In the expression 3⁵, the number 3 is called the ‘base’ and the number 5 is called the ‘index’; but what is the entire thing — 3⁵ as a whole — called?

I found myself asking this question last year when I was creating some resources for teaching indices. There came a sense of déjà vu; I had had exactly the same problem the year before when I was working on calculus resources for a different company.

A digression into calculus

Feel free to skip this section if you’re unfamiliar with the calculus, it just provides a bit of background motivation.

Differentiation is a linear operator, which means that

and

Using these basic properties of differentiation becomes second nature, and so you can forget that these things needed to be proved! When I’m creating solution videos for calculus problems as part of my job, I like to make it explicit whenever I use one of these rules.

When applying the first rule, I think there are basically two things I could say:

‘d by d x of … f of x plus g of x … equals d by d x of f of x … plus d by d x of g of x.’ ‘The derivative of a sum of functions is equal to the sum of their derivatives.’

I don’t know about you, but of those two options I prefer the latter.

While the former may work well trying to find the derivative of x² + sin(x) (just write down the rule, define f(x) = x² and g(x) = sin(x), and Bob’s your uncle), finding ḟ(t) where f(t) = t² + 3t⁵+ sin(t) we run into three problems:

f has already been defined.

We’re differentiating with respect to t, not x.

We have a sum of three functions, not just two.

Being explicit about the fact that the derivative of a sum of functions is equal to the sum of their derivatives is especially important because the same cannot be said for the product of functions:

‘d by d x of … f of x times g of x … is not equal to d by d x of f of x … times d by d x of g of x.’ ‘The derivative of a product of functions is not equal to the product of their derivatives.’

How about putting the following rule into words?

Or, more generally:

This was something that stumped me. With the previous rules we could make use of the terminology surrounding addition and multiplication, but with exponentiation it isn’t so clear how to do this. I thought it would help to know the answer to my question: ‘In the expression 3⁵, the number 3 is called the ‘base’ and the number 5 is called the ‘index’; but what is the entire thing — 3⁵ as a whole — called?’

Back to indices

I still didn’t have an answer and wanted to get to the bottom of this, so I went to twitter for help.

Twitter was useless. (Or perhaps, more accurately, I was useless at twitter.)

However, a couple of days later, a colleague suggested that the word I was looking for was ‘power’. Checking the copy of The Concise Oxford Dictionary of Mathematics in the office seemed to confirm this:

Definition of ‘power’ from The Concise Oxford Dictionary of Mathematics, 5th ed.

How could I have overlooked this?

Growing up in the British education system, I knew that the term ‘power’ had something to do with exponentiation, but if I had ever seen a definition I’d never taken it in. Its usage certainly didn’t make its meaning clear to me:

3⁵ is read as ‘3 to the power of 5’

of 5’ 10, 100, 1000, 10 000, … are powers of 10.

of 10. ln(1 + x) = x − ½x² + ⅓x³ − … is a power series.

series. f(x) = a·xᵇ is the general form of a power function.

The word ‘power’ always seemed to be used in one of these phrases and never on its own; so, just as I could define ‘waiting with bated breath’ but not ‘bated’, and ‘spick and span’ but neither ‘spick’ nor ‘span’, I could define ‘3 to the power of 5’ and ‘powers of 10’ but not ‘power’ alone.

The fact that the dictionary definition includes what is presumably a common misconception — ‘When aᵖ is formed, p is sometimes called the power, but is more correctly called the index to which a is raised’ — suggests that I wasn’t the only one unsure of the definition of ‘power’.

So, is this case closed? In the expression 3⁵, is 3 the base, 5 the index, and 3⁵ itself the power?

Is ‘sum’ : ‘addition’ :: ‘difference’ : ‘subtraction’ :: ‘product’ : ‘multiplication’ : ‘quotient’ : ‘division’ :: ‘power’ : ‘exponentiation’?

It’s not quite that simple.

Exponents of American terminology, and others

Before continuing to discuss the terminology used for exponentiation in the UK, I thought I should acknowledge that different terminology is used elsewhere in the English-speaking world.

For example, ‘exponent’ is the preferred term in the USA for what we prefer to call an ‘index’ in the UK. Why this should be so is an interesting question but not the subject of this article, not least because unlike the case of the trapezoid and trapezium I’m not actually sure what the answer is! I will just note that in the dictionary responsible for the Americans usage of ‘trapezoid’, the term ‘exponent’ seems to be preferred.

Taken from page 634 of ‘A Mathematical and Philosophical Dictionary’ vol. 1, Charles Hutton, 1795

Although this article will focus on terminology used in the UK, I hope that people used to other terminology will get something out of it.

Reopening the case

For several months I was sure in my understanding: in the expression 3⁵, 3 is the base, 5 is the index, and 3⁵ itself is the power.

Scene from Bruce Almighty via Giphy

Then I saw a video entitled ‘Negative Powers’ which gave 3⁻⁵ as an example. Negative power? Negative index maybe, but 3⁻⁵ is definitely positive. The explanation was simple, this video was taking ‘powers’, ‘indices’ and ‘exponents’ to mean the same thing.

