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In conventional binary number systems, the base, or radix, is 2; thus the rightmost bit represents 2 0, the next bit represents 2 1, the next bit 2 2, and so on. However, a binary number system with base −2 is also possible. The rightmost bit represents (−2) 0 = +1, the next bit represents (−2) 1 = −2, the next bit (−2) 2 = +4 and so ...
For example, a two's complement signed 16-bit integer can hold the values −32768 to 32767 inclusively, while an unsigned 16 bit integer can hold the values 0 to 65535. For this sign representation method, the leftmost bit ( most significant bit ) denotes whether the value is negative (0 for positive or zero, 1 for negative).
This table illustrates an example of an 8 bit signed decimal value using the two's complement method. The MSb most significant bit has a negative weight in signed integers, in this case -2 7 = -128. The other bits have positive weights. The lsb (least significant bit) has weight 2 0 =1. The signed value is in this case -128+2 = -126.
A simple arithmetic calculator was first included with Windows 1.0. [5]In Windows 3.0, a scientific mode was added, which included exponents and roots, logarithms, factorial-based functions, trigonometry (supports radian, degree and gradians angles), base conversions (2, 8, 10, 16), logic operations, statistical functions such as single variable statistics and linear regression.
For example, if six bits are used to represent the number "00 1010" (decimal positive 10) and the sign extends operation increases the word length to 16 bits, then the new representation is simply "0000 0000 0000 1010". Thus, both the value and the fact that the value was positive are maintained.
Signed-digit representation can be used to accomplish fast addition of integers because it can eliminate chains of dependent carries. [1] In the binary numeral system, a special case signed-digit representation is the non-adjacent form, which can offer speed benefits with minimal space overhead.
A more efficient encoding can be designed using the fact that the exponent range is of the form 3×2 k, so the exponent never starts with 11. Using the Decimal32 encoding (with a significand of 3*2+1 decimal digits) as an example (e stands for exponent, m for mantissa, i.e. significand):
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.