Introduction to the calculus of powers

Historical remarks

When you evaluate the product: a ·4 = a + a + a + a, you are not at all conscious of the fact that you are dealing with an abbreviated computation and mode of writing. There do not only arise sums of equal terms, but quite often also products of equal factors such as a·a·a·a·a, which also have an abbreviated notation and calculus. You have encountered earlier products a·a and a·a·a during computations of areas and volumes, which with reference to the square and cube are called squares and cubes. These numbers were already known 3000 B.C.; tables of them have been found on tablets at sites near Babylon.

The Greeks, who linked the theory of numbers to geometry, did not go beyond squares and cubes; in fact, in the work of Pythagoras (570 - 497) and Euclid (~ 300 b.C.), you only find squares and cubes. In order to arrive at higher powers, you must leave Geometry. This step, which was extraordinarily difficult for the Greeks, was taken by the Diophantos of Alexandria (250 p.C.), who employed powers up to order 6. The Indians were also ahead of everyone in this field.

The notation changed. For example, the German arithmetician Adam Riese (1492 - 1559) used the notation x¹ = rc = radix, x² = z = zensus , x³ = cb = cubus, x⁴etc= zensus de zensus, up to x⁹. Michael Stifel (1487 - 1567) had the fortunate idea to use letters following the alphabet and explained how to write any power with his symbols in very abbreviated form. Nevertheless, all these abbreviations were rather clumsy. In the 16th Century, there arose the subscript notation a₃, a₄, which Descartes (1596 - 1650) converted to the present day superscript notation a³, a⁴, . . ., still retaining aa for a². Gauß still used aa, because it did not occupy more space than a². Newton was the first to employ powers in which a general number was the exponent. With the aid of Descartes' abbreviated notation, the calculus of powers reached the present, rather clear format.

Definition: A power is a product of equal factors.

Terminology: aⁿ is called a power, a its base, n its exponent. You write.

By definition, n is a number, a an arbitrary, absolute or also relative number. Since we are talking of a product and there must always be at least two factors; a¹ is not defined and a¹ = a. You should not interchange the base and the exponent, because 3² 2³!

If a is an absolute number, the value of aⁿ rises, remains equal or drops, depending on whether a < or = or > 1.

In the case of relative numbers, you must take care of the leading signs:

Positive factors yield a positive product.

Negative factors yield a positive product for even exponents,
a negative product for odd exponents.

If n is an arbitrary integer, then

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