1. Chapter 1
Number System
RGGP, Narwana

2. CHAPTER 1 PREVIEW Counting in Decimal and Binary Place Value
Binary to Decimal Conversion
Decimal to Binary Conversion
Electronic Translators
Hexadecimal Numbers
Octal Numbers

3. COUNTING IN DECIMAL AND BINARY
Number System -
Code using symbols that refer to
a number of items.
Decimal Number System -
Uses ten symbols (base 10 system)
Binary System -
Uses two symbols (base 2 system)

4. Introduction to Numbering Systems
We are all familiar with the decimal number system (Base 10). Some other number systems that we will work with are:
Binary  Base 2
Octal  Base 8
Hexadecimal  Base 16

5. PLACE VALUE Numeric value of symbols in different positions.
Example - Place value in binary system:
Place Value 8s 4s 2s 1s
Binary
Yes
Yes No No
Number 1 1
RESULT: Binary 1100 = decimal = decimal 12

6. BINARY TO DECIMAL CONVERSION
Convert Binary Number to a Decimal Number:
Binary
Decimal
= 51

7. TEST 9 Binary 1001 = 15 Binary 1111 = 2 Binary 0010 =
Convert the following binary numbers into decimal numbers: 9
Binary = 15
Binary = 2
Binary =

8. DECIMAL TO BINARY CONVERSION
Divide by 2 Process
Decimal #
13 ÷ 2 = remainder 1
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1 1 1 1

9. Convert the following decimal numbers into binary:
TEST
Convert the following decimal numbers into binary:
1011
Decimal =
0100
Decimal =
10001
Decimal =

10. HEXADECIMAL NUMBER SYSTEM
Uses 16 symbols -Base 16 System
0-9, A, B, C, D, E, F
Decimal 1 9 10 15 16
Binary
0001
1001
1010
1111
10000
Hexadecimal 1 9 A F 10

11. HEXADECIMAL AND BINARY CONVERSIONS
Hexadecimal to Binary Conversion
Hexadecimal C 3
Binary
Binary to Hexadecimal Conversion
Binary
Hexadecimal E A

12. DECIMAL TO HEXADECIMAL CONVERSION
Divide by 16 Process
Decimal #
47 ÷ 16 = 2 remainder 15
2 ÷ 16 = 0 remainder 2 2 F

13. HEXADECIMAL TO DECIMAL CONVERSION
Convert hexadecimal number 2DB to a decimal number
256s 16s 1s
Place Value
D B
Hexadecimal
(256 x 2) (16 x 13) (1 x 11)
= 731
Decimal

14. TEST Convert Hexadecimal number A6 to Binary A6 =
Convert Hexadecimal number 16 to Decimal
16 =
22 (Decimal)
Convert Decimal 63 to Hexadecimal
63 =
3F (Hexadecimal)

15. Uses 8 symbols -Base 8 System
OCTAL NUMBERS
Uses 8 symbols -Base 8 System
0, 1, 2, 3, 4, 5, 6, 7
Decimal 1 6 7 8 9
Binary
001
110
111
Octal 1 6 7 10 11

16. Binary-Coded Decimal (BCD)
Digit
Bit pattern
0000 1
0001 2
0010 3
0011 4
0100 5
0101 6
0110 7
0111 8
1000 9
1001
Four bits per digit
Note: the following bit patterns are not used:

17. Example
= ? (in BCD)

18. ASCII Features 7-bit code 8th bit is unused (or used for a parity bit)
27 = 128 codes
Two general types of codes:
95 are “Graphic” codes (displayable on a console)
33 are “Control” codes (control features of the console or communications channel)

19. ASCII Chart

20. Addition (decimal)

21. Addition (binary)

22. Addition (binary)

23. Addition (binary)

24. Multiplication (decimal)

25. Multiplication (binary)

26. Multiplication (binary)
It’s interesting to note that binary multiplication is a sequence of shifts and adds of the first term (depending on the bits in the second term.
is missing here because the corresponding bit in the second terms is 0.

27. Methods for representing signed ints.
signed magnitude
1’s complement (diminished radix complement)
2’s complement (radix complement)

28. Signed magnitude Ex. 4-bit signed magnitude 1 bit for sign
3 bits for magnitude

29. 1’s Complement 1’s complement of N is defined as (2n -1)-N.
If n=4 have (2n -1) being = 1111
So for n=4 would subtract any 4-bit binary number from 1111.
This is just inverting each bit.
Example: 1’s compliment of is

30. 2’s complement The 2’s complement is defined as 2n-N
Can be done by subtraction of N from 2n or adding 1 to the 1’s complement of a number.
For 6 = 0110
The 1’s complement is 1001
The 2’s complement is 1010

31. Operation with 2’s complement
Add 4 and -6
Will use the 2’s complement of -6 or 1010
1110
And taking the 2’s complement of 1110 get = 0010

32. ELECTRONIC TRANSLATORS
Devices that convert from decimal to binary numbers and from binary to decimal numbers.
Encoders -
translates from decimal to binary
Decoders -
translates from binary to decimal

33. ELECTRONIC ENCODER - DECIMAL TO BINARY
Binary output
Decimal input
Decimal to
Binary
Encoder 3 7 5
Encoders are available in IC form.
This encoder translates from decimal input to binary (BCD) output.

34. ELECTRONIC DECODING: BINARY TO DECIMAL
Binary input
Decimal output
Binary-to-
7-Segment
Decoder/
Driver
Electronic decoders are available in IC form.
This decoder translates from binary to decimal.
Decimals are shown on an 7-segment LED display.
This decoder also drives the 7-segment display.