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. To 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.

Contents

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:

(setqnumber#b1101001); #b1101001  —  105(ashnumber-1); #b0110100  —  105 >> 1 ⇒ 52(ashnumber-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.

If first digit isEvenEvenEvenEvenEvenOddOddOddOddOdd
And second digit is0 or 12 or 34 or 56 or 78 or 90 or 12 or 34 or 56 or 78 or 9
Write0123456789

Example: 1738/2=?

Write 01738. We will now work on finding the result.

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

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References

  1. ↑ Steele, Robert (1922), The Earliest arithmetics in English, Early English Text Society, vol. 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, vol. 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, JSTOR   224785, S2CID   144157808 .
  5. 1 2 Wadleigh, Kevin R.; Crawford, Isom L. (2000), Software optimization for high-performance computing, Prentice Hall, p.  92, ISBN   978-0-13-017008-8 .
  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. Retrieved 2009-10-11.
  8. ↑ Programming languages — C, International Standard ISO/IEC 9899:1999, Section 7.12.6.6.
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