Number System

Author: Ravi Poswal

Number System

A number system is simply a way of writing and representing numbers using symbols. We use numbers every day to count, calculate, and measure things.

Computers, however, are electronic machines built from millions of tiny switches. A switch can only be in one of two states — ON or OFF. Because of this, computers cannot understand our normal (decimal) numbers directly, and they need a number system with only two symbols. This is why the Binary Number System is so important in computers.

There are two broad types of number systems:

  1. Non-positional number system
  2. Positional number system

1. Non-Positional Number System

In this system, every symbol has a fixed value. It means the same thing no matter where it is placed in the number.

Example — Tally-mark style counting

I = 1     II = 2     III = 3     IIII = 4     IIIII = 5

Here, the symbol "I" always equals 1 — whether it comes first, second, or last in the group. The symbols are just added together to get the total value.

Problem with this system

It becomes very difficult to do arithmetic (addition, subtraction, multiplication) using such symbols. That is why this system is not practical for computers or for everyday calculations.

2. Positional Number System

In this system, a symbol's value changes depending on where (which position) it is placed in the number. This is the system we normally use.

Example — Why position matters

In the number 55:

The first '5' (left side) means 5 TENS = 50

The second '5' (right side) means 5 ONES = 5

Same symbol, but different value — because of its position!

The value of every digit in a positional number system depends on THREE things:

  1. The digit itself
  2. The position of the digit in the number
  3. The base of the number system

Base = the total number of different symbols (digits) used in that number system.

Important Rule: The largest value a single digit can have is always ONE LESS than the base. For example, in the decimal system (base 10), the largest single digit is 9.

Computers and computer users commonly work with four positional number systems: Decimal, Binary, Octal and Hexadecimal. Let's look at each one.

i. Decimal Number System (Base 10)

  • The number system we use in daily life
  • Has 10 symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
  • Base = 10, so the largest single digit is 9
  • Each position represents a power of 10

Example

2586₁₀ = (2×10³) + (5×10²) + (8×10¹) + (6×10⁰)

        = 2000 + 500 + 80 + 6

        = 2586

ii. Binary Number System (Base 2)

  • This is the number system used inside computers
  • Has only 2 symbols: 0 and 1 (matches computer's OFF/ON switches)
  • Base = 2, so the largest single digit is 1
  • Each position represents a power of 2

What is a "Bit"?

"Bit" stands for Binary digIT. Each 0 or 1 in a binary number is called one bit. A binary number made up of n bits is called an n-bit number. For example, 10101 has 5 bits, so it is a 5-bit number.

Example

10101₂ = (1×2⁴) + (0×2³) + (1×2²) + (0×2¹) + (1×2⁰)

        = 16 + 0 + 4 + 0 + 1

        = 21₁₀

iii. Octal Number System (Base 8)

  • Has 8 symbols: 0, 1, 2, 3, 4, 5, 6, 7
  • Base = 8, so the largest single digit is 7
  • Since 8 = 2³, exactly 3 binary bits are enough to represent any octal digit

Example

2057₈ = (2×8³) + (0×8²) + (5×8¹) + (7×8⁰)

       = 1024 + 0 + 40 + 7

       = 1071₁₀

iv. Hexadecimal Number System (Base 16)

  • Has 16 symbols: 0–9 and A, B, C, D, E, F
  • A=10, B=11, C=12, D=13, E=14, F=15
  • Base = 16, so the largest single digit is 15 (shown as 'F')
  • Since 16 = 2⁴, exactly 4 binary bits are enough to represent any hex digit

Example

1AF₁₆ = (1×16²) + (A×16¹) + (F×16⁰)

       = (1×256) + (10×16) + (15×1)

       = 256 + 160 + 15

       = 431₁₀
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