Number System – JSS1 Mathematics Lesson Note (Second Term)

Converting a number from one base to another is a two-stage process. First, change the given number to base ten, then change the result to the required base. Numbers in base two can also be added, subtracted and multiplied using simple rules for the digits 0 and 1.

Behavioural objectives

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

• convert a number from one base to another through base ten;
• add numbers in base two;
• subtract numbers in base two;
• multiply numbers in base two.

How do you convert a number from one base to another?

This is a two-stage process. First, convert the given base to base ten. Then convert the result to the required base.

Example 1: Convert 2135₆ to a number in base two

Step one: Convert to base ten.

2135₆ = (2 × 6³) + (1 × 6²) + (3 × 6¹) + (5 × 6⁰)
= (2 × 216) + (1 × 36) + (3 × 6) + (5 × 1)
= 432 + 36 + 18 + 5
= 491₁₀

Step two: Convert the answer in base ten to a number in base two by dividing repeatedly by 2. Read the remainders from the bottom upwards.

Therefore 2135₆ = 111101011₂

Example 2: Express 10111₂ as a number in base three

10111₂ = (1 × 2⁴) + (0 × 2³) + (1 × 2²) + (1 × 2¹) + (1 × 2⁰)
= 1 × 16 + 0 × 8 + 1 × 4 + 1 × 2 + 1 × 1
= 16 + 0 + 4 + 2 + 1
= 23₁₀

Now divide repeatedly by 3 and read the remainders from the bottom upwards.

Therefore 10111₂ = 212₃

How do you add and subtract in base two?

Use the following rules.

Addition in base two

• 0 + 0 = 0
• 1 + 0 = 1
• 1 + 1 = 10
• 1 + 1 + 1 = 11

Subtraction in base two

• 0 – 0 = 0
• 1 – 0 = 1
• 10 – 1 = 1
• 11 – 1 = 10

Example 3: Add the following

1. 1101₂ and 1111₂
2. 11011₂, 10101₂ and 1001₂

Solution:

(1) Using the addition rule 1 + 1 = 10, write 0 down and carry the 1 to the next column. Carry the 1 in the same way in each column until you reach the last one.

 1101₂
+ 1111₂
= 11100₂

(2) Add the three numbers in the same way.

 11011₂
 10101₂
+ 1001₂
= 111001₂

Example 4: Subtract (a) 1011₂ from 10101₂ (b) 11011₂ from 111010₂

Solution:

(a) Using the subtraction rule 1 – 1 = 0, write 0 down and move to the next column.

 10101₂
– 1011₂
= 1010₂

(b)

 111010₂
– 11011₂
= 11111₂

How do you multiply binary numbers?

Multiply by each digit of the second number, move each row one place to the left, and then add the rows using the rules for addition in base two.

Example 5: Find the product of 1011₂ and 101₂

Solution:

 1011₂
× 101₂
 1011
 0000 (moved one place left)
 1011 (moved two places left)
= 110111₂

Example 6: Multiply 1111₂ by 110₂

Solution:

 1111₂
× 110₂
 0000
 1111 (moved one place left)
 1111 (moved two places left)
= 1011010₂

Evaluation

1. Add the following: (a) 1001₂ + 1111₂ (b) 101110₂ + 10010₂
2. Subtract 1011₂ from 1101101₂.
3. Multiply 110101₂ by 1011₂.
4. Find the value of the following binary numbers: 1011₂ + 10₂ – 111₂