Video thumbnail from ‘Negative Powers’, Nagwa 2018

I mentioned this to a friend, expecting her to be repulsed by the conflation of powers and indices. She wasn’t. The restrictive definition of power in the dictionary wasn’t the one that she used and saw used in her decades of experience in the classroom. This is a friend whose mathematical taste I value above my own, so I knew there had to be more to this story.

Diving into the textbooks

The same friend had access to lots of maths textbooks, and sent me some pictures of some relevant sections.

Here’s a helpful notation box from one A level textbook:

Taken from page 2 of ‘Edexcel AS and A level Mathematics Pure Mathematics Year 1/AS Textbook’, Pearson 2017

Ok, so if ‘power’ is defined to mean the same thing as ‘index’ and ‘exponent’, then we’re still left with the question of what to call ‘x⁵’. Surely we can’t call this the ‘power’ too?

Taken from page 63 of ‘MST124 Essential Mathematics Open University Unit 01 Algebra’

Apparently we can.

(Henri Poincaré once wrote ‘Mathematics is the art of giving the same name to different things’. I don’t think this is quite what he had in mind.)

The explanation in this excerpt from an Open University resource is probably the most lucid description I’ve seen of the terminology surrounding exponentiation, so I encourage you to read the whole thing if you haven’t already. What I initially got out of it was that, in the context of the expression 3⁵, ‘power’ has two meanings:

It can refer to just the raised number in the smaller font: 5. It can refer to the whole thing: 3⁵.

We’re assured that confusion between these two meanings of ‘power’ can be avoided by taking into account the context in which the term is used. In the UK textbooks I’ve looked at, the convention seems to be that the word ‘index’ is used for meaning 1 whenever the word ‘power’ is needed for meaning 2. ‘Exponent’ doesn’t seem to be used, despite often being defined. For example, have a look at these ‘key points’ from a GCSE textbook:

Taken from pages 9 and 10 of ‘Edexcel GCSE (9–1) Mathematics: Higher Student Book’, Pearson 2015

However, when the convention of using the word ‘index’ is broken, the result is sub-optimal:

Taken from page 11 of ‘Edexcel GCSE (9–1) Mathematics: Higher Student Book’, Pearson 2015

A productive analogy

What I asked for in my tweet was an analogue for exponentiation that acts like ‘sum’ does for addition, ‘difference’ does for subtraction, ‘product’ does for multiplication, and ‘quotient’ does for division.

We’ve discovered that ‘power’ has two meanings in the context of exponentiation, but I’d argue that neither of those meanings is analogous to ‘sum’ / ‘difference’ / ‘product’ / ‘quotient’.

We say:

‘the sum of 3 and 5 is 8’,

of 3 and 5 is 8’, ‘the difference of 3 and 5 is 2’ (or −2? A topic for another article.),

of 3 and 5 is 2’ (or −2? A topic for another article.), ‘the product of 3 and 5 is 15’,

of 3 and 5 is 15’, ‘the quotient of 3 and 5 is ⅗’,

but not ‘the power of 3 and 5 is 243’.

We always talk about the sum/difference/product/quotient of two numbers, but a power of one number. Focusing on making an analogy to multiplication, the way we use ‘power’ is much closer to the way we use ‘multiple’ than the way we use ‘product’.

The dictionary definition of ‘power’ says (emphasis my own):

When a real number a is raised to the index p to give aᵖ, the result is a power of a.

Compare this to ‘When a whole number a is multiplied by another whole number p to give ap, the result is a multiple of a.’

Similarly, from the Open University resource above (emphasis my own):

we say that 2³, or 8, is a power of 2.

Compare this to ‘we say that 2×3, or 6, is a multiple of 2.’

How did we get here?

In common usage today, ‘power’ is both the analogue of ‘multiple’ and a synonym for ‘index’. How did we end up in this situation?

Here’s my conjecture:

Initially, the word ‘power’ was only used as an analogue of ‘multiple’. This allowed people to talk about, for example, ‘the powers of 3’: {3, 3², 3³, 3⁴, 3⁵, 3⁶, …}.

The need then arose to single out a particular ‘power of 3’, say ‘3⁵’. The form of words chosen was ‘the 5th power of 3’. This makes sense; if you put the powers of 3 in order: 3, 3², 3³, 3⁴, 3⁵, 3⁶, …; then 3⁵ is indeed ‘the 5th power of 3’. You could imagine a world in which we didn’t have the word ‘product’ and so we had to refer to 3×5 as ‘the 5th multiple of 3’.

As 3⁵ is ‘the 5th power of 3’, it’s natural to say that 5 is the ‘index’ of 3⁵. Here, the word ‘index’ is being used as it is in computer science, where we talk about the index of an element in a list or array.

Referring to 3⁵ as ‘the 5th power of 3’ is a bit awkward; in symbolic notation the base 3 comes before the index 5, but this phrase swaps the order. As a result, ‘the 5th power of 3’ became something like ‘3 raised to the 5th power’ or just ‘3 to the 5th power’, a form of words which makes the logic of it being the 5th power of 3 less obvious.

