Rational Approximation and its Applications in Mathematics by Jacek Gilewicz, Maciej Pindor, Wojciech Siemaszko

By Jacek Gilewicz, Maciej Pindor, Wojciech Siemaszko

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The diminished-radix complement is also known as the one’s complement in binary notation. The MSB contains the sign bit, as with sign magnitude, while the rest of the number is either the absolute value of the number or its bit-by-bit complement. The decimal number 4 would appear as 0100 and -4 as 1011. As in the foregoing method, two forms of zero would result: 0000 and 1111. The radix complement, or two’s complement, is the most widely used notation in microprocessor arithmetic. It involves using the MSB to denote the sign, as in the other two methods, with zero indicating a positive value and one meaning negative.

To present algorithms for the four basic operations of mathematics-addition, subtraction, multiplication, and division-this chapter will concentrate on integeronly arithmetic. The operations for fixed-point and integer-only arithmetic are essentially the same; the former simply involves some attention to the placement of the radix point. This chapter begins with the basic operations and the classic algorithms for them, followed by some more advanced algorithms. The classic algorithms aren’t necessarily the fastest, but they are so elegant and reflect the character of the binary numbering system and the nature of arithmetic itself so well that they are worth knowing.

This is done in a single stroke, with the bits of one operand shifted directly into the other, easing normalization of long integers and making for fast binary multiplies and divides. In recent years we’ve seen the introduction of microprocessors and microcontrollers that are specially designed to handle floating-point as well as fixed-point arithmetic. These processors have significantly enhanced real-time control applications and digital signal processing in general. One such microprocessor is the TMS34010; a microcontroller with a similar aptitude is the 80C196.

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