DLD Lecture 03(Number conversion & arithmetic).pptx

Number conversion & arithmetic

1. Easy Approach: Numbering Systems And Arithmetic Operations On Hex, Binary, And Octal Lecture-03 Teacher: Dr. Syed Ali Asgher Email: [email protected]

2. 2 12 Department of Computer Science

3. Exercise – Convert ... 3 Don’t use a calculator! Decimal Binary Octal Hexa- decimal 33 1110101 703 1AF Skip answer Answer Department of Computer Science

4. Exercise – Convert … 4 Decimal Binary Octal Hexa- decimal 33 100001 41 21 117 1110101 165 75 451 111000011 703 1C3 431 110101111 657 1AF Answer

5. What is the two’s complement? A two's-complement system is a system in which negative numbers are represented by the two's complement of the absolute value. 5

6. Example Suppose we're working with 8 bit and suppose we want to find how -28 would be expressed in two's complement notation. First we write out 28 in binary form. 0 0 0 1 1 1 0 0 1 1 1 0 0 1 0 0 1 1 1 0 0 0 1 1 Then we invert the digits. 0 becomes 1, 1 becomes 0. Then we add 1. That is how one would write -28 in 8 bit binary. 6

7. 7 An 8-bit unsigned integer number can represent the values 0 to 255. However for signed number a two's complement 8-bit number can only represent non-negative integers from 0 to 127, because the rest of the bit represent the negative integers −1 to −128. 2 8 = 256

8. Common Powers (1 of 2) Base 10 8 Power Preface Symbol 10 -12 pico p 10 -9 nano n 10 -6 micro  10 -3 milli m 10 3 kilo k 10 6 mega M 10 9 giga G 10 12 tera T Value .000000000001 .000000001 .000001 .001 1000 1000000 1000000000 1000000000000

9. Common Powers (2 of 2) Base 2 9 Power Preface Symbol 2 10 kilo k 2 20 mega M 2 30 Giga G Value 1024 1048576 1073741824 What is the value of “k”, “M”, and “G”? In computing, particularly w.r.t. memory , the base-2 interpretation generally applies.

10. 10

11. Example 11 / 2 30 = 1. Double click on My Computer 2. Right click on C: 3. Click on Properties

12. Review – multiplying powers For common bases, add powers 12 2 6  2 10 = 2 16 = 65,536 or… 2 6  2 10 = 64  2 10 = 64k a b  a c = a b+c

13. Arithmetic Operations on Hex, Binary, Octal 13

14. Binary Addition Rules: 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 2 = 10 2 = 0 with 1 to carry 1 + 1 + 1 = 3 = 11 2 = 1 with 1 to carry 14

15. Binary Addition Two n -bit values Add individual bits Propagate carries E.g., 15 10101 21 + 11001 + 25 101110 46 1 1

16. Binary Addition 1 1 1 1 1 1 0 1 1 1 + 0 1 1 1 0 0 1 0 1 0 0 1 1 16 Verification 55 10 + 28 10 83 10 64 32 16 8 4 2 1 1 0 1 0 0 1 1 = 64 + 16 + 2 +1 = 83 10

17. Binary Addition Example Verification 1 0 0 1 1 1 + 0 1 0 1 1 0 + ___ 1 1 1 1 0 1 128 64 32 16 8 4 2 1 = = 17

18. Multiplication (1 of 3) Decimal (just for fun) 18 35 x 105 175 000 35 3675

19. Multiplication (2 of 3) Binary, two 1-bit values 19 A B A  B 1 1 1 1 1

20. Multiplication (3 of 3) Binary, two n -bit values As with decimal values E.g., 20 1110 x 1011 1110 1110 0000 1110 10011010

21. Fractions Decimal to decimal (just for fun) 21 3.14 => 4 x 10 -2 = 0.04 1 x 10 -1 = 0.1 3 x 10 = 3 3.14

22. Fractions Binary to decimal 22 10.1011 => 1 x 2 -4 = 0.0625 1 x 2 -3 = 0.125 0 x 2 -2 = 0.0 1 x 2 -1 = 0.5 0 x 2 = 0.0 1 x 2 1 = 2.0 2.6875

23. Fractions Decimal to binary 23 3.14579 .14579 x 2 0.29158 x 2 0.58316 x 2 1.16632 x 2 0.33264 x 2 0.66528 x 2 1.33056 etc. 11.001001...

24. Exercise – Convert ... 24 Don’t use a calculator! Decimal Binary Octal Hexa- decimal 29.8 101.1101 3.07 C.82 Skip answer Answer

25. Decimal Addition 111 3758 + 4657 8 4 1 5 25 What is going on? 1 1 1 (carry) 3 7 5 8 + 4 6 5 7 14 11 15 10 10 10 (subtract the base) 8 4 1 5

26. Octal Addition 0-7= 8 1 1 6 4 3 7 8 + 2 5 1 0 8 1 1 1 4 7 8 26 Example: 3 5 3 6 8 + 2 4 5 7 8 6 2 1 5

27. Hexadecimal Addition 1 1 Example: 7 C 3 9 16 + 3 7 F 2 16 B 4 2 B 16 27 8 A D 4 16 + 5 D 6 16 9 0 A A

28. Binary Subtraction 1 0 1 0 0 1 1 - 0 1 1 1 0 0 1 1 0 1 1 1 28 Verification 83 10 - 28 10 55 10 64 32 16 8 4 2 1 1 1 0 1 1 1 = 32 + 16 + + 4 + 2 +1 = 55 10

29. Exercise Hexadecimal: 650E + 08C1 = 4A6 + 1B3 = 4A6 – 1B3 = Octal: 162 + 537 = 456 + 123 = 456 – 173 = 29 Website: https://madformath.com/calculators/digital-systems/

30. Solutions HD 650E + 08C1 = 6DCF 4A6 + 1B3 = 659 4A6 – 1B3 = 2F3 OCTAL 162 + 537 = 721 456 + 123 = 601 456 – 173 = 263 30