1. Sistem Bilangan
Oleh :
Sri Heranurweni

2. Analogue vs Digital Analogue * Continuous range of value
* Precision limited by Noise
Digital
* Discrete range of values
* Precision limited by number of “Bit”

3. Analogue vs Digital
Analogue
Digital

4. Analogue vs Digital
The real world is analogue ( by because all signal in world be shape analogue)
But in controlling, Digital one had using for process.
Both of signal had been converter each other

5. Analoge vs Digital Analogue Analogue Digital Processing D to A A to D
Why Digital Only by using in Processing?
^ Adventure in integrated Circuit has made the complex processing of
digital data.
^ Digital Control processing has made easier than analogue
^ Digital circuits are inherently more noise resistant

6. Digital and Boolean Digital represented by boolean logic
Boolean is the name of mathematician’s expert
Now boolean is called by conventional logic because there is new logic that called by fuzzy logic
But all electronic still using boolean logic to processing the controlling system

7. Why Boolean
It is convenient in electrical system to use a two-value system to represent value true/false, on/off, yes/no and 1/0
* Two voltage or current levels can be used
* Easier to process and distribute reliably (diandalakan)
* Don’t think of them as numbers (even though we often represent them as 0/1 for brevity(ketangkasan))
The need for binary numbers
* Multi-value quantities need to be represented in the digital system. Therefore need numbers made up from the simple two value system

8. Positional Number System
Decimal point
7x10-1
7x10-2
8x10-3
8 x 100
7 x 101
5 x 102
3 x 103
Base 10, weigthing are powers of 10

9. Unsigned binary numbers
Binary point
1 x 2-1 = 0.500
0 x 2-2 = 0.000
1 x 2-3 = 0.125
0 x 20= 0.000
0 x 21= 0.000
1 x 22= 4.000
1 x 23= 8.000
Each bit of the
Number may be
Representaed by
A Boolean value
Binary, weightings are powers of 2

10. Multi-precision Arithmatic
Additional of A and B A1 B1 A2 B2 + B3
Carry
Flag
Out In

11. Multi-precision Arithmatic
Carry
Flag
Carry
Out
Carry In A1 B1 A2 B2 - A2 B3

12. Hexadecimal Numbers
660 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 2 3 4 5 6 7 8 9 A B C D E F 4
: 16 41 9
: 16 2
Hexadecimal : Hex
215 13
: 16 7
Hexadecimal : 7D Hex

13. Hexadecimal Numbers 660 215 1 2 3 4 5 6 7 8 9 A B C D E F 0000 00011 2 3 4 5 6 7 8 9 A B C D E F
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
660
215 D

14. Decimal to Binary Generetee each digit by successive division
Or multiplication.
There is no guarantee the fraction will be
finite
Number =
36.375
Base = 2
Decimal Number
Binary Digits
Converter Number
0.5 1
0.75
0.375 36 18
010010 9
01001 4
0100
010 01
Fractional part – Multiplication by base
Whole part – divition by base

15. Binary Additional Easy Layaou ? 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1
= 0
= 1
= 1
= 0 carry 1
Easy Layaou ?

16. Binary Addition 190 + 141 =331 Carry out of Each column 1 1 1 1 1 1 1 1 1 1
Carry out of
8-bit number

17. Binary Subtraction
A borrow-out of 1 from
This column becomes a borrow in
of 2 in this column
229 – 46 = 183 2 2 2 2 2
Borrow in from
Left column 1 1 1 1 1
Borrow out 1 1 1 1 1 1
Both rows subtracted

18. Exercise Convert to 8-bit binary and do the arithmetic operation
* *
* 224 – 134 *
* 112 – 89 * 111 – 25
Convert back to decimal and check the result

