CIS 234: Numbering Systems

Dr. Ralph D. WestfallApril, 2010

Problem 1 computers only understand binary

coded data (zeros and ones) 00000000, 11111111, 01010101

people like to count in decimals00000000=0, 11111111=255,

01010101=85 1st problem: it is extremely hard for

people to work with binary data

Problem 2 (other Powerpoint) since computers only work with

numbers, they need to use numbers to identify letters to print or show on screen e.g., 01000001=65=A

people who don't read English also use computers

2nd problem: what kind of numbering should be used for different languages?

Problem 1 Solution making binary easier to work with create numbering systems that

are: compatible with binary numbers easier to read than binary

Numbering Systems all numbering systems have a "base"

digit position = base raised to a power any number raised to power of zero = 1

decimal system is "base 10"123 = 1 * 10^2 + 2 * 10^1 + 3 * 10^0 = 1 * 100 + 2 * 10 + 3 * 1

binary system is "base 2"1010 = 1 * 2^3 + 0 * 2^2 + 1 * 2^1 + 0 * 2^0 = 1 * 8 + 0 * 4 + 1 * 2 + 0 * 1

Can Convert Between Bases

converting binary to decimal1111 = 1 * 2^3 + 1 * 2^2 + 1 * 2^1 + 1 *

2^0 15 = 1 * 8 + 1 * 4 + 1 * 2 + 1 * 1

binary/decimal conversion is awkward binary & decimal numbers don't

match up well (sizes are inconsistent when convert between them)7=111 (3 bits), 8=1000 (4 bits)1001=9 (1 digit), 1010=10 (2 digits)

Hexadecimal Numbers (error) hex means 6, decimal means 10 hexadecimal means base 16 first 9 digits of "hex" same as in decimal

0 is 0, 1 is 1 … 9 is 9 in either system 10 in decimal is a in hex

11=b, 12=c, 13=d, 14=e, 15=f in hex 16=10 in hex

example: color codes in HTML

Conversions hex to decimal is easier than binary

2f = 2 * 16^1 + 15 * 16^0 = 473a = 3 * 16^1 + 10 * 16^0 = 58

hex to binary is even easier one hex digit for every 4 bits

f=1111, a=1010, 8=1000 2 hex digits = 1 byte

ff=11111111=255 (Start>All

Programs>Accessories>Calculator)(or Start>Run>calc>OK)

Octal Numbers octal means 8

octopus has 8 legs octal system uses base 8 has not caught on as much as hex

2 octal digits = 6 bits 3 octal digits = 9 bits 1 byte = 8 bits

Data Storage why is the range of the byte data

type from –128 to + 127? 8 bits = 1 byte

00000000=001111111=12710000000 =-128binary arithmetic11111111=-1 ( =127 - 128) sign is in left bit (0=plus, 1=minus)

Numbering System Calculations

01 binary+01=10 in binary

101 + 11 =1000

1 hexadecimal+9= a in hex

a+1= b

Microsoft Calculator Does Hex Start>All Programs>Accessories>

Calculator View>Scientific (or Start>Run>calc>OK)

click Hex radio button and try calculating with hex #s (including A through F keys)

click Bin and do binary calculations using only ones and zeros

Review Questions What two digits are used in

computer hardware? What does binary mean?

Give an example of a binary number What does hexadecimal mean?

Give an example of a hexadecimal number

Name some advantages of hex numbers over binary ones

Review Questions - 2 Convert 1010 into a decimal number

Into a hexadecimal number Convert binary 11 to a decimal

number What are the decimal values of the

following hexadecimal numbers? 5 a f

Review Questions - 3 Add 10 plus 11 in binary Is 10000000 a positive or negative

number in binary (on a PC)? What is the hexadecimal result of:

adding 2 to a? adding 5 + 5 + 5?

worksheet