Objective 1.02 Understand Numbering Systems Computer Programming I

Number Systems Number systems we will talk about: Decimal (Base10) Binary (Base2) Hexadecimal (Base16) Computer Programming I 2/19/2019

Decimal 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. 102= 100 Base squared = 100 2 3 7 1 6 8 9 4 5 Computer Programming I 2/19/2019

Decimal 4 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 11 21 31 41 51 61 71 81 91 101 12 22 32 42 52 62 72 82 92 102 13 23 33 43 53 63 73 83 93 103 14 24 34 44 54 64 74 84 94 104 15 25 35 45 55 65 75 85 95 105 16 26 36 46 56 66 76 86 96 106 17 27 37 47 57 67 77 87 97 107 18 28 38 48 58 68 78 88 98 108 19 29 39 49 59 69 79 89 99 109

Binary Binary is Base 2 2 one digit numbers 0 and 1 For example: Base 10 of 4 = Binary 0100 or 100 102=100 – works in binary too! 2 squared = 4 Computer Programming I 2/19/2019

Machine Language OOOO OOOO 128 64 32 16 8 4 2 1 Binary or Base2: 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 OOOO 128 64 32 16 8 4 2 1 Ex: 0000 1001= The right most digit (1) = 1 (20) The two middle digits are 0 therefore have no value. The left most digit (1) = 8 (23 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

Why Binary? 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. 27 128 26 64 25 32 24 16 23 8 22 4 21 2 20 1 Computer Programming I 2/19/2019

Powers of 2 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? Computer Programming I 2/19/2019

Powers of 2 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 1 1 22 4 21 2 20 1 Computer Programming I 2/19/2019

The 1’s 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? 1 1 1 1 1 1 1 1 27 128 26 64 25 32 24 16 23 8 22 4 21 2 20 1 1 28 256 27 128 26 64 25 32 24 16 23 8 22 4 21 2 20 1 Computer Programming I 2/19/2019

Let’s Try This… 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. 27 128 26 64 25 32 24 16 23 8 22 4 21 2 20 1 Computer Programming I 2/19/2019

Example Binary Figure out the following numbers in binary… Decimal 56 100 198 64 18 84 231 27 128 26 64 25 32 24 16 23 8 22 4 21 2 20 1 Computer Programming I 2/19/2019

Example Binary Answers Figure out the following numbers in binary: Dec Binary (Answer) 56  111000 100 1100100 198 1100110 64  1000000 18  10010 84  1010100 231 11100111 Computer Programming I 2/19/2019

Hexadecimal 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: 102=100 (162=256 in decimal) This works in Hex as it does for ANY number system. Computer Programming I 2/19/2019

Hexadecimal 15 Contains 16 digits starting with 0-9 & A-F containing the values from 0 – 15. Hex Dec Hex 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 = 30 utilize less disk space. F = 15 1F = 31

Hexadecimal Conversion 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… 9F Base 16 = 1001 1111 Base 2 23 8 22 4 21 2 20 1 23 8 22 4 21 2 20 1 9F Base 16 = 159 Base 10 161 16 160 16 Computer Programming I

Hexadecimal 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. Computer Programming I 2/19/2019

Another Conversion to Hexadecimal Figure out the following Decimal numbers to Hex: Decimal 56 100 198 64 18 128 231 3 8 162 256 161 16 160 1 Ask “How many of ‘256’ can come out of 56 (our decimal number)? 0 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 Ask “How many of ‘1’ can come out of 8 (the left over)? 8 with 0 left over Computer Programming I 2/19/2019

Example Hex Answers Figure out the following Decimal numbers to Hex: Dec Hex 56  38 100 64 198 C6 64  40 18  12 128 80 256 100 Computer Programming I 2/19/2019

Conclusion In this lesson we learned about number systems used in Programming. Decimal Binary Hexadecimal Computer Programming I 2/19/2019

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