1. COMPUTER PROGRAMMING I Objective 1.02 Understand Numbering Systems

2. Number Systems 2/29/2016 Computer Programming I 2 Number systems we will talk about:  Decimal (Base 10 )  Binary (Base 2 )  Hexadecimal (Base 16 )

3. Decimal 2/29/2016 Computer Programming I 3 The number system we use in math and in life. Base 10: ten one digit numbers:  0,1,2,3,4,5,6,7,8,9  After 9 comes 10 (the first two digit number) of course.  10 2 = 100  Base squared = 100 0 1 23 45 6 7 8 9

4. Decimal What can we say about the decimal system? It is our day to day number system… The Decimal system has 10 digits; values are from 0 to 9. 0 10 20 30 40 50 60 70 80 90 100 1 11 21 31 41 51 61 71 81 91 101 2 12 22 32 42 52 62 72 82 92 102 3 13 23 33 43 53 63 73 83 93 103 4 14 24 34 44 54 64 74 84 94 104 5 15 25 35 45 55 65 75 85 95 105 6 16 26 36 46 56 66 76 86 96 106 7 17 27 37 47 57 67 77 87 97 107 8 18 28 38 48 58 68 78 88 98 108 9 19 29 39 49 59 69 79 89 99 109 4

5. Binary 2/29/2016 Computer Programming I 5 Binary is Base 2 2 one digit numbers  0 and 1 For example: Base 10 of 4 = Binary 0100 or 100 10 2 =100 – works in binary too! 2 squared = 4

6. Machine Language Binary or Base 2 : Only contains 2 types of digits; 1 or 0. The power of 2. Each digit from the right to the left is increased by power of 2. Each one (1) digit has a value representing on and each zero (0) digit do not hold a value representing off. OOOO 128 64 32 16 8 4 2 1 Ex: 0000 1001= The right most digit (1) = 1  (2 0 ) The two middle digits are 0 therefore have no value. The left most digit (1) = 8 (2 3 or 2x2x2). The other digits have no value. The total value of all numbers would = 9. (8+0+0+1) ex: 0000 1111  8+4+2+1 = 15 in decimal amount 1111 1111  128+64+32+16+8+4+2+1 = 255 6

7. Why Binary? 2/29/2016 Computer Programming I 7 Computers operate on a series on electric impulses. If the current is flowing the circuit is complete (1), otherwise the current is off (0) Write down the powers of 2 from 0-128. 201201 2122122 224224 238238 2 4 16 2 5 32 2 6 64 2 7 128

8. Powers of 2 2/29/2016 Computer Programming I 8 Remember from math the powers of 2: 1, 2, 4, 8, 16, 32, 64, 128 (first 8) Remember any number to the zero power is 1 and any number to the 1 power is that number. So if Decimal 4= 100 in binary, what does decimal 5 equal in binary?

9. Powers of 2 2/29/2016 Computer Programming I 9 Remember from math the powers of 2: 1, 2, 4, 8, 16, 32, 64, 128 (first 8) Remember any number to the zero power is 1 and any number of the 1 power is that number. So if Decimal 4= 100 in binary, what does decimal 5 equal in binary?  A: 101 201201 2122122 224224 110

10. The 1’s 2/29/2016 Computer Programming I 10 So let’s go beyond our basic example. Remember the most right digit has the least significant value and the most left digit has the most significant value. What is 1111 1111 in Decimal? That would be 255. So… 1 0000 0000 would be 256, right? 201201 2122122 224224 238238 2 4 16 2 5 32 2 6 64 2 7 128 11111111 201201 2122122 224224 238238 2 4 16 2 5 32 2 6 64 2 7 128 00000000 2 8 256 1

11. Let’s Try This… 2/29/2016 Computer Programming I 11 On your paper draw 8 columns Above each column label a power of 2, starting at 128 in the first (left most) column. Finish with 1 in the last (right most) column. 201201 2122122 224224 238238 2 4 16 2 5 32 2 6 64 2 7 128

12. Example Binary 2/29/2016 Computer Programming I 12 Figure out the following numbers in binary… Decimal 56 100 198 64 18 84 231 201201 2122122 224224 238238 2 4 16 2 5 32 2 6 64 2 7 128

13. Example Binary Answers 2/29/2016 Computer Programming I 13 Figure out the following numbers in binary: Dec Binary (Answer) 56  111000 100  1100100 198  1100110 64  1000000 18  10010 84  1010100 231  11100111

14. Hexadecimal 2/29/2016 Computer Programming I 14 Hex is Base 16 There are fifteen one digit numbers:  0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F What comes after F? Remember our rule: 10 2 =100 (16 2 =256 in decimal)  This works in Hex as it does for ANY number system.

15. Hexadecimal Contains 16 digits starting with 0-9 & A-F containing the values from 0 – 15. Hex DecHex Dec-Hex 20-2F and so on… 0 = 0 10 = 16-Each digit carries a value of 16. 1 = 1 11 = 17-Hex = 6 + Decimal = 10 (Hexadecimal =16) 2 = 2 12 = 18-Hexadecimal is only 4 bits (binary value) 3 = 3 13 = 19-ex: 1111 = 15 in decimal “F” in Hex value 4 = 4 14 = 20-Another ex: 1001 1100 = 9C in Hex and 5 = 5 15 = 21-156 in Decimal value. 6 = 6 16 = 22 7 = 7 17 = 23 8 = 8 18 = 24 9 = 9 19 = 25 Remember! Hex is only 4 bits long and its A = 10 1A = 26 highest value is F in Hex or B = 11 1B = 27 15 in decimal or 1111 in binary. C = 12 1C = 28 D = 13 1D = 29 Mainframe computers use Hexadecimal to E = 14 1E = 30utilize less disk space. F = 15 1F = 31 15

16. Hexadecimal Conversion Computer Programming I 16 Hex 9F to binary 9 F 1 0 0 1 1 1 1 1 Hex 9F to Decimal 9 F (16 * 9) + (1 * 15) = 159 in Decimal Add the values… 201201 2122122 224224 238238 201201 2122122 224224 238238 16 0 16 16 1 16 9F Base 16 = 1001 1111 Base 2 9F Base 16 = 159 Base 10

17. Hexadecimal 2/29/2016 Computer Programming I 17 Think if you had 3 hands. You would have 15 fingers right? That is what hex has! So after 9 comes A (10), B (11), C (12), D (13), E (14) and F (15) Let try our example again in Hex.

18. Another Conversion to Hexadecimal 2/29/2016 Computer Programming I 18 Figure out the following Decimal numbers to Hex: Decimal 56 100 198 64 18 128 231 16 0 1 16 1 16 16 2 256 1.Ask “How many of ‘256’ can come out of 56 (our decimal number)? 0 2.Ask “How many of ‘16’ can come out of 56? 3 (3 * 16 = 48 with 8 left over) Put the 3 in the 16’s spot 3.Ask “How many of ‘1’ can come out of 8 (the left over)? 8 with 0 left over 3 8

19. Example Hex Answers 2/29/2016 Computer Programming I 19 Figure out the following Decimal numbers to Hex: Dec Hex 56  38 100  64 198  C6 64  40 18  12 128  80 256  100

20. Conclusion 2/29/2016 Computer Programming I 20 In this lesson we learned about number systems used in Programming. Decimal Binary Hexadecimal

21. For More Information http://software2i.com/viewthread.php?tid=56509&e xtra=page%253D1&page=1 http://software2i.com/viewthread.php?tid=56509&e xtra=page%253D1&page=1 http://www.tpub.com/neets/book13/53e.htm http://www.plcs.net/chapters/number23.htm 24