1. COMS 161 Introduction to Computing
Title: Digital Numbers
Date: February 2, 2005
Lecture Number: 10

2. Announcements
Homework 3
Due 2/04/05
Paper 1 proposal

3. Review Numbers Items verses bits Numeric ranges
Binary to Decimal Conversion
Decimal to Binary Conversion
Repeated division

4. Outline Numbers Decimal to Binary Conversion Adding binary numbers
Subtraction Method
Adding binary numbers
Signed numbers
Hexadecimal
Binary Coded Decimal (BCD)

5. Decimal to Binary Conversion
Algorithm:
Subtraction Method
Start with a positional number that is greater than the decimal number
If (the positional number is less than the decimal number)
Subtract the positional number from the decimal number
Write down a 1
Else
Write down a 0
Continue until the remainder is 0

6. Decimal to Binary Conversion
Positional value
Subtraction
Remainder
Binary
27 = 12810
10510 02
26 = 6410
4110
012
25 = 3210
910
0112
24 = 1610
01102
23 = 810
110
011012
22 = 410
21 = 210
20 = 110
010

7. Adding Binary Numbers Binary addition is just like decimal addition
Can only use the 2 (1 and 0) symbols of the binary alphabet
Carry ahead if needed

8. Adding Binary Numbers Rules of addition 0 + 0 = 1 0 + 1 = 1 1 + 0 = 1
1 + 1 = 10
= 11 1 1 1 17 11
+ 7
+ 1 2 4 1

9. 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]

10. Signed Magnitude
One less bit to represent the magnitude

11. 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

12. 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

13. 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

14. 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

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

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 left 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. 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

22. 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

23. 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