Powers with positive and negative exponents

Explanation and Rules

A power is a product of equal factors. If you want to form a power, you must have at least two factors; by definition, the lowest power is a². Since, in addition, the exponent indicates the frequency of the factor, it must be a number. For a¹, we have agreed to write a¹ = a. So far we have considered powers of the form aⁿ, n 1. Since a is arbitrary, only the exponents are subject to restrictions.

Great advances in mathematics have resulted often through attempts to get rid of restrictions. You have already seen this. Impossible subtractions and divisions were made possible by the introduction of relative numbers and fractions. Also, in the case of powers, there existed the trend to remove restrictions by sensible rules, indeed , as earlier, by retaining as many rules as possible and removal of contradictions (Law of permanence).

The motive for the introduction of relative numbers was the inversion of addition - subtraction -, of fractions the inversion of multiplication - division; we will now try to gain an extension for powers, based on multiplication, by inversion of multiplication - division.

If you take into account in the sequence of powers

the direction, in which you advance, you will recognize that proceeding to the right, every power arises out of the preceding one by multiplication by a, proceeding to the left, every power arises by division of the preceding one by a, whence you obtain the pattern

We can now define:

and continue

You see that the definitions for a⁺ⁿ and aⁿ correspond; beyond this we obtain the

The zero power of any number is 1, a power with a negative exponent is the reciprocal of the power with the same positive exponent.

The earlier Rules for powers are without difficulty extended to aⁿ and a^--n; we ask now whether these definitions were chosen so well, that the earlier Rules also apply to them. We will not prove all of them, but just look, as an example, at the third Rule:

If you admit relative numbers as exponents, this distinction is no longer necessary, since a³/a⁵= a^-2.

Examples:

From Arithmetic

The design of our number system is related to the powers.

Natural numbering of position values does not start from the decimal point (hundred is three positions to the left, hundredth is three position to the right) but from the units!

The units are the starting point or
zero point of our position value numbering.

From the unit position, the hundreds are located 2 positions to the left , hundredths two positions to the right, thousands 3 positions to the right, thousandths 3 positions to the left, etc. Then, in agreement with the Rule: If you multiply by 100, every digit moves two positions to the left, if you divide by 100, every digit moves two positions to the right, etc.

The development of the methods of arithmetic is closely linked to the position value system. Total mechanisation of the position value led to clearer computations. The only significance of the decimal point is that it separates the units from the fractions. You might be regretted that the numbering of the position value starts from the decimal point instead of from the units!

Graphical representation

The new definitions lead to new power functions. Since the graphical representations of y=x⁺ⁿ and y = xⁿ correspond to each other, we only have to augment the earlier steps for the power functions:

y = xⁿ and y = x^-n.

y = x⁰: For each value of x this function y has the value 1. Thus you obtain the points with the ordinate 1, that is a parallel to the abscissa at the distance 1.

y = x ^-1. We know the image of this function from the Function of inverse proportionality y = m/x, since it must be the same as that of y = 1/x - a hyperbola.

I. Two branches.

We note here as a special feature two parts or two branches. In the function y = 1/x, x can have all values except for 0, because you may not divide by zero.Therefore you do not get a value of y for x=0. The regular course of the curve is interrupted at this location. This explains the two parts or branches of the curve.

II. Asymptotes: While the distance of a parabola of any order from the coordinate axes increases with growing |x|, the branches of the hyperbola approach the axes without a limit, but never reach them. Straight lines, which are approached without limit but are never reached, are called asymptotes (Greek: never coinciding).

The x- and y- axes are asymptotes
of the hyperbola.

III. Axial and central symmetry.

If you draw the two bisectors of the coordinate axes, you can fold the hyperbola into itself by flipping about either of the angle bisectors. This establishes also its central symmetry with the origin as centre.

The image of y = x ^-1 or xy = 1 is called a hyperbola of second order, that of y = x^-2 or yx² = 1 a hyperbola of order three. All odd exponents of x in y = x^-n generate hyperbolae like the one shown in the figure above, even exponents hyperbolae like those in the last figure.

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