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CIS 234: Numbering Systems

This article discusses the different numbering systems used in computing, including binary, decimal, hexadecimal, and octal. It covers conversions between these systems and their advantages and disadvantages. It also provides example calculations using Microsoft Calculator.

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CIS 234: Numbering Systems

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

  2. 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

  3. 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?

  4. Problem 1 Solution • making binary easier to work with • create numbering systems that are: • compatible with binary numbers • easier to read than binary

  5. 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

  6. 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)

  7. 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

  8. 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)

  9. 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

  10. 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)

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

  12. 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

  13. 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

  14. 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

  15. 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

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