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 i TRANSFORMATIONS 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 <-- lsb 18/2 = 9 r=0 9/2 = 4 r=1 4/2 = 2 r=0 2/2 = 1 r=0 1/2 = 0 r=1 <-- msb 36 (base 10) == 100100 (base 2) 14/2 = 7 r=0 <-- lsb 7/2 = 3 r=1 3/2 = 1 r=1 1/2 = 0 r=1 <-- msb 14 (base 10) == 1110 (base 2) binary --> 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 3 hex --> binary (trivial!) just write down the 4 bit binary code for each hexadecimal digit example: 3 9 c 8 0011 1001 1100 1000 octal --> binary (just like hex to binary!) (trivial!) just write down the 8 bit binary code for each octal digit hex --> octal do it in 2 steps, hex --> binary --> octal decimal --> 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 notation normalization to keep a unique form for every representable noninteger, they are kept in NORMALIZED form. A normalized number will follow the following rule: 1 <= mantissa < base In this form, the radix point is always placed one place to the right of the first significant symbol (as above). on precision, accuracy, and significant digits These terms are often used incorrectly or ignored. They are important! A measurement (in a scientific experiment) implies a certain amount of error, depending on equipment used. Significant digits tell about that error. For example, a number given as 3.6 really implies that this number is in the range of 3.6 +- .05, which is 3.55 to 3.65 This is 2 significant digits. 3.60 really implies that this number is in the range of 3.6 +- .005, which is 3.595 to 3.605 This is 3 significant digits. So, the number of significant digits given in a number tells about how accurately the number is known. The larger the number of significant digits, the better the accuracy. Computers (or calculators, a more familiar machine) have a fixed precision. No matter what accuracy of a number is known, they give lots of digits in an number. They ignore how many significant digits are involved. For example, if you do the operation 1.2 x 2.2. given that each number has 2 significant digits, a correct answer is 1.2 x 2.2 ----- 24 + 24 ----- 264 --> 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: 101.001 (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 <-- most significant (f ) .6 1.2 1 -1 .2 0.4 0 .4 0.8 0 .8 (it must repeat from here!) ---- .8 is .1100 NON-BINARY FRACTIONS -------------------- same as with binary, only the base changes! f f . . . f f f . f f f . . . n-1 n-2 2 1 0 -1 -2 -3 | | radix point The decimal value is calculated in the same way as for non-fractional numbers, the exponents are now negative. example: 101.001 (octal) 1 x 8**2 + 1 x 8**0 + 1 x 8**-3 64 + 1 + 1/512 65 1/512 = 65.0019 (approx) 13.a6 (hexadecimal) 1 x 16**1 + 3 x 16**0 + a x 16**-1 + 6 x 16**-2 16 + 3 + 10/16 + 6/256 19 166/256 = 19.64 (approx) CONVERSION WITH OTHER BASES --------------------------- another base to decimal: Just plug into the summation. 134 (base 5) 1 x 5**2 + 3 x 5**1 + 4 x 5**0 25 + 15 + 4 44 (base 10) decimal to another base: Keep dividing by the base, same algorithm. 100 (base 10) to base 5 rem 100/5 20 0 20/5 4 0 4/5 0 4 100 (base 10) = 400 (base 5)