Division by two

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An orange that has been sliced into two halves.

In mathematics, division by two, also called halving, mediation, or dimidiation, is common in formulas and as a step in arithmetical calculations; it is equivalent to multiplication by one half.[1] Starting with an arbitrary number or quantity x, its division by two can be written as any of the following equivalent expressions: x÷2, x/2, x2, 12x, 0.5x.

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.

## Contents

* 1 Binary
* 2 Binary floating point
* 3 Decimal
* 4 See also
* 5 References

## 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

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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

1. ↑ Steele, Robert (1922), The Earliest arithmetics in English, Early English Text Society, 118, Oxford University Press, p. 82 .
2. ↑ 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 .
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 .
4. ↑ Waters, E. G. R. (1929), "A Fifteenth Century French Algorism from Liége", Isis 12 (2): 194–236, doi:10.1086/346408 .
5. ↑ 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 .
6. ↑ 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 .
7. ↑ "Math.scalb". Java Platform Standard Ed. 6. http://java.sun.com/javase/6/docs/api/java/lang/Math.html#scalb(double, int). Retrieved 2009-10-11.
8. ↑ Programming languages — C, International Standard ISO/IEC 9899:1999 , Section 7.12.6.6.

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