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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: 0-9     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	    n-1            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: 0-7     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: 0-9, a-f     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 n-bit number (in weighted positional notation):       S     S    .  .  .   S    S    S        n-1   n-2            2    1    0    given a base b, this is the decimal value	  the summation (from i=0 to i=n-1) of S  x  b**i						iTRANSFORMATIONS BETWEEN BASES-----------------------------any 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 --> hexon representing nonintegers---------------------------what 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  . . .        n-1   n-2            2    1    0     -1   -2   -3                                         |                                           |  					  binary point       The decimal value is calculated in the same way as for       non-fractional 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

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