CIS 234: Numbering Systems

CIS 234: Numbering Systems Dr. Ralph D. Westfall April, 2010

Problem 1 • computers only understand binary coded data (zeros and ones) • 00000000, 11111111, 01010101 • people like to count in decimals 00000000=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 decimal 1111 = 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 = 47 3a = 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=0 01111111=127 10000000 =-128 binary arithmetic 11111111=-1 ( =127 - 128) • sign is in left bit (0=plus, 1=minus)

01 binary +01 =10 in binary 101 + 11 =1000 1 hexadecimal +9 = a in hex a +1 = b Numbering System Calculations

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

CIS 234: Numbering Systems