Week 5: DATA REPRESENTATION

Topics:

Binary Addition

Binary Subtraction

Study Topics

BINARY ADDITION

Binary addition is easier than decimal addition. You are adding the numbers 0 and 1 together two digits at a time. The rules are the same as with decimal addition except when you are adding together 1 + 1 which is a 2 in decimal and a 10 in binary. (1 in the 2’s position and a 0 in the 1’s position). Just like with decimal addition, you will bring down the 0 and carry the 1 to the next position. 

Rules of Binary Addition

0 0 1 1 

+0 +1 +0 +1

0 1 1 0 (total 2 which is 01 in binary - bring down the 0 

The only other situation you will see when adding two binary numbers together is the add together 3 one bits. You are adding two one bits and have a 1 carry bit from the previous position.

1

1

+ 1 

1. (total 3 which is 11 in binary – bring 

For example: Add the two 8-bit binary numbers
 
 

01010011

+ 11010100

Position1: (1 + 0) = 1 Position 2: (1 + 0) = 1 Position 3: (0 + 1) = 1

01010011 01010011 01010011

+ 11010100 + 11010100 + 11010100

1 11 111

Position 4: (0 + 0) = 0 Position 5: (1 +1) = 10 Position 6: (0 + 0 + 1) = 1

(bring down the 0, carry the 1)

01010011 01010011 01010011

+ 11010100 + 11010100 + 11010100

0111 00111 100111 

Position 7: (1 + 1) = 10 Position 8: (1 + 0 + 1)

01010011 01010011 

+ 11010100 + 11010100

0100111 100100111 

For example: Add the two 8-bit binary numbers

01001101

+ 01101101

Position 1: (1 + 1) = 10 Position 2: (1 + 0 + 0) = 1 Position 3: (1 + 1) = 10

01001101 01001101 01001101

+ 01101101 + 01101101 + 01101101

0 10 010

Position 4: (1 + 1 + 1) = 11 Position 5: (1 + 0 + 0) = 1 Position 6: (0 + 1) = 1

01001101 01001101 01001101

+ 01101101 + 01101101 + 01101101

1010 11010 111010
 
 

Position 7: (1 + 1) = 10 Position 8: (1 + 0 + 0) = 1
 
 

01001101 01001101 

+ 01101101 + 01101101

0111010 10111010

BINARY SUBTRACTION

Binary subtraction involves adding a positive and negative number together. You will convert the negative number to a negative form and then add the two numbers together. It is difficult to tell the value of a number when it is in a negative form. Step 5 will allow you to verify the answer to the problem, if the number is negative.

Steps of two’s complement subtraction (8 bit numbers) 

1. Convert the numbers to binary
2. Complement the bits of the bottom number
3. Add one to the bottom number
4. Add the two numbers together
5. If there are 8 bits in the result, the answer is negative. Flip the bits and add one to check the result.

If there are 9 bits in the result, the answer is positive. Drop the rightmost bit.
 
  For example:

45₁₀

- 24₁₀

21₁₀ 

1. Convert the decimal numbers to binary.

 

 
 

45 = 00101101₂

24 = 00011000₂
 
 

2. Complement the bits of the bottom number. (Change the 1 bits to 0 and the 0 bits to 1)

 

 
 

24 = 00011000₂ (Original number)

24 = 11100111₂ (Complemented number)
 
 

3. Add one to the complemented bottom number

 

 
 

11100111₂

00000001₂

11101000 (new bottom number)
 
 

4. Add the top number and new bottom number together.

 

 
 

00101101₂

11101000₂

100010101
 
 

5. There are 9 bits in the result, the answer is positive. Drop the rightmost bit.

For example:

50₁₀

- 64₁₀

-14₁₀ 

1. Convert the decimal numbers to binary.

 

 
 

50 = 00110010₂

64 = 01000000₂
 
 

2. Complement the bits of the bottom number. (Change the 1 bits to 0 and the 0 bits to 1)

 

 
 

64 = 01000000₂ (Original number)

64 = 10111111₂ (Complemented number)
 
 

3. Add one to the complemented bottom number

 

 
 

10111111₂

00000001₂

11000000 (new bottom number)
 
 

4. Add the top number and new bottom number together.

 

 
 

00110010₂

11000000₂

11110010₂
 
 

5. There are 8 bits in the result, the answer is negative. Flip the bits and add 1 to the number. Check the result by converting the binary number to a decimal number.

11110010₂ = original number = -14₁₀
 
 

00001101₂

00000001₂

00001110₂

  Result = 00001110₂ = 14₁₀ This last step was performed only to check the answer to see that it was a 14₁₀. The original number 11110010₂ is the way the number is stored inside the computer system.

STUDY TOPICS 

• List the characteristics of numbering systems
• List the characteristics of each numbering system
• Describe the purpose of each numbering system
• Convert each number type to binary
• Add binary numbers
• Subtract binary numbers

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