JSS Number Bases Lesson

JSS Mathematics: Number Bases

This lesson explains number bases in a simple way for JSS students. It includes worked examples from base 10 to other bases and from base 2 to base 10.

1. Meaning of Number Bases

What are number bases?

A number base tells us how many digits a counting system uses. The common bases are base 10, base 2, base 5, base 7, and base 8.

Base 10 uses digits 0–9. Base 2 uses only 0 and 1.

2. Base 10 to Another Base

Method

Divide the number repeatedly by the new base. Write the remainders from bottom to top.

Example: 657₁₀ to base 7

3. Base 2 to Base 10

Method

Expand the binary number using powers of 2, starting from the right.

1011₂ = 1×2³ + 0×2² + 1×2¹ + 1×2⁰

Worked Examples: Base 10 to Other Bases

Example 1

Convert 657₁₀ to base seven.

657 ÷ 7 = 93 remainder 6
93 ÷ 7 = 13 remainder 2
13 ÷ 7 = 1 remainder 6
1 ÷ 7 = 0 remainder 1

Answer: 657₁₀ = 1626₇

Example 2

Convert 27₁₀ to binary.

27 ÷ 2 = 13 remainder 1
13 ÷ 2 = 6 remainder 1
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1

Answer: 27₁₀ = 11011₂

Example 3

Convert 41₁₀ to base two.

41 ÷ 2 = 20 remainder 1
20 ÷ 2 = 10 remainder 0
10 ÷ 2 = 5 remainder 0
5 ÷ 2 = 2 remainder 1
2 ÷ 2 = 1 remainder 0
1 ÷ 2 = 0 remainder 1

Answer: 41₁₀ = 101001₂

Example 4

Convert 24₁₀ to binary.

24 ÷ 2 = 12 remainder 0
12 ÷ 2 = 6 remainder 0
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1

Answer: 24₁₀ = 11000₂

Example 5

Convert 75₁₀ to base five.

75 ÷ 5 = 15 remainder 0
15 ÷ 5 = 3 remainder 0
3 ÷ 5 = 0 remainder 3

Answer: 75₁₀ = 300₅

Example 6

Convert 1253₁₀ to base eight.

1253 ÷ 8 = 156 remainder 5
156 ÷ 8 = 19 remainder 4
19 ÷ 8 = 2 remainder 3
2 ÷ 8 = 0 remainder 2

Answer: 1253₁₀ = 2345₈

Example 7

Convert 37₁₀ to base five.

37 ÷ 5 = 7 remainder 2
7 ÷ 5 = 1 remainder 2
1 ÷ 5 = 0 remainder 1

Answer: 37₁₀ = 122₅

Example 8

Convert 37₁₀ to binary.

37 ÷ 2 = 18 remainder 1
18 ÷ 2 = 9 remainder 0
9 ÷ 2 = 4 remainder 1
4 ÷ 2 = 2 remainder 0
2 ÷ 2 = 1 remainder 0
1 ÷ 2 = 0 remainder 1

Answer: 37₁₀ = 100101₂

Example 9

Convert 54₁₀ to binary.

54 ÷ 2 = 27 remainder 0
27 ÷ 2 = 13 remainder 1
13 ÷ 2 = 6 remainder 1
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1

Answer: 54₁₀ = 110110₂

Worked Examples: Binary to Base 10

Example 10

Change 11011₂ to base ten.

11011₂ = 1×2⁴ + 1×2³ + 0×2² + 1×2¹ + 1×2⁰
= 16 + 8 + 0 + 2 + 1

Answer: 11011₂ = 27₁₀

Example 11

Change 11111₂ to base ten.

11111₂ = 1×2⁴ + 1×2³ + 1×2² + 1×2¹ + 1×2⁰
= 16 + 8 + 4 + 2 + 1

Answer: 11111₂ = 31₁₀

Example 12

What is the value of 1111₂ in base ten?

1111₂ = 1×2³ + 1×2² + 1×2¹ + 1×2⁰
= 8 + 4 + 2 + 1

Answer: 1111₂ = 15₁₀

Example 13

Change 11101₂ to base ten.

11101₂ = 1×2⁴ + 1×2³ + 1×2² + 0×2¹ + 1×2⁰
= 16 + 8 + 4 + 0 + 1

Answer: 11101₂ = 29₁₀

Example 14

Change 100100₂ to base ten.

100100₂ = 1×2⁵ + 0×2⁴ + 0×2³ + 1×2² + 0×2¹ + 0×2⁰
= 32 + 4

Answer: 100100₂ = 36₁₀

Quick Rule Summary

Type of Conversion	Method
Base 10 to another base	Keep dividing by the new base and write the remainders from bottom to top.
Binary to Base 10	Expand using powers of 2, starting from the right.
Understanding bases	The base shows how many digits are used in the number system.

Class Exercises

1. Convert 46₁₀ to binary.
2. Convert 98₁₀ to base seven.
3. Convert 154₁₀ to base eight.
4. Change 10110₂ to base ten.
5. Change 110101₂ to base ten.
6. Convert 63₁₀ to base five.
7. Convert 45₁₀ to binary.
8. Change 100011₂ to base ten.

Teacher’s note: Encourage learners to write each division step neatly and to read remainders from the bottom upward.

JSS Mathematics Lesson on Number Bases