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CONVERTING BASE 8 TO DECIMAL CALCULATOR
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Chapter 4  Number Systems
about number systems.here's the decimal number system as an example: digits (or symbols) allowed: 09 base (or radix): 10 the order of the digits is significant 345 is really 3 x 100 + 4 x 10 + 5 x 1 3 x 10**2 + 4 x 10**1 + 5 x 10**0 3 is the most significant symbol (it carries the most weight) 5 is the least significant symbol (it carries the least weight)here's a binary number system: digits (symbols) allowed: 0, 1 base (radix): 2 each binary digit is called a BIT the order of the digits is significant numbering of the digits msb lsb n1 0 where n is the number of digits in the number 1001 (base 2) is really 1 x 2**3 + 0 x 2**2 + 0 x 2**1 + 1 x 2**0 9 (base 10) 11000 (base 2) is really 1 x 2**4 + 1 x 2**3 + 0 x 2**2 + 0 x 2**1 + 0 x 2**0 24 (base 10)here's an octal number system: digits (symbols) allowed: 07 base (radix): 8 the order of the digits is significant 345 (base 8) is really 3 x 8**2 + 4 x 8**1 + 5 x 8**0 192 + 32 + 5 229 (base 10) 1001 (base 8) is really 1 x 8**3 + 0 x 8**2 + 0 x 8**1 + 1 x 8**0 512 + 0 + 0 + 1 513 (base 10)here's a hexadecimal number system: digits (symbols) allowed: 09, af base (radix): 16 the order of the digits is significant hex decimal 0 0 1 1 . . . 9 9 a 10 b 11 c 12 d 13 e 14 f 15 a3 (base 16) is really a x 16**1 + 3 x 16**0 160 + 3 163 (base 10)given all these examples, here's a set of formulas for the general case. here's an nbit number (in weighted positional notation): S S . . . S S S n1 n2 2 1 0 given a base b, this is the decimal value the summation (from i=0 to i=n1) of S x b**i iTRANSFORMATIONS BETWEEN BASESany base > decimal just use the definition give above.decimal > binary divide decimal value by 2 (the base) until the value is 0 example: 36/2 = 18 r=0 octal 1. group into 3's starting at least significant symbol (if the number of bits is not evenly divisible by 3, then add 0's at the most significant end) 2. write 1 octal digit for each group example: 100 010 111 (binary) 4 2 7 (octal) 10 101 110 (binary) 2 5 6 (octal)binary > hex (just like binary to octal!) 1. group into 4's starting at least significant symbol (if the number of bits is not evenly divisible by 4, then add 0's at the most significant end) 2. write 1 hex digit for each group example: 1001 1110 0111 0000 9 e 7 0 1 1111 1010 0011 1 f a 3hex > binary (trivial!) just write down the 4 bit binary code for each hexadecimal digit example: 3 9 c 8 0011 1001 1100 1000octal > binary (just like hex to binary!) (trivial!) just write down the 8 bit binary code for each octal digithex > octal do it in 2 steps, hex > binary > octaldecimal > hex do it in 2 steps, decimal > binary > hexon representing nonintegerswhat range of values is needed for calculations very large: Avogadro's number 6.022 x 10 ** 23 atoms/mole mass of the earth 5.98 x 10 ** 24 kilograms speed of light 3.0 x 10 ** 8 meters/sec very small: charge on an electron 1.60 x 10 ** (19)scientific notation a way of representing rational numbers using integers (used commonly to represent nonintegers in computers) exponent number = mantissa x base mantissa == fraction == significand base == radix point is really called a radix point, for a number with a decimal base, its called a decimal point. all the constants given above are in scientific notationnormalization to keep a unique form for every representable noninteger, they are kept in NORMALIZED form. A normalized number will follow the following rule: 1 2.64 > 2.6 or 2.6 + .05 a calculator will most likely give an answer of 2.640000000, which implies an accuracy much higher than possible. The result given is just the highest precision that the calculator has. It has no knowledge of accuracy  only precision.BINARY FRACTIONS f f . . . f f f . f f f . . . n1 n2 2 1 0 1 2 3   binary point The decimal value is calculated in the same way as for nonfractional numbers, the exponents are now negative. example: 1011 (binary) 1 x 2**2 + 1 x 2**0 + 1 x 2**3 4 + 1 + 1/8 5 1/8 = 5.125 (decimal) 2**1 = .5 2**2 = .25 2**3 = .125 2**4 = .0625 etc.converting decimal to binary fractions Consider left and right of the decimal point separately. The stuff to the left can be converted to binary as before. Use the following algorithm to convert the fraction: fraction fraction x 2 digit right of point .8 1.6 1