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What is Zero Power Zero?

I have been teaching mathematics in an Australian High School since 1982, and I am a contributing author to mathematics text books.

When we observe the sun’s travel across the sky, we can relate to our Medieval ancestors’ belief that the Earth was a stationary object about which the sun dutifully revolved once a day. It was only with Renaissance thinking that this geocentric model of planetary motion was replaced by Heliocentrism, which proposed the notion of an orbiting Earth around a fixed sun.

what-is-zero-power-zero

The point being made is that hypotheses and conjectures are based on available evidence and not on ignorance. This can be exemplified in the arena of mathematics.

In early years of schooling, we are taught about natural numbers that consist of positive whole numbers which we can identify with because they are used for counting. After all, there is a one-to-one correspondence between counting on our fingers and counting how many objects we see.

what-is-zero-power-zero

Later, zero and negative whole numbers are introduced to establish the set of integers. Further study has students faithfully commit to memory that the square root of a number can only be found if the number is not negative. However, this mathematical indoctrination is soon supplanted by the appearance of complex numbers, where the existence of the square root of a negative number is permitted.

We gratefully, and somewhat naively, embrace knowledge as exhaustive until newer and more valid information comes along. Let’s introduce a spanner in this framework of passive acceptance by examining the index law bp in conjunction with limits.

The base raised to a power

A number can be written in index form as bp, where b is the base and p is the power.

For example, 8 = 23 because 2 × 2 × 2 = 8 and 625 = 54 because 5 × 5 × 5× 5 = 625.

Similarly, 76 = 7 × 7 × 7 × 7 × 7 × 7 = 117,649.

A calculator’s power function can also be used for decimals. For example, 1.628 = 47.43731683, 43.6 =147.0333894 and 2.95.1 =228.1548942.

Limit From the Right Using a Fixed Base

What happens when the base is fixed and the power approaches zero?

Let’s use the number 2 as the base.

The table below shows different values for the power, which are in descending order.

what-is-zero-power-zero

The graph of the data in the table shows that as the value of the power, p, approaches 0 from the right (that is, descending values of p), 2p approaches the value 1.

Using limit notation, this can be written as

what-is-zero-power-zero

The graph for the data in the table above is shown below.

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what-is-zero-power-zero

Limit From the Left Using a Fixed Base

Approaching p = 0 from the left requires negative values of p. The value of 2p for some negative values of p are given in the following table.

what-is-zero-power-zero

We can also view the information in the table graphically.

what-is-zero-power-zero

As the value of p approaches 0 from the left (increasing values of p), 2p approaches the value 1. Using limit notation, this can be written as

what-is-zero-power-zero

Thus we have the limits:

what-is-zero-power-zero

These two limits can be combined in the form

what-is-zero-power-zero

This reads as ‘the limit of 2p as p approaches zero is 1’, accommodating the conditions that zero is approached from the left and from the right, as shown graphically.

what-is-zero-power-zero

Letting the base approach zero

In bp, what happens when the base, b, approaches zero?

Let’s examine the situation when p = 2. The results and the graph for various values of b approaching zero both from the left and from the right are shown below.

what-is-zero-power-zero
what-is-zero-power-zero

It can be seen from the graph that as the base, b, approaches zero from either side of zero, b2 approaches zero. We can describe this as the limit

what-is-zero-power-zero

When the base is the same as the Power

Particularly interesting will be to let b = p and see what happens.

When the base and the power are the same, ab becomes aa or bb. What do you think the shape of the graph will look like as a approaches zero from the right?

what-is-zero-power-zero

The values in the table above show that as a decreases, the value of aa is not consistent. This is made clear by inspecting the graph below, which covers all values of a from 0 to 1.

what-is-zero-power-zero

On this graph, notice that the value of aa decreases up to point P and then increases to reach a value of 1 when a = 0.

Point P is the minimum value of aa whose exact coordinates can be found using differentiation, since the derivative is zero at a turning point.

To differentiate, first let y = aa and then take the natural log of both sides.

loge (y) = loge(aa).

This simplifies to loge (y) = aloge(a).

what-is-zero-power-zero

Let the derivative equal zero and solve the equation for a.

what-is-zero-power-zero

Since aa is never zero, we must have loge(a) + 1 = 0 or a = e-1.

Now substitute e-1 in y = aa to get

what-is-zero-power-zero

This means the turning point, P, on the graph has coordinates

what-is-zero-power-zero

This is approximately (0.37, 0.69), a remarkable result since intuitively we would think that for all values of a, aa would approach zero as a becomes smaller.

We conclude that as a approaches zero, aa approaches 1. Hence we have

what-is-zero-power-zero

Note that 00 cannot definitively be stated to be equal to 0, even though the limit indicates that it does have a value of zero when the infinite case is considered. It may seem counterintuitive to say that we can get infinitely close to zero and that if we ever reach it, the value of 0a will not be 0.

For this reason, the value assigned to 00 is to some extent a function of the context in which it is used by mathematicians. That is, to be consistent with a particular calculation, the mathematician may assign the value 0, 1 to 00 or state that the value is indeterminate.

Approaching Zero to the Power Zero from the Left

Now consider what happens as we approach a = 0 from the left. For simplicity, let’s look at the case when a is a negative integer.

Some results are shown in the following table and in its accompanying graph.

what-is-zero-power-zero
what-is-zero-power-zero

Notice that for even values of a, aa is positive, and for odd value of a, aa is negative.

The positive values of aa and the negative values of aa are connected by separate smooth curves. Using non-integer even values lie on the curve above the a axis, and non-integer odd values lie on the curve below the a axis.

Values of a between 0 and -1

For values of a that lie between 0 and -1, real solutions do not exist. However, solutions can be obtained over the set of complex numbers.

For example, for a = -0.5, (-0.5)-0.5 is approximately the two complex numbers 1.4i and -1.4i, where i2 = -1. Similarly, for a = -0.25, (-0.25)-0.25 is -1+i and -1- i.

By allowing complex solutions and by considering the cartesian plane as the complex number plane, the graph below illustrates the behaviour of the points as a = 0 is approached from the left.

what-is-zero-power-zero

By combining this graph with the earlier graph that shows the approach from the right, the following is obtained.

what-is-zero-power-zero

Conclusions

We can now make the following observations.

1. The value of aa as a approaches 0 from the right is 1.

2. The value of aa as a approaches 0 from the left has two limits; 1 and -1.

3. As we approach 0 from the right, the graph has a minimum turning point.

4. As we approach 0 from the left, the graph has a maximum turning point and a minimum turning point.

5. The graph is not continuous at a = 0 because there are two different limits as we approach a = 0.

I hope that this review goes some way in demonstrating that the obvious is not always obvious. A calculator will not supply an answer to 00, and our discussion provided some explanation as to why this is so.


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