Number Systems Part 2

1. Number Systems Part 2 Numerical Overflow Right and Left Shifts Storage Methods Subtraction Ranges

2. Numerical Overflow • An overflow is when something doesn’t fit in a certain space • Numeric overflow is when the storage for a calculation is too small to hold the result • For example we have an 8 bits register, if we add two binary numbers and the result turns out to be 9 bits it would not fit in the register

3. Example • Let’s say we have an 8 bit register • Add the following; • Do we have an overflow?

4. Numerical Overflow • When we have a numeric overflow we will have an error in our calculation • When we have an overflow we would need to remove the extra bit at the start of the number • Lets say we had a 7 bit register and the result of a calculation is 11001100 the actual answer would be 1001100

5. Example • Let’s say we have a 7 bit register • Add the following; • Do we have an overflow? • Actual answer =

6. Working • Suppose we have a 4 bit registers. • Perform the additions • Is there and overflow? • What is the actual answer?

7. Answers

8. What is Bit Shifting? • Bit shifting is the process of moving all the bits in a binary number • We have two shifts • A right shift • A left shift • The right shift would divide the number while the left would multiply it

9. Right Shift • The right shift is used to perform divisions • Hence if we where to perform three right shifts we would be diving the number by 2 three times • For example if we shift 101100112, = 17910 right by three places, we get 000101102= 2210 • 179 /2 = 89 • 90/2 = 44 • 45/2 = 22

10. How Right Shifts work • Lets say we have 11001012 and we wish to perform 2 right shifts 1st Shift right LOST 0 ADDED 1 2nd Shift right 0 ANS =

11. Working • Shift the numbers to the right by the number of shifts indicated in the brackets; • 1000110 (2) • 1110001 (4) • 1111000 (1) • 01010101 (4) • 0000111 (3) Change to decimal to check your answers

13. Left Shift • The left shift is used to perform multiplications • Hence if we where to perform four left shifts we would be multiplying the number by 2 four times • If any bits are lost when shifting left the result is no longer accurate • For example if we shift 011001012, = 10110 left by one place, we get 110010102 = 20210 • 101 x 2 = 202

14. How Left Shifts work • Lets say we have 011001012 and we wish to perform 2 left shifts 1st Left Shift LOST 0 2nd Left Shift Answer no longer correct as a 1 was lost 1

15. Working • Shift the numbers to the left by the number of shifts indicated in the brackets, also state which answers are not correct. • 0000110 (4) • 1110001 (3) • 0011110 (1) • 01010101 (2) • 0000111 (3) Change correct answers to decimal to check your answers

17. Storage Methods • When we have a fixed register we might want to store out binary number in a certain way • The three ways are using; • sign and magnitude(first digit is 0 = positive, first digit is1 = negative) • one’s complement • two’s complement.

18. Example 1 • Lets say we have a register of 8-bits and we wish to store the number 1410 • Since the number is positive, all we need to do is convert it to binary and store it • It would be the same in the three different storage methods, 1410= 000011102

19. Example 2 • Lets say I have an 8 bit register and want to store -1410 • The number now changes since it is negative, we first change it to binary = 000011102

20. Working • Convert the following to; • Sign and magnitude • One’s Complement • Two’s Complement Size of register is indicated in the brackets. • 3310 (8) • -6010 (9) • -4410 (7) • 1010 (5) • -7410 (10)

21. Subtraction • We are able to subtract using binary numbers thanks to Two’s Complement • When we need to subtract in binary we first need to change the negative number to Two’s Complement • if we have 22 – 10 we have to change 10 to Two’s Complement then perform an addition • Binary numbers must be of the same length in order to subtract in binary

22. Example • We want to perform the following subtraction in binary; Change to Binary Two’s Complement Overflow

23. Working • Perform the following subtractions in binary; • 30-20 • 100 – 50 • 50 – 25 • 5 – 2 • 66- 60

24. Ranges • A ranges determines the lowest and the highest values which can be represented by a certain register • We will be going through two types of ranges using binary; • normal binary • two’s complement.

25. Normal Binary • What is the range of numbers which can be represented using 8-bits? • Since we have 8 Bits, the maximum value represented in binary is 11111111 and the smallest is 00000000. • If we convert these numbers to decimal we end up with: 0 to 255

26. Example • What is the range of numbers which can be represented using 5-bits? • Since we have 5 Bits, the maximum value represented in binary is 11111 and the smallest is 00000. • If we convert these numbers to decimal we end up with: 0 to 31

27. Two’s Complement • What is the range of numbers which can be represented using 8-bits in two’s complement? • The first number in two’s complement identifies the number as a negative number, the highest possible positive number is 011111112 = 12710 • To find the smallest number we convert the positive number to negative so in this case -12710. • The range is: -127 to 126 (126 due to the 0)

28. Example • What is the range of numbers which can be represented using 10-bits in two’s complement? • 01111111112= 51110 • 51110 becomes negative to show the smallest number helps = -51110. • The range is: -511 to 510