# Computer Representation of Numbers

Most computers represent integers as binary numbers (see the ``math refresher'' handout) with a certain number of bits. A computer with 16-bit integers can represent integers from 0 to 65,535 (that is, from 0 to 2<sup>16</sup>-1), or if it chooses to make half of them negative, from -32,767 to 32,767. (We won't get into the details of how computers handle negative numbers right now.) A 32-bit integer can represent values from 0 to 4,294,967,295, or +-2,147,483,647.

Most of today's computers represent real (i.e. fractional) numbers using exponential notation. (Again, see ``math refresher'' handout. Actually, deep down inside, computers usually use powers of 2 instead of powers of 10, but the difference isn't important to us right now.) The advantage of using exponential notation for real numbers is that it lets you trade off the range and precision of values in a useful way. Since there's an infinitely large number of real numbers (and in three directions: very large, very small, and very negative), it will never be possible to represent all of them (without using potentially infinite amounts of space).

Suppose you decide to give yourself six decimal digits' worth of storage (that is, you decide to devote an amount of memory capable of holding six digits) for each value. If you put three digits to the left and three to the right of the decimal point, you could represent numbers from 999.999 to -999.999, and as small as 0.001. (Furthermore, you'd have a resolution of 0.001 everywhere: you could represent 0.001 and 0.002, as well as 999.998 and 999.999.) This would be a workable scheme, although 0.001 isn't a very small number and 999.999 isn't a very big one.

If, on the other hand, you used exponential notation, with four digits for the base number and two digits for the exponent, you could represent numbers from 9.999 x 10<sup>99</sup> to -9.999 x 10<sup>99</sup>, and as small as 1 x 10<sup>-99</sup> (or, if you cheat, 0.001 x 10<sup>-99</sup>). You can now represent both much larger numbers and much smaller; the tradeoff is that the absolute resolution is no longer constant, and gets smaller as the absolute value of the numbers gets larger. The number 123.456 can only be represented as 123.4, and the number 123,456 can only be represented as 123,400. You can't represent 999.999 any more; you have to settle for 999.9 (9.999 x 10<sup>2</sup>) or 1000 (1.000 x 10<sup>3</sup>). You can't distinguish between 999.998 and 999.999 any more.

Since superscripts are difficult to type, computer programming languages usually use a slightly different notation. For example, the number 1.234 x 10<sup>5</sup> might be indicated by 1.234e5, where the letter e replaces the ``times ten to the'' part.

You will often hear real, exponential numbers numbers referred to on computers as ``floating point numbers'' or simply ``floats,'' and you will also hear the term ``double'' which is short for ``double-precision floating point number.'' Some computers also use ``fixed point'' real numbers, (which work along the lines of our ``three to the left, three to the right'' example of a few paragraphs back), but those are comparatively rare and we won't need to discuss them.

It's important to remember that the precision of floating-point numbers is usually limited, and this can lead to surprising results. The result of a division like 1/3 cannot be represented exactly (it's an infinitely repeating fraction, 0.333333...), so the computation (1 / 3) x 3 tends to yield a result like 0.999999... instead of 1.0. Furthermore, in base 2, the fraction 1/10, or 0.1 in decimal, is also an infinitely repeating fraction, and cannot be represented exactly, either, so (1 / 10) x 10 may also yield 0.999999.... For these reasons and others, floating-point calculations are rarely exact. When working with computer floating point, you have to be careful not to compare two numbers for exact equality, and you have to ensure that ``round off error'' doesn't accumulate until it seriously degrades the results of your calculations.

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