4.12 Data Representation

1. 4.12 Data Representation Two’s Complement & Binary Arithmetic

2. Learning Objectives: • Describe and use two's complement and sign and magnitude to represent positive and negative integers. • Perform integer binary arithmetic: addition and subtraction.

3. Representing Negative Numbers - Sign & Magnitude • As there is no third symbol available to store a negative symbol explicitly we must use a bit to show if a number is negative or not. • We name this bit the ‘Sign Bit’ • We use the leftmost bit. • If the ‘Sign Bit’ is 1 then the number is negative, if it is 0 then it is positive.

4. Representing Negative Numbers- Sign & Magnitude • At first glance it may appear to be simple from this point, for example: 3 = 0 0000011 so-3 = 1 0000011 Sign Bits

5. Binary – Decimal Spreadsheet Converter 2 Try using it to ‘play’ with sign and magnitude binary numbers. 29/11/2014 5

6. Binary Arithmetic Rules • 0 + 0 = 0 • 0 + 1 = 1 • 1 + 0 = 1 • 1 + 1 = 0 (carry 1) • 1+1+1 = 1 (carry 1)

7. Problems withSign & Magnitude • Addition and subtraction calculations give incorrect results: • e.g. +3 + -3 = 0 but 0 0000011 0 00000011 +1 0000011 -1 00000011 1 0000110 i.e. not 0 000000000 • This is due to the fact that the sign bit does not have a place value. • Rules to correct this error: • If the signs are the same, add the magnitudes as unsigned numbers and, if there is a “carry out”, use it as the MSB (last bit). • If the signs differ, subtract the smaller magnitude from the larger, and keep the sign of the larger e.g. http://www.cs.uwm.edu/classes/cs315/Bacon/Lecture/HTML/ch04s11.html To correct this error:

8. Problems withSign & Magnitude Also note that this method allows two representations of 0: 1 0000000 0 0000000 This will cause problems for a processor. 29/11/2014 8

9. Two’s Complement • This is a better way to represent negative numbers. • Imagine a km clock in a car set at 00000000 kilometres. • If the car goes forward one km the reading becomes 00000001. • If the meter was turned back one km the reading would be 99999999 km. • This could be interpreted as ‘-1’ km.

10. Using Two’s Complement • Negative denary to binary • Work out the binary number as if it were positive. • From the left, flip all bits up to the last ‘1’, leave this and any other bits after that alone. • Flip means change 0 to 1 or 1 to 0. • Negative binary to denary • Reverse of above

11. Two’s Complement • So: • 0 0000011 = 3 • 0 0000010 = 2 • 0 0000001 = 1 • 0 0000000 = 0 • 1 1111111 = -1 • 1 1111110 = -2 • 1 1111101 = -3 Sign Bit

12. Binary – Decimal Spreadsheet Converter 2 • Try using it to ‘play’ with two’s complement binary numbers.

13. -5 • Work out the binary number as if it were positive. • 5 = 0 0000101 • From the left, flip all bits up to the last ‘1’, leave this and any other bits after that alone. • -5 = 11111011 Don’t flip the last 1. 1 1 1 1 1 0 1 1

14. 11111011 • From the left, flip all bits up to the last ‘1’, leave this and any other bits after that alone. • 00000101 • Work out the decimal number as if it were positive. • 0 0000101 = 5 • Add the minus sign. • 11111011 = -5

15. Alternative way of using Two’s Complement • The MSB stays as a number, but is made negative. This means that the column headings are • -128 64 32 16 8 4 2 1 • +117 does not need to use the MSB, so it stays as 01110101. • -117 = -128 + 11 • = -128 + (8 + 2 + 1) • Fitting this in the columns gives 10001011

16. Two’s Complement • Now addition and subtraction calculations give the correct results: • 0 0000011 = 3 + 1 1111101 = -3 0 0000000 = 0 11 111111 <- carries • Notes: • The last ‘carry’ of 1 (carry in and out of the Most Significant Bit – MSB) has to be ignored unless an overflow has occurred (see next slide). • The arithmetic works here, as all the bits, including the sign bit, in this method have a place value. • There is only one representation for zero. • 00000000 = 0 • 10000000 = -128 (not 0 as in sign & magnitude)

17. Problem with Two’s Complement • +102 = 01100110 • +117 = 01110101 • 102+117 = 219 but • 01100110 + 01110101 11011011 = -37 111 <- carries • The original numbers are positive but the answer is negative! • There has been an overflow from the positive part of the byte to the negative. • To solve this error: • If an "overflow" occurs add an extra bit and use this as the new sign bit. • An overflow in a two's complement sum has occurred if: • The sum of two positive numbers gives a negative result. • The sum of two negative numbers gives a positive result. • e.g. For the example above: 011011011 = 219 (which is correct). New Sign bit (-256) old sign bit = +128

18. Two’s Complement Also note that: There is only one representation for zero. 29/11/2014 18

19. Plenary • A particular computer stores numbers as 8 bit, two’s complement, binary numbers. • 01011101 and 11010010 are two numbers stored in the computer. • Write down the decimal equivalent of 11010010. • Add the two binary values together and comment on your answer.

20. Plenary • -46 • 00101111 = +47 • A positive and negative have been added together and the result is positive. • Because the larger value was positive. • There was carry in and out of MSB therefore ignore carry out, (result is correct).