1. COMS 161 Introduction to Computing
Title: Digital Numbers
Date: January 26, 2005
Lecture Number: 7

2. Announcements Homework 2 Read the paper “The World Wide Web”
Be prepared to discuss it
Why do sea-gulls fly over the sea? Because if they flew over the bay they would be bagels!
A man left home running. He ran a ways and then turned left, ran the same distance and turned left again, ran the same distance and turned left again. When he got home there were two masked men. Who were they? A catcher and an umpire

3. Review
Digital information
Advantages
Disadvantages

4. Outline Numbers Non-positional Positional Decimal Octal Hexadecimal
Definition
Decimal
Octal
Hexadecimal
Unsigned and signed numbers
Binary Coded Decimal (BCD)

5. Digital Domain Converting analog to digital information
We need a digital representation of the information
Recall, digital is a discrete system
Where symbols are numbers
The digital advantages
Therefore, we need a numerical encoding of the data
Numerical means numbers

6. Numbers Two types of notion used to represent numbers
Non-positional notation
No special significance is given to order
Counting numbers on your fingers
Tick mark counting method
The number of items is important, not the order
My honey do list
Not important which task I do first, just so I get them all done

7. Numbers Positional notation
Significance is given to order the digits appear in the number
The decimal numbering system uses positional notation
This is the system we use
365 is not the same a 653
These are completely different numbers
They use the same digits

8. Positional notation 365 means
Three hundreds
Six tens
Five ones
Each digit is multiplied by a power of 10

9. Decimal number system Synonyms
Decimal notation
Base-10 system
Both digits and their location in the number are important
Ten unique symbols (digits)
0, 1, 2, …, 9

10. Octal number system
Decimal is not the only positional number system available
Octal
Positional,base-8 system
Each digit is multiplied by a power of 8
Eight unique symbols (digits)
0, 1, 2, …, 7

11. Binary number system Binary Positional, base-2 system
Each digit is multiplied by a power of 2
Two unique symbols (digits), 0 and 1

12. Binary number system Digital and binary relationship
The language of computers use binary digits
Only 2 possible values
0 and 1
Much simpler to make electronics that distinguish between one of two values
Distinguishing between more than two values is very difficult

13. Binary number system Since binary digits have two possible values
Binary digits are called bits
They only contain a little “bit” of information
Numbers represented in binary form will (most likely) require more digits (bits) than the decimal form

14. Binary Number System How do 2 bits represent 4 values? Bit Pattern
Numeric Value
Item Represented 00
Black 01 1
White 10 2
Red 11 3
Green

15. Binary Number System How do 3 bits represent 8 values? Bit Pattern
Numeric Value
Item Represented
000
Black
001 1
White
010 2
Red
011 3
Green
100 4
Blue
101 5
Purple
110 6
Magenta
111 7
Sky blue

16. Binary Number System How do 4 bits represent 16 values? Bit Pattern
0000
1000 8
0001 1
1001 9
0010 2
1010 10
0011 3
1011 11
0100 4
1100 12
0101 5
1101 13
0110 6
1110 14
0111 7
1111 15

17. Binary Number System
Bit patterns and numeric values are consistent with other slides
Acceptable to add leading 0’s if desired

18. Binary number system To represent more information
Lump together multiple bits called strings
One bit: 2 values
Two bits: 4 values (00, 01, 10, 11)
Three bits: 8 values (000, 001, …, 111)
Four bits: 16 values (0000, …, 1111)
Five bits: 32 values
Six bits: 64 values
Seven bits: 128 values
Eight bits: values

19. Binary number system number of values = 2number of bits In general
1 = 20
2 = 21
4 = 22
8 = 23
16 = 24
32 = 25
64 = 26
128 = 27
256 = 28

20. Binary number system Common grouping
4 bits: nibble
8 bits: byte
One byte represents 256 different values or items

21. Binary Number System
We can represent the non-negative numbers (unsigned number)
How about representing negative numbers?
Let the left most bit represent the sign (+, -) of the number
Called signed magnitude representation
[s][mag]
0 – 2number of bits - 1

22. Signed Magnitude
One less bit to represent the magnitude

23. Signed Magnitude Problems Two values of 0 Incorrect arithmetic
More difficult to detect than one value of 0
Incorrect arithmetic
2 – 1 = 2 + (-1) = 1

24. Two’s Complement Representation
Sign bit in a sense
Positive numbers
The leading bit (left most) is zero
The same as signed magnitude
Negative numbers
The leading bit is one
Defined so that when added to their corresponding positive number the answer is zero

25. Two’s Complement Representation
Bit Pattern
Value
0000
1000 -8
0001 1
1001 -7
0010 2
1010 -6
0011 3
1011 -5
0100 4
1100 -4
0101 5
1101 -3
0110 6
1110 -2
0111 7
1111 -1

26. Two’s Complement Representation
Problems with signed magnitude representation are solved with the two’s complement representation
There is only value of zero
Arithmetic is correct
Solution is in two’s complement form
2 – 1 = 2 + (-1) = 1

27. Binary number system Letters in the English language…
Z = 9010 =
a = 9710 =
Z = =
Numbers are still left over for punctuation

28. Binary number system Precision
The number of bits used to represent an item
Letter: precision of 8 bits
Integer (whole number): precision of 32 or 64 bits
Always finite
Computers have finite precision
Presents some limitations

29. Hexadecimal number system
Sometimes called hex
Positional,base-16 system
Each digit is multiplied by a power of 16
Sixteen unique symbols (digits)
0, 1, 2, …, 15
Symbol a or A for 10
Symbol b or B for Symbol e or E for 14
Symbol c or C for Symbol f or F for 15
Symbol d or D for 13

30. Hexadecimal number system
A hex number can represent 16 different items
Equivalent to 4 bits
Makes it easy to convert between binary and hex
Group bits by 4’s from the left end
Substitute the hex symbol
9010 = = 5A16
Is the base 16 really needed?
6610 = = 4216

31. Hexadecimal number system
Use the backwards conversion to convert hex to binary
One hex digit is equivalent to 4 bits
Substitute the binary nibble
Always start at the right end
Add zeros to the left end as necessary to fill in 4 bits

32. Hexadecimal number system
BIN
0000 1
0001 2
0010 3
0011 4
0100 5
0101 6
0110 7
0111 8
1000 9
1001 A 10
1010 B 11
1011 C 12
1100 D 13
1101 E 14
1110 F 15
1111

33. Digitization The process of converting analog information into binary
Discrete forms are unambiguous
Text and numbers are discrete
Conversion of discrete to digital
Come up with a mapping
As we did with the letters

34. Binary Coded Decimal Integers (whole numbers)
One mapping is to use its binary equivalent
Binary Coded Decimal (BCD)
010 = 00002
110 = 00012 …
910 = 10012
Need a minimum of 4 bits to represent 10 different values
Some 4 bit quantities are wasted

35. Binary Coded Decimal String of decimal digits
Each decimal digit is represented by 4 bits
The number of bits needed to represent different numbers vary
Performing arithmetic is complicated