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.
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:
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.
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]
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 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.
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.