded Decimal (BCD): Notes will be made later.

Positional notation

When we write decimal (Base₁₀) numbers, positional notation is used. Each digit is multiplied by an appropriate number of 10 depending on its position in the number.

Number System	Example	Positional Expansion	Base-10 Conversion
Base 10	`17₁₀`	`1 × 10¹ + 7 × 10⁰`	`10 + 7 = 17₁₀`
Base 10	`1216₁₀`	`1 × 10³ + 2 × 10² + 1 × 10¹ + 6 × 10⁰`	`1000 + 200 + 10 + 6 = 1216₁₀`
Binary	`1011₂`	`1 × 2³ + 0 × 2² + 1 × 2¹ + 1 × 2⁰`	`8 + 0 + 2 + 1 = 11₁₀`
Octal	`731₈`	`7 × 8² + 3 × 8¹ + 1 × 8⁰`	`448 + 24 + 1 = 473₁₀`
Hexadecimal	`AD14₁₆`	`10 × 16³ + 13 × 16² + 1 × 16¹ + 4 × 16⁰`	`40960 + 3328 + 16 + 4 = 44308₁₀`

Digit	Positional Value
`1`	`1 × 2²`
`0`	`0 × 2¹`
`1`	`1 × 2⁰`
`.`	Binary point
`0`	`0 × 2⁻¹`
`1`	`1 × 2⁻²`
`1`	`1 × 2⁻³`

Place-Value, Binary

Decimal	Binary
Ones, tens, hundredths, etc.	Ones, twos, fours, eighths, etc.

Weight	64	32	16	8	4	2	1
Value	1	1	0	1	0	1	1

8	4	2	1	Digit
0	0	0	1	1
0	0	1	0	2
0	0	1	1	3
0	1	0	0	4
0	1	0	1	5
0	1	2	0	6
0	1	1	1	7
1	0	0	0	8

Binary Coded Decimal (BCD)

BCD splits the digits of a decimal into 4-bit binaries.

Decimal	BCD
`23.12₁₀`	`00100011.00010010₂`
`79.92₁₀`	`01111001.10010010₂`

Octal → Binary

Octal digit	Binary
`3`	`011`
`5`	`101`
`7`	`111`
Therefore: `357₈` → `011101111₂`

Number-System Conversion

Number System	Representation
Decimal	`239₁₀`
Octal	`357₈`
Hexadecimal	`EF₁₆`
Binary	`11101111₂`
BCD	`001000111001`