Number Base – JSS1 Mathematics Lesson Note (Second Term)

A number base is the number of digits a counting system uses. The decimal system is base ten and uses the digits 0 to 9. The binary system is base two and uses only 0 and 1. In any base, each place in a number has a value that is a power of the base.

Behavioural objectives

By the end of the lesson, students should be able to:

• state the digits used in the decimal and binary systems;
• write numbers in other bases in expanded notation;
• convert numbers in other bases to base ten;
• convert numbers in base ten to base two.

What is the decimal system?

The usual system of counting today is called the decimal or denary system. It is also called base ten. It lets us write small or large numbers using combinations of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.

How do you write a number in expanded notation?

Each digit is multiplied by the base raised to the power of its place. The power is 0 for the last digit on the right and increases by 1 for each place to the left.

Example 4.1: Write the following in expanded notation.

1. 1101100₂
2. 2135₆
3. 4567₈

Solution

1. 1101100₂ = (1 × 2⁶) + (1 × 2⁵) + (0 × 2⁴) + (1 × 2³) + (1 × 2²) + (0 × 2¹) + (0 × 2⁰)
2. 2135₆ = (2 × 6³) + (1 × 6²) + (3 × 6¹) + (5 × 6⁰)
3. 4567₈ = (4 × 8³) + (5 × 8²) + (6 × 8¹) + (7 × 8⁰)

How do you convert a number to base ten?

Continue from the expanded notation and evaluate it. Convert the numbers in Example 4.1 to base ten.

Solution

1. 1101100₂ = (1 × 2⁶) + (1 × 2⁵) + (0 × 2⁴) + (1 × 2³) + (1 × 2²) + (0 × 2¹) + (0 × 2⁰)
   = (1 × 64) + (1 × 32) + (0 × 16) + (1 × 8) + (1 × 4) + (0 × 2) + (0 × 1)
   = 64 + 32 + 0 + 8 + 4 + 0 + 0 = 108₁₀
2. 2135₆ = (2 × 6³) + (1 × 6²) + (3 × 6¹) + (5 × 6⁰)
   = (2 × 216) + (1 × 36) + (3 × 6) + (5 × 1)
   = 432 + 36 + 18 + 5 = 491₁₀
3. 4567₈ = (4 × 8³) + (5 × 8²) + (6 × 8¹) + (7 × 8⁰)
   = (4 × 512) + (5 × 64) + (6 × 8) + (7 × 1)
   = 2048 + 320 + 48 + 7 = 2423₁₀

What is the binary system?

In the binary system, the greatest digit used is 1, so the two digits available are 0 and 1. Each digit in a binary number has a place value.

How do you convert a number in base ten to base two?

Divide the number by 2 again and again, writing down the remainder each time, until the quotient is 0. Then read the remainders from the bottom to the top.

Examples:

1. Convert 29₁₀ to base 2.
2. Convert 79₁₀ to base 2.
3. Convert 145₁₀ to base 2.

Solution

(a) 29₁₀

29₁₀ = 11101₂

(b) 79₁₀

79₁₀ = 1001111₂

(c) 145₁₀

145₁₀ = 10010001₂

Evaluation

1. Expand the following with their bases.
   1. 3531₈
   2. 101010₂
   3. 1110110242
2. Convert the following numbers to base 2.
   1. 356₁₀
   2. 47₁₀
   3. 218₁₀