1. COMS 161 Introduction to Computing
Title: Digital Numbers
Date: February 4, 2005
Lecture Number: 11

2. Announcements

3. Review Numbers Decimal to Binary Conversion Adding binary numbers
Subtraction Method
Adding binary numbers

4. Outline Numbers Signed numbers Hexadecimal Binary Coded Decimal (BCD)
Binary to hexadecimal conversion
Binary Coded Decimal (BCD)

5. Binary Number System How about representing negative numbers?
Let the left most bit represent the sign (+, -) of the number
Called signed magnitude representation
[s][mag]

6. Signed Magnitude
One less bit to represent the magnitude

7. 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

8. 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

9. 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

10. Two’s Complement Representation
Both problems with the signed magnitude representation are solved with 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

11. 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

12. 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
Combatable but extends the ASCII standard

13. 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

14. 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

15. ASCII Table

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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 = =

22. 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

23. 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

24. 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 =