The power of zero

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Dan Meyer recently published a blog post on how to teach the idea that any number raised to the power of zero is simply one. This is an important result that crops up in many different areas and so it’s critical that students know it.

Meyer asked us to choose between two strategies. The first he described as ‘teaching tricks’ and I agree that it’s not very enlightening. The presenter of the video simply tells you that anything raised to the power of zero is one. And that’s it.

Meyer is keen on a different pattern-based activity that he describes as ‘sensemaking’. I think this approach is better than the first but it still doesn’t really explain anything. In my view, it’s more of a pattern-based trick. I’m not quite sure what ‘sensemaking’ means – which is ironic – but I thought this would be a good cue to explain the way that I would teach the concept.

The first important point is that I think this needs to be taught over and over again to students of different ages, whenever exponentials arise. So I’ve taught it in Years 8, 9, 10, 11 and 12. With older students studying Maths Methods – our abstract maths course – I would rely more on pronumerals in the explanation although I would still always illustrate with natural numbers.

Secondly, this would be a small part of my explanation of index laws rather than something that sits on its own. I just see the power of zero as an interesting result.

And I haven’t worked any of this out myself – it comes from an amalgam of textbook explanations, discussions with colleagues and so on. So this is not the patent Ashman method.

Prior knowledge

As ever with mathematics, there are some things that students need to understand in order to grasp the explanation. I won’t outline how I would teach these underpinning ideas because that might make the post rather long. Suffice it to say that there would be a process like the one I’m about to describe. It would be critical that they understood the following three points and so I would check this through questioning:

1. Fractions can be composed and decomposed in the following way (I might also discuss non-examples involving sums):

$\frac{3 \times 4\times 13}{5 \times 7} =\frac{3}{7}\times\frac{4}{5}\times 13$

2. Indices are a shorthand for repeated multiplication of the same term:

$3\times 3\times 3\times 3 = 3^4$

3. Any number divided by itself is one – I usually remind students of this by saying something like, “How many fives [point to denominator] are there in five [point to numerator]”:

$\frac{5}{5} = 1$

This enables us to work out some laws for indices. I would explicitly lead students through one of these and I would pepper the exposition with questions to the students as I went along. I don’t ask for them to raise their hands – as far as my students are concerned, I call on them randomly – and I tend to ask them to work out any of the component parts that relate to prior knowledge.

I would start by looking at something like this result:

$\frac{3^5}{3^2} =\frac{3\times 3\times 3\times 3\times 3}{3\times 3} \\\\=\frac{3}{3}\times\frac{3}{3}\times 3\times 3\times 3\\\\ = 1\times 1\times 3\times 3\times 3\\\\ = 3\times 3\times 3 = 3^3$

With younger students, I would give a couple more examples with natural numbers and with older ones I might use a pronumeral such as ‘a’. To make it simple, I’d have more repeated terms in the numerator than denominator. The fact that we can then partner up each term on the bottom with one on the top means that, in order to get the final power, we can do a subtraction of the two original powers. This is similar to one of the ways that subtraction is explained to students in earlier number work.

As an aside, you might think that I should talk about ‘canceling’ here and I certainly introduce the shorthand of canceling at some point. However, I tend to persist with the kind of decomposition I’ve used above for a quite a while so that students can see where canceling comes from – even if I fade it out and only go through the whole process every third or fourth example. At this point, I might not rewrite as separate quotients but still talk of ‘three divided by three is one’ as I draw my cancelling lines through the terms.

I don’t think you can avoid the error of over-generalisation that many students eventually make that leads them to cancelling the two in an expression like this:

$\frac{2x+a}{2}$

However, when they do this you can return them to decomposition as an explanation of why that doesn’t work.

Back to the original point, we have now derived a rule that looks something like this:

$\frac{a^m}{a^n} = a^{m-n}$

This is incredibly powerful and should also lead to a discussion and some examples and exercises that include negative powers, but that’s not the objective here.

Armed with this rule, we can reverse engineer a zero power. How can we construct one? Well, ‘m‘ and ‘n‘ would need to be equal:

$2^0 =\frac{2^3}{2^3} \\\\=\frac{2\times 2\times 2}{2\times 2\times 2}\\\\ =\frac{2}{2}\times\frac{2}{2}\times\frac{2}{2}\\\\\ =1\times 1\times 1 =1$

If the students really need convincing then you can give a few more examples just to show that it always comes out as one.

I also use index laws to show the equivalence of fractional powers and roots.

14 thoughts on “The power of zero ”

1. So what’s the line on anything to the power of zero = 1 = 0.999… (recurring)?

1. 10 x 0.999… – 0.999… = 9 x 0.999…
But
10 x 0.999… – 0.999… = 9
So
9 x 0.999… = 9
Therefore
0.999… = 1

2. Richard I says:

You can also reverse engineer introductions to

i) negative powers …….. if (a^m) / (a^n) = a^(m-n) , what choices of m and n would give a final answer of a^(-3) ……. cancelling highlights the fact that ‘cancelling’ is not a mathematical operation, it is shorthand for division and that is why we have 1 as a numerator.

ii) fractional powers ……. if (a^m)^n = a^(mn), then what choice of m would give (a^m)^2 = a ….

