2. Learning program: Mechanic – electrician Name of the program: Numerical systems II. class Hexadecimal numerical system Made by: Mgr. Holman Pavel Projekt Anglicky v odborných předmětech, CZ.1.07/1.3.09/04.0002 je spolufinancován Evropským sociálním fondem a státním rozpočtem České republiky.

3. Numerical systems

4. Hexadecimal system– is expressed by the symbol H or by the index (16). Hexadecimal system is a position system and therefore like in other systems each number can be expressed as a sum of products, which consist of numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and letters A, B, C, D, E, F and power of the basis 16, which again determines order or in other words the value. According to positions in the positional system we can characterize the hexadecimal system this way: 16 n ; 16 n-1 ;…; 16 5 = 1048576; 16 4 = 65536; 16 3 = 4096; 16 2 = 256; 16 1 = 16; 16 0 = 1; 16 -1 = 0,0625; 16 -2 = 0,0039; 16 -3 = 0,0002; 16 -1 = 0,125; …; … ; 16 -(n-1) ; 16 -n Exercise: Express the number1A56 (16) in the hexadecimal system according to individual orders and coefficients of the product. 1A5C = 1*16 3 + 10*16 2 + 5*16 1 + 12*16 0

5. The value of the number in the hexadecimal system can be easily expressed in the decimal system. You just need to add up individual values of elements in the orderly record of the number. Exercise: Convert the number 1B3 (16) from the hexadecimal system to the decimal numerical system. 1*16 2 + 11*16 1 + 3*16 0 = 1*256 + 11*16 + 3*1 = 256 + 176 + 3 = 435 Exercise: Convert the number12CA (16) from the hexadecimal system to the decimal numerical system. 1*16 3 + 2*16 2 + 12*16 1 + 10*16 0 1*4096 + 2*256 + 12*16 + 10*1 4096 + 512 + 192 + 10 = 4810

6. PowerVarianceResult 16 3 = 40966358 – 4096 = 22621 16 2 = 2562265 – 8*256 = 2148 16 1 = 16214 – 13*16 = 6D 16 0 = 16 – 6*1 = 06 Sequential subtraction method This method is easily usable for the changeover from one basis to another. The original number is divided by the sequential subtraction of tailing off powers of the new basis, where a desired power of the new basis is smaller or equal to the remaining part of the original number. Exercise: Convert the number 6358 (10) to the hexadecimal numerical system.

7. Sequential division method is often considered the basic method of division of the given decimal number by the basis of the hexadecimal system. After the division by the basis 16 we write the result of it by the division to the decimal integers and in the same time we have to determine, what the remainder of the division is. The value of the remainder can be 0 to 15. For values of the remainder 10 to 15 we use letters A = 10, B = 11, C = 12, D=13, E = 14 a F = 15. In another step we repeat this procedure by division of the previous result by the basis of the system. Again we write down the result of the integer division and the value of the remainder. We repeat this procedure until the result of the division by the system is the number 0. We write down the value of all reminders and record the result. Remainders are written into the result in the reverse order. Exercise: Express the number 958 (10) in the hexadecimal system. CalculationPartial quotie nt Reminde r 9548 : 16 = 59659612 = C 596 : 16 = 37374 37 : 16 = 225 2 : 16 = 002 1005 (10) = 254C (16)

8. Conversion from the hexadecimal to the decimal numerical system Conversion from the decimal to the hexadecimal numerical system 1B2 (16) = A2C3 (16) = B1A2 (16) = A1B2 (16) = ABC (16) = 1BC (16) = A2C (16) = B1A (16) = A1B (16) = BAC (16) = 7B (16) E7 (16) FF (16) 64 (16) 4D (16) 444 (10) 2604 (10) 2842 (10) 2587 (10) 2988 (10) 123 (10) = 231 (10) = 255 (10) = 100 (10) = 77 (10) = 1234 (10) = 4321 (10) = 1278 (10) = 1434 (10) = 2012 (10) = 4D2 (16) 10E1 (16) 4FE (16) 59A (16) 7DC (16) 434 (10) 41667 (10) 45474 (10) 41394 (10) 2784 (10)

9. Question chart: 1 22 2 3 for 100for 500for 300 ABCD EFGH Prémie 3 3 1 1 The End

10. Question for 100 How many symbols are used in the hexadecimal system?

11. What is the numerical basis used in the hexadecimal system? Question for 100

12. How many letters are used in the hexadecimal system? Question for 100

13. What is the value of the hexadecimal number 1A3 (16) in decimal system? Question for 300

14. What is the value of the hexadecimal number A13 (16) in decimal system? Question for 300

15. What is the value of the hexadecimal number A2 (16) in decimal system? Question for 300

16. What is the value of the decimal number 123 (10) in the hexadecimal system? Question for 500

17. What is the value of the decimal number 248 (10) in the hexadecimal system? Question for 500

18. What is the value of the decimal number 1234 (10) in the hexadecimal system? Question for 500

19.  Mužík, J. Management ve vzdělávání dospělých. Praha: EUROLEX BOHEMIA, 2000. ISBN 80-7361-269-7.  Operační program Vzdělávání pro konkurenceschopnost, ESF 2007 – 2013.  Dostupné na: http://www.msmt.cz/eu/provadeci-dokument-k-op-vzdelavani-pro- konkurenceschopnosthttp://  MALINA, V. Digitální technika. České Budějovice: KOPP, 1996  KRÝDL, M. Číslicová technika. Dubno, 1999  PODLEŠÁK, J., SKALICKÝ, P. Spínací a číslicová technika. Praha, 1994  PECINA, J. Ing. PaedDr. CSc.; PECINA, P. Mgr. Ph.d. Základy císlicové techniky. Brno, 2007