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e=5; s=1.234571 − e=5; s=1.234567 ----- e=5; s=0.000004 e=−1; s=4.000000 (after rounding and normalization) The floating-point difference is computed exactly because the numbers are close—the Sterbenz lemma guarantees this, even in case of underflow when gradual underflow is supported.
The number e is a mathematical constant approximately equal to 2.71828 that is the base of the natural logarithm and exponential function.It is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can invite confusion with Euler numbers, or with Euler's constant, a different constant typically denoted .
2.3434E−6 = 2.3434 × 10 −6 = 2.3434 × 0.000001 = 0.0000023434. The advantage of this scheme is that by using the exponent we can get a much wider range of numbers, even if the number of digits in the significand, or the "numeric precision", is much smaller than the range. Similar binary floating-point formats can be defined for computers.
While base ten is normally used for scientific notation, powers of other bases can be used too, [25] base 2 being the next most commonly used one. For example, in base-2 scientific notation, the number 1001 b in binary (=9 d) is written as 1.001 b × 2 d 11 b or 1.001 b × 10 b 11 b using binary numbers (or shorter 1.001 × 10 11 if binary ...
The decimal number 0.15625 10 represented in binary is 0.00101 2 (that is, 1/8 + 1/32). (Subscripts indicate the number base.) Analogous to scientific notation, where numbers are written to have a single non-zero digit to the left of the decimal point, we rewrite this number so it has a single 1 bit to the left of the "binary point". We simply ...
The number 123.45 can be represented as a decimal floating-point number with the integer 12345 as the significand and a 10 −2 power term, also called characteristics, [11] [12] [13] where −2 is the exponent (and 10 is the base). Its value is given by the following arithmetic: 123.45 = 12345 × 10 −2.
For example, in duodecimal, 1 / 2 = 0.6, 1 / 3 = 0.4, 1 / 4 = 0.3 and 1 / 6 = 0.2 all terminate; 1 / 5 = 0. 2497 repeats with period length 4, in contrast with the equivalent decimal expansion of 0.2; 1 / 7 = 0. 186A35 has period 6 in duodecimal, just as it does in decimal. If b is an integer base ...
1 ⁄ 7 = 0.142857142857... 1318 ⁄ 185 = 7.1243243243... Every time this happens the number is still a rational number (i.e. can alternatively be represented as a ratio of an integer and a positive integer). Also the converse is true: The decimal expansion of a rational number is either finite, or endlessly repeating.