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Dale & Lewis Chapter 3 Data Representation Representing Real numbers • Need to represent numbers such as 0.543 or -17.45 • The decimal system uses the “decimal point” • Remember the definition of a number system, but extended to negative exponents dn-1 x bn-1 + dn-2 x bn-2 + … + d2 x b2 + d1 x b1 + d0 x b0 + d-1 x b-1 + d-2 x b-2 + … • So 0.543 is 0 x 100 + 5 x 10-1 + 4 x 10-2 + 3 x 10-3 • The decimal point position delimits where the exponents become negative Conversion from binary to decimal • Use the definition of a number in a positional number system with base 2 • Evaluate the formula using decimal arithmetic • Example 10.1011 = 1 x 21 + 0 x 20 + 1 x 2-1 + 0 x 2-2 + 1 x 2-3 + 1 x 2-4 = 2 + 0.5 + 0.125 + 0.0625 = 2.6875 • Generalizing, the “decimal point” is called radix point in non-decimal number systems Conversion from decimal to binary • Integer part: convert separately, as done before • Fractional part: − Repeatedly multiply by 2 − Integer part is the next digit − Fraction is developed left to right • Example: convert 3.14579 to binary • Integer part is 11 .14579 x 2 = 0.29158 .0 .29158 x 2 = 0.58316 .00 .58316 x 2 = 1.16632 .001 .16632 x 2 = 0.33264 .0010 etc... 3.14579 in binary is 11.0010… Rounding errors • Try representing .1 (decimal) into binary • How would one represent fractions with base 5 or 3? • Infinite sequences, a computer with fixed number of bits will make an approximation in the representation • Needs to be taken into account, or it will lead to rounding errors Patriot missile system Floating point numbers • Real numbers are represented using a concept inherited from scientific notation • Scientific notation is a way of writing numbers that are too big or too small to be conveniently written in decimal form • The following all represent +1234.56 +123456.0 +12345.6 +1234.56 +123.456 +12.3456 +1.23456 +0.123456 +0.0123456 x10-2 x10-1 x100 x101 x102 x103 x104 standard representation x105 Floating point numbers • The representations differ in that the decimal point “floats” to the left or right by adjusting the exponent • Any real number x can be written as x = ±f x be where b > 1 is the base, e is an integer for the exponent, and the fraction is 1/b ≤ f < 1 (normalized) • We know how to express each part in binary, but convention for the integer e uses Excess notation Excess notation • Another way to express integers (we’ve seen signmagnitude and two’s complement) • This shifts the list of numbers represented by bits to include negative numbers, with 10…0 representing 0 • Convert binary excess-n to decimal: (binary value)-n − Example for excess-8: 0101 5-8 = -3 • Convert decimal to binary excess-n: (decimal value)+n − Example for excess-8: 6 6+8 = 1110 Excess notation Floating point numbers in binary • An example for “single precision” binary floating point number (32 bits) − S is the sign: 0 positive, 1 negative − E is the exponent (here in 8 bits, 256 values, excess-128 notation) − M is the manitssa (fractional part) has an implicit radix point at the beginning (here 23 bits, 8,388,608 values) SEEEEEEEEMMMMMMMMMMMMMMMMMMMMMMM 10010100101010011101001010001101 is -.01010011101001010001101 x 2-87 Floating point number examples • Using 8-bit floating point representation (SEEEMMMM) − Convert 3.14579 to binary 3.14579 = 11.0010… 11.0010… x 20 .110010… x 2+2 01101101 (rounded) − Convert 0.1 to binary 0.1 = 0.0001100110011… .0001100110011… x 20 .1100110011… x 2-3 00011101 (rounded)