Positional number systems

Number system is a way of writing numbers using certain characters.

Let’s look at the most common positional systems – depending on the location (digit) in the notation of the number, the same sign has different meanings.

Integer “X” in the positional number system can be expressed as follows:

• b – base of the system
• a_k are the digits of the number (0 ≤ a_k ≤ b-1)
• k – number of digits

An expanded form of writing an integer:

Binary number system: base – 2

Used in discrete mathematics, computer science and programming. Contains only two digits – 0 and 1. The number written in this system is indicated by the letter B at the end (prefix).

examples:

• 10101₂ = 10101B = 1×2⁴+ 0 × 2³+ 1 × 2²+ 0 × 2¹+ 1 × 2⁰ = 16+4+1= 21
• 10111₂ = 10111B = 1×2⁴+ 0 × 2³+ 1 × 2²+ 1 × 2¹+ 1 × 2⁰ = 16+4+2+1= 23
• 100011₂ = 100011B = 1×2⁵+ 0 × 2⁴+ 0 × 2³+ 0 × 2²+ 1 × 2¹+ 1 × 2⁰ =32+2+1= 35

Octal number system: base – 8

Eight digits are used to write a number – from 0 to 7.

examples:

• 27₈ = 2 × 8¹+ 7 × 8⁰ = 16+7 = 23
• 30₈ = 3 × 8¹+ 0 × 8⁰ = 24
• 4307₈ = 4 × 8³+ 3 × 8²+ 0 × 8¹+ 7 × 8⁰= 2247

Decimal number system: base -10

The most common system that is used everywhere. Contains numbers from 0 to 9.

Example:

2538₁₀ = 2 × 10³+ 5 × 10²+ 3 × 10¹+ 8 × 10⁰

Hexadecimal number system: base – 16

Numbers from 0 to 9 are used, as well as letters from A to F. Numbers are denoted by a prefix H. The system is used in computer science and programming.

examples:

• 28₁₆ = 28H = 2×16¹+ 8 × 16⁰ = 40
• 2F₁₆ = 2FH = 2×16¹+ 15 × 16⁰ = 47
• BC12₁₆ = BC12H = 11×16³+ 12 × 16²+ 1 × 16¹+ 2 × 16⁰= 48146

Table of correspondence of numbers of number systems