1. COMS 161 Introduction to Computing
Title: Digital Numbers
Date: February 7, 2005
Lecture Number: 12

2. Announcements Test next (not this) Wednesday First draft of paper
Due 2/11/05

3. Review
Numbers
Signed numbers
Two’s complement numbers

4. Outline Numbers Numeric representation of letters Hexadecimal
Binary to hexadecimal conversion
Hexadecimal to binary conversion
Binary Coded Decimal (BCD)

5. Digital Letters Digital system All entities are represented as numbers
How do we represent the letters in the English language
The letters form a discrete set (unique unambiguous, precise)
No sampling is needed
Simply need a mapping from each letter to a numerical representation
A = 65
B = 66

6. Digital Letters Important that all converters use the same mapping
Otherwise the inverse process (converting a number to a letter) would give incorrect results
Computers in the US primarily use the American Standard Code for Information Interchange (ASCII)
Unicode is an international standard
Compatible but extends the ASCII standard

7. ASCII mapping
How many bits will I need to encode the letters of the English alphabet?
Upper case
Lower case
Decimal digits
Punctuation
Arithmetic symbols
Printer control characters

8. ASCII mapping Letters in the English language A = 6510 = 0100 00012
B = 6610 = …
Z = 9010 =
a = 9710 =
z = =
Numbers are still left over for punctuation

9. ASCII Table

10. 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

11. 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

12. 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 right end
Substitute the hex symbol
9010 = = 5A16
Is the base 16 really needed?
6610 = = 4216

13. 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

14. 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

15. Hexadecimal to decimal conversion
Same procedure as converting a binary number to a decimal number
The digits of the hex number are the coefficients of the corresponding positional weighting factor
ABC16 = 0xABC = A * B * C * 160
= A * B * 16 + C * 1
= 10 * * * 1 = =

16. 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

17. 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

18. 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
Why?
15910 =