Transcript AVOP-ELEKTRO-HOL-004

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.
Numerical systems
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:
2n; 2n-1;…; 24 = 16; 23 = 8; 22 = 4; 21 = 2; 20 = 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*25 + 0*24 + 1*23 + 0*22 +1*21 + 1*20 + 1*2-1 + 0*2-2 +1*2-3
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.
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.
.
Power
Variation
Result
27 = 128
151 – 128 = 23
1
26 = 64
23 – 64 = - 41
0
25 = 32
23 – 32 = - 9
0
24 = 16
23 – 16 = 7
1
23 = 8
7 – 8 = -1
0
22 = 4
7–4=3
1
21 = 2
3–2=1
1
20 = 1
1–1=0
1
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.
105(10) = 1101001(2)
Calculation
Partial
quotient
Remai
nder
105 : 2 = 52
52
1
52 : 2 = 26
26
0
26 : 2 = 13
13
0
13 : 2 = 6
6
1
6:2=3
3
0
3:2=1
1
1
1:2=0
0
1
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.
Calculation
Partial result
Result
0,725 x 2 =
1,45
1
0,45 x 2 =
0,9
0
0,9 x 2 =
1,8
1
0,8 x 2 =
1,6
1
0,6 x 2 =
1,2
1
0,2 x 2 =
0,4
0 etc.
Number 0,725(10) = 0,101110…(2)
The End
Question chart:
Numerical projection
Numerical projection
Numerical projection
for 100
for 300
for 500
Prémie
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Prémie
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1
2
2
2
3
3
3
Prémie
A
B
C
D
E
F
G
H
Binary system for 100
How many numbers does the binary system use?
Binary system for 100
What is the other name for the binary system?
Binary system for 100
What is the numerical basis used in the binary system?
Binary system for 300
What is the value of the binary number 101(2) in the decimal
system?
Binary system for 300
What is the value of the binary number 111(2) in the decimal
system?
Binary system for 300
What is the value of the binary number 1101(2) in the decimal
system?
Binary system for 500
What is the value of the decimal number 123(10) in the binary
system?
Binary system for 500
What is the value of the decimal number 321(10) in the binary
system?
Binary system for 500
What is the value of the decimal number 1234(10) in the binary
system?
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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-prokonkurenceschopnost
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