Number Bases.

Numbers are the foundation of all mathematics, but did you know that the way we represent them depends on the base we use? The base of a number system tells us how many digits are used before the digits start repeating in patterns. The most common base we use in everyday life is Base 10, also called the Decimal System. However, there are many other bases such as Binary (Base 2), Octal (Base 8), and Hexadecimal (Base 16) — all widely used in computing and digital systems.

What Is a Number Base?

A number base (or radix) is the number of unique digits used to represent numbers in a positional number system.

For example:

  • In Base 10, we use digits 0 to 9.

  • In Base 2, we use digits 0 and 1 only.

  • In Base 8, we use digits 0 to 7.

  • In Base 16, we use digits 0 to 9 and letters A to F (representing values 10 to 15).

So, the value of each digit in a number depends on its position and the base being used.

Example of Positional Value (Base 10)

Let’s take the number 345 in base 10.

It can be expanded as:

(3×102)+(4×101)+(5×100)(3 × 10^2) + (4 × 10^1) + (5 × 10^0)
=300+40+5=345= 300 + 40 + 5 = 345

Each digit’s position determines how much it’s worth — that’s what we mean by positional value.

Example in Base 2 (Binary)

Consider the binary number 1011₂.
This means:

(1×23)+(0×22)+(1×21)+(1×20)(1 × 2^3) + (0 × 2^2) + (1 × 2^1) + (1 × 2^0)
=8+0+2+1=1110= 8 + 0 + 2 + 1 = 11_{10}

So, 1011₂ = 11₁₀.

Example in Base 8 (Octal)

Take 572₈:

(5×82)+(7×81)+(2×80)(5 × 8^2) + (7 × 8^1) + (2 × 8^0)
=320+56+2=37810= 320 + 56 + 2 = 378_{10}

So, 572₈ = 378₁₀.

Example in Base 16 (Hexadecimal)

Let’s convert 2AF₁₆ to base 10.

Remember: A = 10, F = 15.

(2×162)+(10×161)+(15×160)(2 × 16^2) + (10 × 16^1) + (15 × 16^0)
=512+160+15=68710= 512 + 160 + 15 = 687_{10}

So, 2AF₁₆ = 687₁₀.

Conversion Between Number Bases

1. Converting from Base 10 to Another Base

To convert from base 10 to another base, we divide the number by the base repeatedly, recording the remainders until the quotient is zero. The remainders (read from bottom to top) give the new base representation.

Example: Convert 25₁₀ to base 2.

Steps:
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

Now, read the remainders from bottom to top: 11001₂.
So, 25₁₀ = 11001₂.

2. Converting from Another Base to Base 10

To convert from any base to base 10, multiply each digit by its place value and add them up.

Example: Convert 1101₂ to base 10.

(1×23)+(1×22)+(0×21)+(1×20)(1 × 2^3) + (1 × 2^2) + (0 × 2^1) + (1 × 2^0)
=8+4+0+1=13= 8 + 4 + 0 + 1 = 13

So, 1101₂ = 13₁₀.

3. Converting Between Non-Decimal Bases

When converting between two bases that are not base 10, it’s easiest to first convert the number to base 10, then to the new base.

Example: Convert 10111₂ to base 8.

Step 1: Convert to base 10.

(1×24)+(0×23)+(1×22)+(1×21)+(1×20)(1 × 2^4) + (0 × 2^3) + (1 × 2^2) + (1 × 2^1) + (1 × 2^0)
=16+0+4+2+1=2310= 16 + 0 + 4 + 2 + 1 = 23_{10}

Step 2: Convert 23₁₀ to base 8.
23 ÷ 8 = 2 remainder 7
2 ÷ 8 = 0 remainder 2

Read remainders bottom to top → 27₈.
So, 10111₂ = 27₈.

                         Assignment

  1. Convert the following from base 10 to base 2:
    a) 18
    b) 45
    c) 67

  2. Convert the following from base 2 to base 10:
    a) 1100
    b) 10101
    c) 11111

  3. Convert the following from base 8 to base 10:
    a) 235₈
    b) 407₈
    c) 765₈

  4. Convert the following from base 16 to base 10:
    a) 1F₁₆
    b) A2₁₆
    c) 7C₁₆

  5. Convert the following from base 2 to base 8:
    a) 11001₂
    b) 101011₂
    c) 111100₂

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