3. Chester Draws says:

Indices are a shorthand for repeated multiplication of the same term

Good luck explaining 9^1.5 = 27 that way. (Yes, I can explain why, just not using “repeated multiplication”).)

And then we strike e^(πi) = -1 and we’re really stuck. I’m willing to see a “common sense” explanation of that, using indices as “repeated multiplication”, if anyone’s game.

I thought about commenting on Meyer’s blog with that, and then thought again. His tolerance for comments too far off message is short.

Thing is, I don’t know any good teachers who don’t teach x^0 = 1 by some sort of explanation using patterns or cancellation. It’s so obvious, and important, that we all do it. The textbooks I have make a point of it. I would do it in revision even.

I’m sure some bad teachers just teach the rule, as a rule. But the problem is really that they are bad teachers, and making them teach x^0 some other way won’t fix that. I also tend to think that teaching the rule, as a rule, is actually better than a bad teacher trying to explain “why” and just making a hash of it.

In the end the students have to learn the rule, regardless. Under pressure they are not going to recall a teacher’s inspiring lesson on why.

If you have a teacher not teaching this properly, and I’m not convinced the exact method of example matters very much, then the problem well predates the method of teaching. The problem is that the teacher isn’t really suited to teaching Maths. Getting more student “engagement” won’t cure that.

1. Stan says:

A reply because I like a challenge not because it is a tested answer. For 9^1.5 we can write this as 9^(3/2) and then write this as (9^1/2)^3. But how to describe 9^1/2 in terms of repeated multiplication. The answer is that N^(1/Q) is to N^P as division is to multiplication. To find 9^1/2 we have to ask what number multiplied by itself 2 times gives an answer of 9. That is what M gives MxM=9. If we are asked to only consider positive M then 3 works and we have (9^1/2)^3 = 3^3 = 3x3x3 = 27.

The step from 9^(3/2) to (9^1/2)^3 can be justified in terms of repeated multiplication by pointing out that if we are repeating something MxN times this is the same as repeating something M times and then repeating the result N times. This works with any rectangular array of M x N items and also works if we have M x 1/N items. For example if we had 4 x 1/6 cartoons of eggs.

Once you describe e^(πi) as a rotation through the angle π radians then the repeated multiplication idea still makes sense. e^π seems a little more difficult until you define both e and π in terms of infinite polynomials.

4. It is so simple to define axaxaxa to mean 1xaxaxaxa, and the a is multiplied four times (not 3).
So the tangle resulting from “a times itself once” is meaningfully replaced by “start with 1 and multiply by a”.
Worse is the tangle with “a times itself no times”.
The mess is resolved by the “1”, and then a^3 is axaxa = 1xaxaxa
a^2 is 1xaxa
a^1 is 1xa
and with no magic at all
a^0 is 1
This is the MEANING or definition of “to the power of”

And root(x) times root(x) is x, by the extended definition.
There is no problem at all.

1. Stan says:

I don’t think this is as semantically clear cut as you think. For example if I make 2 copies of something I have 3 of them. It is not the same as counting 2 sets of N things and finding I have 2N of them or counting no sets of N things leaves me with none not one.

1. If I define the “a multiplied by itself three times” I get axaxa
The difficulty is “a multiplied by itself once”, which is close to being meaningless.
This is what the “1” is for.

2. Stan says:

I agree the multiplied by itself N times phrase is clumsy.

But to make the 1 meaningful beyond it being a definition that works with the other definitions (exponent rules) perhaps a concrete example helps.

If we are describing exponential growth such as the number of cells when a blob of cells are splitting and doubling in population every second we have at time zero no doubling and the population is p_0 x 1. After t seconds it is p_0 x 2^t. So in general when we say we are multiplying some quantity by p another quantity integer q we can do this no times, one time, two times and so on. Multiplying p by q no times leaves us with p.

If we take away p and talk in relative quantities then we can say something grew by a multiplicative factor r = p^n and if has not grown at all the multiplicative factor must be 1 for this to correspond to what we are saying. We can describe this in words as the growth is a multiplier p, applied n times over.

Ultimately there is a good reason we use symbols for math. To be specific we end up with a clumsy wording.

3. You put it better than I do !
The “1” problem is a result of the simple way of using axaxa and then, later, creating the exponent notation which results in a^0. It is notation, not something bigger.

5. Dylan Wiliam says:

I remember asking a (high-achieving) 9th grade class about the value of 0^0, and they found convincing reasons why it might be 0, 1 or undefined. We embarked on an analysis of the limits of x^x as x tended towards zero, for positive x, which supported the idea that 0^0 was 1. But then, we looked at the behaviour of x^x for negative x, and things got really wild…

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