19. Binary Number Circle 4 – bit Binary Number Circle 11 1110 - 1 1
In real hardware there is a fixed number
Of bits available. We often ignore leading zeros
But they are still there!
Examlpe :
If we only use 4 bits then the binary
Counting sequence “wraps around”
At 15 ↔ 0
= 10
4 – bit
Binary
Number Circle

20. Binary Number Circle 4 – bit Binary Number Circle 8 1000 - 14 - 1110
Subtracting across the boundary
Still “works” if you think of result
As the distance on the number
Circle.
4 – bit
Binary
Number Circle
(Module arithmetic – ignore
The borrow /carry)
(-1)1010

21. Representing –ve Number
Several choices for natation
* sign + magnitude notation
* 1’s complement
* 2’s complement notation
* various ‘excess codes ‘

22. Sign Number – sign + magnitude Notation
Sign Bit Magnitude
0  +ve
Simple binary number
1  - ve
How about Null or Zero
Problem ?
+ 0 
- 0  1000

23. Signed Numbers – Sign + magnitude Notation
Arithmetic
Difficult to do – have to work out that operation to perform
actually calculate –(6-5) i.e. exchange the operands and do subtraction!
actually calculate –(5+6) i.e. negate the addition of the negated numbers !
Required action depends the signs of the numbers and on which has the large magnitude. Natural for us –a bit hard for the computer since the only way it can work out the bigger number is to do a subtraction!

24. Sign + Magnitude Examples
Value
4-bit sign + magnitude
8-bit sign + magnitude +7
0111 +6
0110 …… +1
0001 +0
0000 -0
1000 -1
1001 -2
1010 -7
1111

25. Sign Numbers – 2’s Complement
As for straight binary numbers but with the weighting of the most significant bit being negative
Example
* 4 bit – weights are -8, 4,2,1
* 8 bit – weights are -128, 64,32,16,8,4,2,1
Need to know how many bits are being used to work out the value of the number – don’t omit leading zeroes

26. Sign Numbers – 2’s Complement
Binary point
1 x 2-1 = 0.500
0 x 2-2 = 0.000
1 x 2-3 = 0.125
0 x 20=
0 x 21=
1 x 22=
1 x 23=
Sign Bit
-4.375
Binary, weightings are powers of 2

27. 2’s Complement Examples
Value
4-bit sign 2’s complement
8-bit sign complement +7
0111 +6
0110 …… +1
0001 +0
0000 -1
1111 -2
1110 -7
1001 -8
1000

28. 2’s Complement Examples
Example : -4 (decimal)
Become 4 = ( binary)
= 1x22 = 4
2’s Complement
-4= 1100 (binary)
= -(23) + 22 =
= -4

29. Exercise
Converse decimal number above into negative (2’s complement) :
-7 ( 4 digit ) (4 digit)
-7 (8 digit) (8 digit)
-12 (8 digit) (8 digit)
-20 (8 digit) (digit)
-100 (8 digit) (digit)

30. Addition 2’s Complement
For 4 digit :
= =7

31. Addition 2’s Complement
For 4 digit
-(8) = -3
Carry out

32. Exercise For 4 Digit : 7 + (-5) -6 + -1 3 + 4 2 + 3 -4 + 7
Converse all item to digital and addition. And then Converse to decimal again

33. Subtraction 2’s Complement
+ 3 (0011)
Discard

34. Subtraction 2’s Complement
(-8)
(-3) =

35. Exercise
for 4 digit . Converse decimal above to digit and subtraction. After that converse to decimal again :
(+3) – (-3)
(-4) – (+2)
(-8)- (+4)
(-3) – (-4)
(7) – (5)

36. Condition code register (CCR)
2’s Complement ALU
Addition and subtraction use the same rules as unsigned binary.
Same hardware may be used for both
Carry (C) is used for unsigned, overflow (v) for signed
C=Carry
V=overflow OP
Signed Numbers
Signed Numbers
The same
hardware OP
C=Carry
Signed Numbers
V=overflow
Arithmetic Flags in
Condition code register (CCR)