2. Learning program: Mechanic – electrician Name of the program: Numerical systems II. class Binary 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. Binary system – is expressed by the symbol B or index (2). Binary system is a position system and same as the decimal system every number can be expressed as an addition of products, which consists of numbers 0 or 1 and a power of the basis 2, which determines the order or the value. According to positions in the position system we can describe by order this way: 2 n ; 2 n-1 ;…; 2 4 = 16; 2 3 = 8; 2 2 = 4; 2 1 = 2; 2 0 = 1; 2 -1 = 0,5; 2 -2 = 0,25; 2 -3 = 0,125; 2 -4 = 0,0625; … ; 2 -(n-1) ; 2 -n Exercise: Describe the number 101011,101(2) in binary system according to individual orders and coefficients of the product. 101011,101 (2) = 1*2 5 + 0*2 4 + 1*2 3 + 0*2 2 +1*2 1 + 1*2 0 + 1*2 -1 + 0*2 -2 +1*2 -3

5. Writing of the number in the binary system is usually done from right to left. It means from the least significant bit LSB to the most significant bit MSB. According to the basis of these powers, which is always 2, is this numerical system called binary. Binary system – Has two states (z=2), use for technical processing of the information using two numbers 0 and 1. Using these two numbers we can project any numerical value, but the number written in the binary system is very confusing for us compared to the one in the decimal system. It is definitely not suitable for practical use in everyday life, but it is very suitable for the numerical processing of the information in technical practice.

6. PowerVariationResult 2 7 = 128151 – 128 = 231 2 6 = 6423 – 64 = - 410 2 5 = 3223 – 32 = - 90 2 4 = 1623 – 16 = 71 2 3 = 87 – 8 = -10 2 2 = 47 – 4 = 31 2 1 = 23 – 2 = 11 2 0 = 11 – 1 = 01 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 the number 151 (10) to the binary 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 binary system. After the division we write the result of it by the division to the 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 or 1. 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 integer and the value of the remainder. We repeat this procedure until the remainder from the original number will be 0. We will write down the value of all remainders and record the result of the number in the binary system. Remainders are written into the result in the reverse order. Exercise: Write the number 105 (10) in the binary system. CalculationPartial quotient Remai nder 105 : 2 = 52521 52 : 2 = 26260 26 : 2 = 13130 13 : 2 = 661 6 : 2 = 330 3 : 2 = 111 1 : 2 = 001 105 (10) = 1101001 (2)

8. Sequential multiplication method This method is used most frequently to express the decimal number smaller than one to the binary system. Exercise: Convert the number 0,725 (10) to the binary system. CalculationPartial resultResult 0,725 x 2 =1,451 0,45 x 2 =0,90 0,9 x 2 =1,81 0,8 x 2 =1,61 0,6 x 2 =1,21 0,2 x 2 =0,40 etc. Number 0,725 (10) = 0,101110… (2)

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 binary system use?

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

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

13. What is the value of the binary number 101 (2) in the decimal system? Binary system for 300

14. What is the value of the binary number 111 (2) in the decimal system? Binary system for 300

15. What is the value of the binary number 1101 (2) 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 binary system?

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

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