As in most English texts, iteration means repeating, and this hold true for mathematics. In the case of numbers, iteration means a procedure in which the result is fed back and the procedure repeated; in some instances, it is repeated over and over. For example, to find the square root of 39, you can use iteration—or repeat a procedure to find the solution. Knowing that the solution must be close to 6 (or the square root of 36, a number close to 39), one can divide 39 by 6 (39/6) and get 6.5. Next, average 6 and 6.5 to get 6.25; then iterate again, dividing 39 by 6.25 (39/6.25) = 6.24 (the actual square root of 39 is 6.244997 …).

One of the most obvious places in which iterations take place is in your calculator or computer. For example, in order to get the square root of 39, as in the example above, a calculator (or computer) automatically uses iteration to calculate the answer to a certain decimal place. The more numbers in a procedure, the more iterations are needed, which is why supercomputing has become such a great asset not only to mathematics, but to many other sciences as well.