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Division by two
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In mathematics, division by two or halving has also been called mediation or dimidiation.[1] The treatment of this as a different operation from multiplication and division by other numbers goes back to the ancient Egyptians, whose multiplication algorithm used division by two as one of its fundamental steps.[2] Some mathematicians as late as the sixteenth century continued to view halving as a separate operation,[3][4] and it often continues to be treated separately in modern computer programming.[5] Performing this operation is simple in decimal arithmetic, in the binary numeral system used in computer programming, and in other even-numbered bases.
Binary
In binary arithmetic, division by two can be performed by a bit shift operation that shifts the number one place to the right. This is a form of strength reduction optimization. For example, 1101001 in binary (the decimal number 105), shifted one place to the right, is 110100 (the decimal number 52): the lowest order bit, a 1, is removed. Similarly, division by any power of two 2k may be performed by right-shifting k positions. Because bit shifts are often much faster operations than division, replacing a division by a shift in this way can be a helpful step in program optimization.[5] However, for the sake of software portability and readability, it is often best to write programs using the division operation and trust in the compiler to perform this replacement.[6] An example from Common Lisp:
(setq number #b1101001) ; #b1101001 — 105 (ash number -1) ; #b0110100 — 105 >> 1 ⇒ 52 (ash number -4) ; #b0000110 — 105 >> 4 ≡ 105 / 2⁴ ⇒ 6
The above statements, however, are not always true when dealing with dividing signed binary numbers. Shifting right by 1 bit will divide by two, always rounding down. However, in some languages, division of signed binary numbers round towards 0 (which, if the result is negative, means it rounds up). For example, Java is one such language: in Java, -3 / 2 evaluates to -1, whereas -3 >> 1 evaluates to -2. So in this case, the compiler cannot optimize division by two by replacing it by a bit shift, when the dividend could possibly be negative.
Binary floating point
In binary floating-point arithmetic, division by two can be performed by decreasing the exponent by one (as long as the result is not a subnormal number). Many programming languages provide functions that can be used to divide a floating point number by a power of two. For example, the Java programming language provides the method java.lang.Math.scalb for scaling by a power of two,[7] and the C programming language provides the function ldexp for the same purpose.[8]
Decimal
The following algorithm is for decimal. However, it can be used as a model to construct an algorithm for taking half of any number N in any even base.
- Write out N, putting a zero to its left.
- Go through the digits of N in overlapping pairs, writing down digits of the result from the following table.
| If first digit is | Even | Even | Even | Even | Even | Odd | Odd | Odd | Odd | Odd |
|---|---|---|---|---|---|---|---|---|---|---|
| And second digit is | 0 or 1 | 2 or 3 | 4 or 5 | 6 or 7 | 8 or 9 | 0 or 1 | 2 or 3 | 4 or 5 | 6 or 7 | 8 or 9 |
| Write | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
Example: 1738/2=?
Write 01738. We will now work on finding the result.
- 01: even digit followed by 1, write 0.
- 17: odd digit followed by 7, write 8.
- 73: odd digit followed by 3, write 6.
- 38: odd digit followed by 8, write 9.
Result: 0869.
From the example one can see that 0 is even.
If the last digit of N is odd digit one should add 0.5 to the result.
See also
- One half
- Median, a value that splits a set of data values into two equal subsets
- Bisection, the partition of a geometric object into two equal halves
- Dimidiation, a heraldic method of joining two coats of arms by splitting their designs into halves
References
- ↑ Steele, Robert (1922), The Earliest arithmetics in English, Early English Text Society, 118, Oxford University Press, p. 82.
- ↑ Chabert, Jean-Luc; Barbin, Évelyne (1999), A history of algorithms: from the pebble to the microchip, Springer-Verlag, p. 16, ISBN 978-3-540-63369-3.
- ↑ Jackson, Lambert Lincoln (1906), The educational significance of sixteenth century arithmetic from the point of view of the present time, Contributions to education, 8, Columbia University, p. 76.
- ↑ Waters, E. G. R. (1929), "A Fifteenth Century French Algorism from Liége", Isis 12 (2): 194–236, doi:10.1086/346408.
- ↑ 5.0 5.1 Wadleigh, Kevin R.; Crawford, Isom L. (2000), Software optimization for high-performance computing, Prentice Hall, p. 92, ISBN 978-0-13-017008-8, https://archive.org/details/softwareoptimiza0000wadl/page/92.
- ↑ Hook, Brian (2005), Write portable code: an introduction to developing software for multiple platforms, No Starch Press, p. 133, ISBN 978-1-59327-056-8.
- ↑ "Math.scalb". Java Platform Standard Ed. 6. http://java.sun.com/javase/6/docs/api/java/lang/Math.html#scalb(double,%20int). Retrieved 2009-10-11.
- ↑ Programming languages — C, International Standard ISO/IEC 9899:1999, Section 7.12.6.6.
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