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Number Systems

Number Systems. Benchmark Companies Inc PO Box 473768 Aurora CO 80047. Number Systems:. Decimal Binary Hexadecimal Octal Binary Coded Decimal (BCD). 10 digits: 0 1 2 3 4 5 6 7 8 9 Counting beyond 9 requires additional place values to begin. This will go on to infinity: i.e.

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Number Systems

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  1. Number Systems Benchmark Companies Inc PO Box 473768 Aurora CO 80047

  2. Number Systems: • Decimal • Binary • Hexadecimal • Octal • Binary Coded Decimal (BCD)

  3. 10 digits: 0 1 2 3 4 5 6 7 8 9 Counting beyond 9 requires additional place values to begin. This will go on to infinity: i.e. … 8,9,10,11……98,99,100,101…999,1000,... The Decimal System The decimal system is a base 10 (modulo 10) number system:

  4. Counting beyond 9 requires additional place values as powers of 10: ______ ______ ______ ______ ______ . _____ 10000 1000 100 10 1 .1 (104) (103) (102) (101) (100) . (10-1) 1 w/4 0’s 1 w/3 0’s 1 w/2 0’s 1 w/1 0’s 1 w/0 0’s . The Decimal System

  5. The Binary System: 2 digits: 0 or 1 (in digital terms, logic 0 or logic 1) Counting beyond 1 requires additional place values. This will go on to infinity: i.e. 0,1,10,11,100,101,110,111,1000,1001,... The Binary System is a base 2 (mod 2) number system:

  6. The Binary System: 2 digits: 0 or 1 (in digital terms, logic 0 or logic 1) Counting beyond 1 requires additional place values as powers of 2: ______ ______ ______ ______ ______ . _____ 16 8 4 2 1 .5 (24) (23) (22) (21) (20) . (2-1) The Binary System is a base 2 (mod 2) number system:

  7. Example:Convert 3710 to binary. It is important to be able to convert binary to decimal and vice-versa. In this example, convert 37 base 10 to it’s binary (base 2) equivalent number. Base 10 >>> Base 2

  8. Example:Convert 3710 to binary. METHOD I: Sum-of-weights: ____ ____ ____ ____ ____ ____ 32 16 8 4 2 1 METHOD II: Repeated-division-by-base (here, base 2) • 37/2 = 18 remainder of 1 This is your LSB • 18/2 = 9 remainder of 0 • 9/2 = 4 remainder of 1 • 4/2 = 2 remainder of 0 • 2/2 = 1 remainder of 0 • ½ = 0 remainder of 1 This is your MSB This process gives you the same result: 3710 is 100101 in binary. 1 0 0 1 0 1

  9. Example:Convert 10110102 to decimal: In this example, convert 1011010 base 2 to it’s decimal (base 10) equivalent number. Base 2 >>> Base 10

  10. Example:Convert 10110102 to decimal: Sum-of-weights uses total of each place value: 1x26 + 0x25 + 1x24 + 1x23 + 0x22 + 1x21 +0x20 1x64 + 0x32 + 1x16 + 1x8 + 0x4 + 1x2 + 0x 1 64 + 0 + 16 + 8 + 0 + 2 + 0 = 9010

  11. The Hexadecimal System “Hexa” = 6 “Decimal” = 10 16 digits: 0123456789 A b C d E F representing decimal 10 through decimal 15 (use of lower case helps differentiate between b and 8 or d and 0 in a digital display) The Hexadecimal system is a base 16 (mod 16) number system:

  12. Convert 5810 to hexadecimal: Sum-of-weights: ____ ____ ____ 256 16 1 Check: 3x16 + 10x1 = 48+10 = 5810 = 3A16 ***Repeated division-by-base is most effective for larger conversions. 0 3 A

  13. Tips for Conversions: THE SHORTCUT FOR CONVERTING BINARY TO HEXADECIMAL HEXADECIMAL TO BINARY Since there is a relationship between 2 and 16 (24 = 16), there is a relationship between the place values in binary and the place values in hexadecimal – look for groups of 4 instead of 3. Example: Convert 101101012 to hexadecimal: 1011 0101 = b516

  14. Tips for Conversions (Continued): Convert 3F716 to binary: *Remember to represent each digit as a 4-bit binary word!* 0011 1111 0111 Drop initial 0’s to simplify. 3F716 = 1111110111

  15. Binary Coded Decimals (BCD) • Uses a 4-bit binary representation of each digit in decimal • Example: 672 in BCD would be 0110 0111 0010 • Example: 1001 0110 0101 1000 is BCD for 9658 • ***In BCD, there will not be values beyond 1001 (decimal 9)

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