2. Learning program: Mechanic – electrician Name of the program: Numerical systems II. class Octal 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. Octal system – is expressed by the symbol O or index (8). The octal system is a position system and like in decimal and binary systems each number can be expressed as an addition of products, which consist of numbers 0, 1, 2, 3, 4, 5, 6 or 7 or the power of the basis 8, which determine the order or value. According to positions in a position system the octal system can be characterized by order this way: 8 n ; 8 n-1 ;…; 8 4 = 4096; 8 3 = 512; 8 2 = 64; 8 1 = 8; 8 0 = 1; 8 -1 = 0,125; 8 -2 = 0,15625; 8 -3 = 0,00195…; 2 -4 = 0,000244…; … ; 2 -(n-1) ; 2 -n Exercise: Express the number 1231,12 (2) in octal system according to individual orders and coefficients of the product. 1231,12 (8) = 1*8 3 + 2*8 2 + 3*8 1 + 1*8 0 +1*8 -1 + 2*8 -2

5. The value of the number in the octal system can be easily expressed in the decimal system. You just have to add individual constituent values in the order writing of the number. Exercise: Convert number 1234 (8) in the octal system into the decimal numerical system. 1*8 3 + 2*8 2 + 3*8 1 + 4*8 0 = 1*512 + 2*64 + 3*8 + 4*1 = 512 + 128 + 24 + 4 = 668 Exercise: Convert number 4321 (8) in the octal system into the decimal numerical system. 4*8 3 + 3*8 2 + 2*8 1 + 1*8 0 = 4*512 + 3*64 + 2*8 + 1*1 = 2048 + 192 + 16 + 1 = 2257

6. PowerVarianceResult 8 3 = 512635 – 512 = 1231 8 2 = 64123 – 64 = 591 8 1 = 859 – 7*8 = 37 8 0 = 13 – 3*1 = 03 Sequential subtraction method This method is easily usable for changeover from one basis to another. Original number is divided by the sequential subtraction of tailing off powers of the new basis, where desired power of the new basis is smaller or equal to the remaining part of the original number. Exercise: Convert number 635 (10) to the octal numerical system.

7. Sequential division method For expressing the conversion of the decimal integer the basis of the conversion is the division of the chosen decimal number by the basis of the octal system. After the division by the basis 8 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 7. 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 rounded on the decimal integer and the value of the remainder. We repeat this procedure until the result of the division by the system basis will be the number 0. We will write down the value of all remainders and record the result. Remainders are written into the result in the reverse order. Exercise: Write the number 1005 (10) in the octal numerical system. CalculationPartial quotient Remin der 1005 : 8 = 1251255 125 : 8 = 15155 15 : 8 = 117 1 : 8 = 001 1005 (10) = 1755 (8)

8. Sequential multiplication method This method is used like in binary and hexadecimal systems most frequently to express the decimal number smaller than one. Exercise: Convert the number 0,725 (10) to the octal system. CalculationPartial resultResult 0,725 x 8 =5,85 0,8 x 8 =6,46 0,4 x 8 =3,23 0,2 x 8 =1,61 etc. Number 0,725 (10) = 0,5631… (8)

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

10. Binary system for 100 How many numbers does the octal system use?

11. What is the other name for octal system? Binary system for 100

12. What is the numerical basis used in the octal system? Binary system for 100

13. What is the value of the octal number 123 (8) in the decimal system? Binary system for 300

14. What is the value of the octal number 135 (8) in the decimal system? Binary system for 300

15. What is the value of the octal number 213 (8) in the decimal system? Binary system for 300

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

17. Binary system for 500 What is the value of the decimal number 321 (10) in the octal system?

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

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