Number Systems Formulas

Mathematical formulas and algorithms for converting between different number systems and positional notations

1Positional Notation System

General Formula

Where:

N = decimal value of the number
dᵢ = digit at position i
r = radix (base) of the number system
i = position index (starting from 0 for rightmost digit)
n = total number of digits

Extended Formula with Fractional Parts

For numbers with fractional parts:

Common Bases

Binary

Base 2

Digits: 0, 1

Octal

Base 8

Digits: 0-7

Decimal

Base 10

Digits: 0-9

Hexadecimal

Base 16

Digits: 0-9, A-F

2Base Conversion Algorithms

Any Base to Decimal

Algorithm: Horner's Method

Pseudocode:

result = 0
for each digit d from left to right:
    result = result × base + d
return result

Decimal to Any Base

Algorithm: Repeated Division

Where qᵢ are quotients, r is the target base, and dᵢ are remainder digits.

Pseudocode:

digits = []
while N > 0:
    remainder = N mod base
    digits.prepend(remainder)
    N = N ÷ base
return digits

3Binary System (Base 2)

Binary to Decimal

Example: (1101)₂ to decimal

Binary Arithmetic Operations

Binary Addition Rules

0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 10 (0 with carry 1)

Binary Multiplication Rules

0 × 0 = 0
0 × 1 = 0
1 × 0 = 0
1 × 1 = 1

Binary-Decimal Conversion Shortcuts

Powers of 2

2⁰ = 1

2¹ = 2

2² = 4

2³ = 8

2⁴ = 16

2⁵ = 32

2⁶ = 64

2⁷ = 128

2⁸ = 256

2⁹ = 512

2¹⁰ = 1024

2¹¹ = 2048

4Octal System (Base 8)

Octal to Decimal

Example: (157)₈ to decimal

Binary-Octal Conversion

Direct Conversion Method

Each octal digit corresponds to exactly 3 binary digits:

0₈ = 000₂

1₈ = 001₂

2₈ = 010₂

3₈ = 011₂

4₈ = 100₂

5₈ = 101₂

6₈ = 110₂

7₈ = 111₂

Powers of 8

8⁰ = 1

8¹ = 8

8² = 64

8³ = 512

8⁴ = 4,096

8⁵ = 32,768

5Hexadecimal System (Base 16)

Hexadecimal to Decimal

Example: (2AF)₁₆ to decimal

Hexadecimal Digits

0₁₆ = 0₁₀

1₁₆ = 1₁₀

2₁₆ = 2₁₀

3₁₆ = 3₁₀

4₁₆ = 4₁₀

5₁₆ = 5₁₀

6₁₆ = 6₁₀

7₁₆ = 7₁₀

8₁₆ = 8₁₀

9₁₆ = 9₁₀

A₁₆ = 10₁₀

B₁₆ = 11₁₀

C₁₆ = 12₁₀

D₁₆ = 13₁₀

E₁₆ = 14₁₀

F₁₆ = 15₁₀

Binary-Hexadecimal Conversion

Direct Conversion Method

Each hexadecimal digit corresponds to exactly 4 binary digits:

0₁₆ = 0000₂

1₁₆ = 0001₂

2₁₆ = 0010₂

3₁₆ = 0011₂

4₁₆ = 0100₂

5₁₆ = 0101₂

6₁₆ = 0110₂

7₁₆ = 0111₂

8₁₆ = 1000₂

9₁₆ = 1001₂

A₁₆ = 1010₂

B₁₆ = 1011₂

C₁₆ = 1100₂

D₁₆ = 1101₂

E₁₆ = 1110₂

F₁₆ = 1111₂

Powers of 16

16⁰ = 1

16¹ = 16

16² = 256

16³ = 4,096

16⁴ = 65,536

16⁵ = 1,048,576

6Fractional Number Conversions

Fractional Part to Decimal

Example: (0.101)₂ to decimal

Decimal Fraction to Any Base

Algorithm: Repeated Multiplication

Pseudocode:

fraction = decimal_part
digits = []
while fraction ≠ 0 and precision not reached:
    fraction = fraction × base
    digit = floor(fraction)
    digits.append(digit)
    fraction = fraction - digit
return "0." + digits

7Arithmetic in Different Bases

Addition in Base r

When the sum of digits ≥ r, carry 1 to the next position and subtract r from the sum.

Multiplication in Base r

Similar to decimal multiplication, but use base r for carries.

Quick Reference Table

Decimal	Binary	Octal	Hexadecimal
0	0	0	0
1	1	1	1
2	10	2	2
3	11	3	3
4	100	4	4
5	101	5	5
6	110	6	6
7	111	7	7
8	1000	10	8
9	1001	11	9
10	1010	12	A
11	1011	13	B
12	1100	14	C
13	1101	15	D
14	1110	16	E
15	1111	17	F
16	10000	20	10

Try Number System Converters