{{Short description|None}} 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''', 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.<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> Starting with an arbitrary number or quantity {{tmath|x}}, its division by two can be written as any of the following equivalent expressions: <math display=block>x \div 2,\ x / 2,\ \frac x 2,\ \tfrac12 x,\ 0.5 x.</math>
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.<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 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 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 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, <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 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 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 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 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}}
Two Category:Elementary arithmetic Category:Binary arithmetic Category:Parity (mathematics) Category:2 (number)