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Here, the product notation indicates a binary floating point representation with the exponent of the representation given as a power of two and with the significand given with three bits after the binary point. To compute the subtraction it is necessary to change the forms of these numbers so that they have the same exponent, and so that when ...
Signed binary angle measurement. Black is traditional degrees representation, green is a BAM as a decimal number and red is hexadecimal 32-bit BAM. In this figure the 32-bit binary integers are interpreted as signed binary fixed-point values with scaling factor 2 −31, representing fractions between −1.0 (inclusive) and +1.0 (exclusive).
For example, rounding x = 2.1784 dollars to whole cents (i.e., to a multiple of 0.01) entails computing 2.1784 / 0.01 = 217.84, then rounding that to 218, and finally computing 218 × 0.01 = 2.18. When rounding to a predetermined number of significant digits , the increment m depends on the magnitude of the number to be rounded (or of the ...
This decimal format can also represent any binary fraction a/2 m, such as 1/8 (0.125) or 17/32 (0.53125). More generally, a rational number a / b , with a and b relatively prime and b positive, can be exactly represented in binary fixed point only if b is a power of 2; and in decimal fixed point only if b has no prime factors other than 2 and/or 5.
Numbers can also be considered "round" in numbering systems other than decimal (base 10). For example, the number 1024 would not be considered round in decimal, but the same number ends with a zero in several other numbering systems including binary (base 2: 10000000000), octal (base 8: 2000), and hexadecimal (base 16: 400).
This rounding rule is biased because it always moves the result toward zero. Round-to-nearest: () is set to the nearest floating-point number to . When there is a tie, the floating-point number whose last stored digit is even (also, the last digit, in binary form, is equal to 0) is used.
For example, 0.1 in decimal (1/10) is 0b1/0b1010 in binary, by dividing this in that radix, the result is 0b0.0 0011 (because one of the prime factors of 10 is 5). For more general fractions and bases see the algorithm for positive bases .
Decimal arithmetic, compatible with that used in Java, C#, PL/I, COBOL, Python, REXX, etc., is also defined in this section. In general, decimal arithmetic follows the same rules as binary arithmetic (results are correctly rounded, and so on), with additional rules that define the exponent of a result (more than one is possible in many cases).