1. Numbering Systems and Conversion 2.01- Understand How Computing Devices Work 1

2. Numbering Systems  Three different numbering systems commonly used:  Decimal (Base 10 ) (everyday life)  Binary (Base 2 ) (programming and networking)  Hexadecimal (Base 16 ) (programming, networking, web design)  While we think in decimal, computers think in binary. Computers are only operate in two states: ON or OFF. We must learn how to count like a computer to properly understand it. 2

3. 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.  10 2 = 100  Base squared = 100 3

4. Decimal 4

5. Binary 5  Binary is Base 2  2 digits  0 and 1  No digit greater than 1 exists, forget about 2,3,4,…  For example: Base 10 4 = Binary 0100 or 100  10 2 = 100 – works in binary too! 2 2 = 4

6. Why Binary? 6  Computers operate on a series of electric impulses. Voltages represent binary values.  The CPU only has to know two values: ON or OFF, TRUE or FALSE, 1 or 0, therefore it only has to measure two voltages.  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

7. Powers of 2 7  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?

8. Powers of 2 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 of the 1 power is that number.  So if Decimal 4 = 100 in binary, what does decimal 5 equal in binary?  A: 101 201201 212212 1102 224224

9. The 1’s 9  So let’s go beyond our basic example.  Remember the most right binary digit (bit) has the least significant value and the most left binary digit (bit) 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

10. Let’s Try This… 10  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

11. Example Binary 11  Convert the following numbers to binary using your chart… Decimal  56  100  198  64  18  84  231 201201 2122122 224224 238238 2 4 16 2 5 32 2 6 64 2 7 128

12. Example Binary Answers 12  Convert the following numbers to binary using your chart: DecimalBinary (Answer)  56  111000  100  1100100  198  1100110  64  1000000  18  10010  84  1010100  231  11100111

13. Hexadecimal 13  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: 10 2 =100 (16 2 =256 in decimal)  This works in Hex as it does for ANY number system.

14. Hexadecimal 14  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.

15. Another Conversion to Hexadecimal 15  Convert 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

16. Example Hex Answers 16  Convert the following Decimal numbers to Hex: Dec Hex  56  38  100  64  198  C6  64  40  18  12  128  80  256  100

17. Binary to Hexadecimal  Half a byte is 4 bits which is called a nibble. A nibble is also one hexadecimal digit. (1111 = 15 or F)  Using this principle converting from Binary to Hex is easy.  What is 1101 1001 in Hex? 17

18. Conversion Process  Spilt the binary digits into groups of 4. So for a 1 byte number there would be two groups of 4. Add 0s if the number is not 8 digits. One group will have powers 128, 64, 32, 16 (left group) and the other will contain 8, 4, 2, 1 (right group).  Write the digits out and simply add and combine the groups. Do NOT add the groups only the 4 numbers in each group. 18 201201 2122122 224224 238238 201201 2122122 224224 238238

19. Conversion Process  Spilt the binary digits into groups of 4. So for a 1 byte number there would be two groups of 4. Add 0s if the number is not 8 digits. One group will have powers 128, 64, 32, 16 (left group) and the other will contain 8, 4, 2, 1 (right group).  Write the digits out and simply add and combine the groups. Do NOT add the groups only the 4 numbers in each group. 19 201201 2122122 224224 238238 201201 2122122 224224 238238 A: 1101= 8+4+1 or D (left group) 1001= 9 (right group) So 11011001 2 = D9 16 or 217 10

20. Hexadecimal to Decimal  In our previous example we arrived at the answer D9. How do we convert that to decimal?  Remember the base of Hex? Just multiply!  D (remember D = 13 in decimal) * 16 = 208 + 9 = 217  Or D times 16 1 + 9 times 16 0  Since D does not exist in decimal we must convert it to the decimal number it represents first. 20

21. Conclusion 21  In this lesson we learned about number systems used in digital electronics and computers.  Decimal  Binary  Hexadecimal

22. Binary Practice Game  http://forums.cisco.com/CertCom/game/binary_game_page.htm ‎ 22