Basic Operations with Binary

Mathematics J.S.S 3 First Term

Theme: Number and Numeration

Sub Theme: Whole Number

WEEK 2

Basic Operations with Binary

Performance Objectives

Students should be able to;

1. Use a computer to do simple mathematical calculations
2. Simplify Word problems

Arithmetic Operations on Binary

Content

Basic operations of addition, subtraction, multiplication and division are carried out in other bases exactly the same as base 10.

To add, subtract and multiply binary number, use the method that you use with base ten numbers. However, you must remember that you are working with powers of two, not powers of ten. The following identities are very useful:

Addition in Base two

To add in a binary number, the following steps are important:

1. Arrange the numbers as in the addition of decimal numbers.
2. Add the elements of the column starting with the rightmost column.
3. Divide the sum by two.
4. Record the remainder, which is either 0 or 1.
5. Add the quotient to the sum of the next column and repeat the process for the next column.

NOTE:

0 + 0 = 0      1 + 0 = 1

0 + 1 = 1        1 + 1 = 10

Examples:

1. Add 111_two to 11_two

Solution:

     
 

 

2. 11011₂ + 10101₂ + 1001₂

Solution:

  

 

3. 1001₂ + 1011₂

Solution

 

Note: 1^st column: 1 + 1 = 10; write down 0, carry 1

2^nd column: as above

3^rd column: 1 + 0 + 0 = 1

4^th column: 1 + 1 = 10; write down 0, carry 1

 

Subtraction in Base Two

Subtractions in binary numbers are the same as subtraction in denary numbers. When the digit of the number to be subtracted is larger than the corresponding digit above it, we transfer one 2 from the next left column. If the immediate next column has zero digits, the transfer will be from the further left column. The same principle is applicable to other bases other than base 2.

Examples:

1. Subtract 101₂ from 111₂

Solution:

 

2. 1110₂ - 101₂

 

Note: 1 st column: 1 from 0’won’t go’. Move the 1 in the 2^nd column to the 1^st column: 10 – 1 = 1; write down 1

2^nd column: the 1 has been removed, leaving 0; 0 – 0 = 0; write down 0

3^rd column: 1 – 1 = 0; write down 0

4^th column: 1 – 0 = 1; write down 1

3. Find the missing numbers in this subtraction in base 2.

Solution:

We subtract 10101₂  from 111111₂ to get the missing numbers.

Hence, 111111₂ – 10101₂ = 101010₂

Therefore, the missing numbers are 101010₂

 

Multiplication in Base Two

Multiplication is repeated addition. This principle is always applied while multiplying binary numbers and other number bases. The important thing we must note is that if we are working in base two and other bases, all the figures we use in the working should be less than 2 or the number base under consideration.

Note

0 X 0 = 0       1 X 0 = 0

0 X 1 = 0       1 X 1 = 1

Examples:

1. Simplify 1101₂ x 111₂

Solution:

          

1101₂ × 111₂ = 1011011_two

 

2. Simplify 1111 x 111

Solution

 

Note: Set out as in a normal long multiplication, multiplying by 1 or 0 as necessary. Take care of placing the digits. Add as explained in part a. All calculations may be checked by converting to base ten.

Division in Base Two

Division in base two is very similar to division in base ten. If the two numbers are in the same base, we divide using the long division method. However, if both numbers are not on the same base, we convert to base ten and then to the base required before solving.

Examples:

1. Divide 10100₂ by 100₂

Solution:

Method 1:

Convert both numbers to base 10 and divide, then convert back to base 2.

10100₂ = 20₁₀ and 100₂ = 4₁₀

20₁₀ ÷ 40₁₀ = 5₁₀

5₁₀ = 101₂

 

Method 2:

Long division.

∴ 10100₂ ÷ 100₂ = 101₂