Representing Integers

1
Representing Integers

• Positive integers
• Adding binary numbers
• Turning addition into subtraction
• Twos complement
• Example

2
Positive Integers

• All data must be stored as binary numbers
• Positive integers have direct conversion
• 000 0 001 1 010 2 011 3
• 100 4 101 5 110 6 111 7

3
Adding Decimal Integers

• Perform addition on each column and record
  carrys
• 456
• 27
• -------
• 483

4
Adding Binary Integers

• Perform addition on each column and record
  carrys
• 111
• 10
• -------
• 1001

5
A Truth Table for Addition

• For each bit, there are two inputs (a, b) and two
  outputs (sum, carry)

6
Logic Gates for Addition

• Basic logic gates are AND, OR, NOT
• Carry is easy ? just an AND gate
• Sum is trickier
• Hint when is the output 1?

7
Subtraction

• Perform subtraction on each space and record
  borrows
• 456
• - 27
• -------
• 429

8
An Easier Way

• Subtraction is addition of a negative number
• Rather than develop a new mechanism, we can
  recycle the previous mechanism
• Reduce hardware complexity!

9
Negative Numbers

• How do we usually represent a negative number?
• Should a computer use this representation?

10
From Addition to Subtraction

• 456 27 456 (-27)
• 456
• -27
• -------
• 429

11
A Different Representation
12
Using Twos Complement

• 7 2 7 (-2)
• 0111 (7)
• 1110 (-2)
• --------
• 0101 (5)

13
Creating a Negative Number

• What is 2?
• 0010
• What is the inverse of 2?
• 1101
• What did we want for 2?

14
Example

• Using twos complement, addition, and an 8-bit
  representation, subtract 13 from 7

15
Summary

• All data is stored as binary numbers
• Internal and external representations do not have
  to be the same
• Twos complement representation simplifies
  computer hardware