NUMBER BASE SYSTEM 1 Lesson Note

Teaching and Learning Resources:

TEACHING MATERIALS

The teaching and learning materials include:

LESSON 1

Conversion of numbers FROM base 10 to other bases

Conversion of numbers in base 10 to any other bases is to continually divide the number by the new base number till we get up to zero level.

Example 1

Convert 36₁₀ to a number in base 5.

Solution:

36 ÷ 5 = 7 remainder 1

7 ÷ 5 = 1 remainder 2

1 ÷ 5 = 0 remainder 1

Therefore:

36₁₀ = 121₅

The answers are written down as indicated by the arrow from down to up.

Example 2

Convert 101₁₀ to binary number.

Solution

Convert 101₁₀ to binary number

101 ÷ 2 = 50 remainder 1

50 ÷ 2 = 25 remainder 0

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

Therefore:

101₁₀ = 1100101₂

Example 3

Convert 25.0625₁₀ to binary number.

Solution

25₁₀ + 0.0625₁₀

For the whole number:

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

Therefore:

25₁₀ = 11001₂

For the fractional part:

0.0625 × 2 = 0.125 → 0

0.125 × 2 = 0.250 → 0

0.250 × 2 = 0.500 → 0

0.500 × 2 = 1.000 → 1

Therefore:

0.0625₁₀ = 0.0001₂

Hence:

25.0625₁₀ = 11001.0001₂

LESSON 2

Conversion of numbers FROM base 10 TO ANOTHER BASE

To convert numbers from other bases to base 10, we can make use of the continuously multiplication method.

Example 1

Change 21423₅ to base 10.

Solution

21423₅ = 2 × 5⁴ + 1 × 5³ + 4 × 5² + 2 × 5¹ + 3 × 5⁰

= 2 × 625 + 1 × 125 + 4 × 25 + 2 × 5 + 3 × 1

= 1250 + 125 + 100 + 10 + 3

= 1488₁₀

Example 2

Convert 111001₂ to base 10.

Solution

111001₂ = 1 × 2⁵ + 1 × 2⁴ + 1 × 2³ + 0 × 2² + 0 × 2¹ + 1 × 2⁰

= 1 × 32 + 1 × 16 + 1 × 8 + 0 × 4 + 0 × 2 + 1 × 1

= 32 + 16 + 8 + 0 + 0 + 1

= 57₁₀

Conversion of numbers involving fraction from other bases to base 10

Conversion of numbers involving decimal fraction from other bases to base 10 means to make use of negative indices.

Example 1

Convert 101.111₂ to base 10.

Solution

101.111₂ = 1 × 2² + 0 × 2¹ + 1 × 2⁰ + 1 × 2⁻¹ + 1 × 2⁻² + 1 × 2⁻³

= 1 × 4 + 0 × 2 + 1 × 1 + 1 × ½ + 1 × ¼ + 1 × ⅛

= 4 + 0 + 1 + ½ + ¼ + ⅛

= 5 + ⅞

= 5 + 0.875

= 5.875₁₀

Example 2

Change 40.241₅ to base 10.

Solution

40.241₅ = 4 × 5¹ + 0 × 5⁰ + 2 × 5⁻¹ + 4 × 5⁻² + 1 × 5⁻³

= 4 × 5 + 0 × 1 + 2 × ⅕ + 4 × ¹⁄₂₅ + 1 × ¹⁄₁₂₅

= 20 + 0 + ²⁄₅ + ⁴⁄₂₅ + ¹⁄₁₂₅

= 20 + ⁵⁰⁄₁₂₅ + ²⁰⁄₁₂₅ + ¹⁄₁₂₅

= 20 + ⁷¹⁄₁₂₅

= 20 + 0.568

= 20.568₁₀

Conversion of numbers from one base to another, other than base 10

Example 1

Convert 21₁₆ to base 12

Solution: Step 1

Convert 21₁₆ to base 10.

21₁₆ = 2 × 16¹ + 1 × 16⁰

= 2 × 16 + 1 × 1

= 32 + 1

= 33₁₀

Step 2

Convert 33₁₀ to base 12.

33 ÷ 12 = 2 remainder 9

2 ÷ 12 = 0 remainder 2

Reading the remainders from bottom to top:

21₁₆ = 29₁₂

Example 2

Convert 124₈ to base 12

Solution: Step 1

Convert 124₈ to base 10.

124₈ = 1 × 8² + 2 × 8¹ + 4 × 8⁰

= 1 × 64 + 2 × 8 + 4 × 1

= 64 + 16 + 4

= 84₁₀

Step 2

Convert 84₁₀ to base 12.

84 ÷ 12 = 7 remainder 0

7 ÷ 12 = 0 remainder 7

Reading the remainders from bottom to top:

124₈ = 70₁₂