Title: Conversion of other bases to base 10.

Objective :

At the end of this lesson, you will be able to convert other bases to base 10.

What you should know :

To fully understand this lesson, you should be familiar with,

• Indices: a³ = a x a x a. 8³ = 8 x 8 x 8.
• Basic arithmetic operations.

Introduction:

Conversion of other bases to base 10 is done differently from the previous lesson. It involves the user of indices, multiplication, and addition.

Presentation /Steps:

Given, 1. convert 4035₈ to base 10. (2) 101101₂ to base 10.

The following steps will be used to solve the above problems.

Steps:

• Rank the digits of the number to be converted starting with 0 from right to left.
• For every digit in the number, multiply each of the digits by the number base value raised to the power of the rank value. I.e digit x (number base value)^{rank value} + digit x (number base value)^{rank value} ..., and simplify the expression gotten.
• Therefore, the result is the new number in base 10.

Soln: 4035₈ to base 10.

=> The number base value is 8.

=> Ranking : 4³ 0² 3¹ 5⁰ , the rank values = 3 2 1 0.

=> digit x (number base value)^{rank value} =

4x8³ + 0x8² + 3x8¹ + 5x8⁰

=> 4x(8x8x8) + 0x(8x8) + 3x8 + 5x1.

=> 4 x 512 + 0 x 64 + 24 + 5.

=> 2045 + 0 + 24 + 5 = 2077.

Therefore, 4035₈ to base 10 = 2077₁₀

Soln 2: 101101₂ to base 10

=> The number base value is 2.

=> Ranking : 1⁵ 0⁴ 1³ 1² 0¹ 1⁰ , the rank values = 5 4 3 2 1 0.

=> 1x2⁵ + 0x2⁴ + 1x2³ + 1x2² + 0x2¹ + 1x2⁰

=> 1x(2x2x2x2x2) + 0x(2x2x2x2) + 1x(2x2x2) + 1x(2x2) + 0x2 + 1x1.

=> 1 x 32 + 0 x 16 + 1 x 8 + 1 x 4 + 0x2 + 1x1.

=> 32 + 0 + 8 + 4 + 0 + 1 = 45.

Therefore, 101101₂ to base 10 = 45₁₀

Summary / Conclusion :

• In indices, any number raised to power 0 is 1.
• In multiplication, any number multiplied by 0 is 0.

Exercise : convert 45₆ to base 10