Editing Division by two

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[[File:Orange_sliced_in_half.jpg | thumb | right | alt=An orange on a white plate that has been divided in half.  | An orange that has been sliced into two halves. ]]
In [[mathematics]], '''division by [[two]]''' or '''halving''' has also been called '''mediation''' or '''dimidiation'''.<ref>{{citation|title=The Earliest arithmetics in English|volume=118|series=Early English Text Society|first=Robert|last=Steele|publisher=Oxford University Press|year=1922|page=82}}.</ref> The treatment of this as a different operation from multiplication and division by other numbers goes back to the ancient Egyptians, whose [[Ancient Egyptian multiplication|multiplication algorithm]] used division by two as one of its fundamental steps.<ref>{{citation|title=A history of algorithms: from the pebble to the microchip|first1=Jean-Luc|last1=Chabert|first2=Évelyne|last2=Barbin|publisher=Springer-Verlag|year=1999|isbn=978-3-540-63369-3|page=16}}.</ref>
Some mathematicians as late as the sixteenth century continued to view halving as a separate operation,<ref>{{citation|title=The educational significance of sixteenth century arithmetic from the point of view of the present time|volume=8|series=Contributions to education|first=Lambert Lincoln|last=Jackson|publisher=Columbia University|year=1906|page=76}}.</ref><ref>{{citation|title=A Fifteenth Century French Algorism from Liége|journal=Isis|volume=12|issue=2|year=1929|first=E. G. R.|last=Waters|pages=194–236|jstor=224785|doi=10.1086/346408|s2cid=144157808}}.</ref> and it often continues to be treated separately in modern [[computer programming]].<ref name="WC00">{{citation|title=Software optimization for high-performance computing|first1=Kevin R.|last1=Wadleigh|first2=Isom L.|last2=Crawford|publisher=Prentice Hall|year=2000|page=[https://archive.org/details/softwareoptimiza0000wadl/page/92 92]|isbn=978-0-13-017008-8|url=https://archive.org/details/softwareoptimiza0000wadl/page/92}}.</ref>
Performing this operation is simple in [[decimal arithmetic]], in the [[binary numeral system]] used in computer programming, and in other even-numbered [[numeral system|base]]s. To [[Division (mathematics)|divide]] an [[odd number]] by [[2]] use the [[mathematical solution]] ((N−1)÷2)+0.5. For example, if N=7, then ((7−1)÷2)+0.5=3.5, so 7÷2=3.5.

==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]] 2<sup>''k''</sup> 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]].<ref name="WC00"/> 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.<ref>{{citation|title=Write portable code: an introduction to developing software for multiple platforms|first=Brian|last=Hook|publisher=No Starch Press|year=2005|isbn=978-1-59327-056-8|page=133}}.</ref> An example from [[Common Lisp]]:

<syntaxhighlight lang="lisp">
 (setq number #b1101001)   ; #b1101001  —  105
 (ash number -1)           ; #b0110100  —  105 >> 1 ⇒ 52
 (ash number -4)           ; #b0000110  —  105 >> 4 ≡ 105 / 2⁴ ⇒ 6
</syntaxhighlight>

The above statements, however, are not always true when dealing with dividing [[Signed number representations|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 (programming language)|Java]] is one such language: in Java, <code>-3 / 2</code> evaluates to <code>-1</code>, whereas <code>-3 >> 1</code> evaluates to <code>-2</code>. 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)|Java programming language]] provides the method <code>java.lang.Math.scalb</code> for scaling by a power of two,<ref>{{cite web
|url=http://java.sun.com/javase/6/docs/api/java/lang/Math.html#scalb(double,%20int)
|title=Math.scalb
|work=Java Platform Standard Ed. 6
|accessdate=2009-10-11
}}</ref> and the [[C (programming language)|C programming language]] provides the function <code>ldexp</code> for the same purpose.<ref>{{citation
|title=Programming languages — C, International Standard ISO/IEC 9899:1999
}}, Section 7.12.6.6.</ref>

==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 and odd numbers|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.

{| class="wikitable"
|-
! 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 [[even and odd numbers|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==
{{reflist}}

[[Category:Division (mathematics)]]
[[Category:Elementary arithmetic]]
[[Category:Binary arithmetic]]
[[Category:Parity (mathematics)]]
[[Category:2 (number)]]