Once the connection with meaning has been lost, it’s just a small step from ‘3 to the 5th power’ to ‘3 to the power of 5’, which appears to be the preferred way of saying 3⁵, at least in the UK:

And of course, once you read 3⁵ as ‘3 to the power of 5’ it’s natural to refer to that 5 as ‘the power’.

So far, so plausible, but of course it’s all conjectural. The true evolution of the terminology surrounding exponentiation will certainly be more messy.

In addition to running the poll above, Jo Morgan has looked at the language in textbooks historically used in either the UK or USA:

Make sure to click through and read the textbook excerpts for yourself!

As far as I can tell, the content of these textbooks doesn’t rule out my conjecture. There is, however, lots of research to be done. I hope to return to this topic one day, so let me know if you can fill in some of the story.

What to do?

Students are taught the importance of the precise use of terminology, but it’s hard to be precise when the terminology itself is so imprecise. Perhaps we should throw away the terminology we currently have and define our own from scratch with the benefit of hindsight.

‘Standards’, by Randall Munroe at XKCD, (CC BY-NC 2.5)

Is it desirable to try to reform the terminology surrounding exponentiation? Is it even possible? We could write resources, textbooks, and dictionaries using the terms we defined and hope that they caught on eventually. In the mean time, at the very least, students would be unfamiliar with the terminology that is actually used by people in the real world.

Taken from page 2 of ‘Edexcel AS and A level Mathematics Pure Mathematics Year 1/AS Textbook’, Pearson 2017

You can see now why textbook authors included three terms for the same concept; that’s how these terms are used, and students need to be prepared for that reality. When we talk about ‘using terminology precisely’ we don’t mean using terminology in a way that conforms to some abstract ideal; we mean using terminology the way other people use it in order to facilitate comprehension.

I do, however, have a couple of changes which might be workable:

Stop using ‘power’ to mean ‘index’ / ’exponent’

Currently there are three words we can use to refer to the ‘5’ in ‘3⁵’: ‘power’, ‘index’, and ‘exponent’. I mentioned before that UK textbooks define ‘exponent’ but tend not to employ it. I suggest that we do the same for this sense of ‘power’, noting that this word is sometimes used to mean the same thing as ‘index’ and ‘exponent’ so that students are aware of this possibility, but steering students away from using it in this way themselves.

More controversially, we could take this opportunity to make the preferred term ‘exponent’ instead of ‘index’, as it is in America. (In return, Americans would be expected to swap their definitions of ‘trapezium’ and ‘trapezoid’ to bring them in line with the UK.)

Suggested terminology after the UK-US terminology trade deal. (Contact me for my graphic design rates⸮)

Say ‘3 to the 5’ instead of ‘3 to the power of 5’

Definition of ‘power’ from The Concise Oxford Dictionary of Mathematics, 5th ed.

The dictionary entry suggests that ‘3 raised to the index 5’ is the correct phrase, but I don’t think I’ve ever seen it in use.

‘3 to the 5’ is more succinct, just as unambiguous, and, most importantly, already in use as a somewhat-frowned-upon contraction of ‘3 to the power of 5’. I suggest embracing this innovation, and bringing it into the fold of precise mathematical terminology. Removing ‘power of’ from ‘3 to the power of 5’ would hopefully sever the link between the word ‘power’ and what, going forward, should properly be called the ‘index’ or ‘exponent’.

Where would this leave us?

The word ‘power’ could be used unambiguously as the analogue of ‘multiple’, and the phrase ‘to the’ would become the analogue of ‘plus’, ‘minus’, ‘divided by’, and ‘times’.

We still wouldn’t have an analogue of ‘product’ though (nor, for that matter, for ‘multiply’) and that was, after all what we were looking for, wasn’t it?

We could, of course, invent new terms.

How about ‘expone’ as the analogue of ‘multiply’? It completes the pattern ‘add’ : ‘addition’ :: ‘subtract’ : ‘subtraction’ :: ‘multiply’ : ‘multiplication’ :: ‘divide’ : ‘division’ :: ‘expone’ : ‘exponentiation’. We could then expone 3 by 5 (in that order) to get 243.

We could choose ‘compound’ to be the analogue of ‘product’. It sounds like the result of some process or operation, and has a pleasing association with ‘compound interest’. We might worry about confusion with ‘compound fractions’ or ‘compound units’, but I think these two topics and exponentiation are sufficiently far apart. Using the same word to mean different things in different areas of mathematics is, after all, quite normal. With this choice, we would find that the compound of 3 and 5 is 243.

Should we go ahead and start using these terms? I think not. John Aldrich’s essay on the origins and sources of mathematical words is excellent on the topic of mathematical terminology in general, and gives good advice on when and how you should introduce new mathematical words. Exponentiation is a mature topic, and if there had been the need for words like ‘expone’ and ‘compound’ then someone would have defined them already. The fact that no-one has previously defined them suggests that we shouldn’t either.

The power of names

Having good terminology is important; anyone who has ever reached for a word in vain will tell you this. However, terminology is just a means to an end, and not an end in itself. My original problem wasn’t to find an analogue of the word ‘product’ for exponentiation, it was to find a natural way of expressing the following in words:

Of course, it is possible to express this verbally even without resorting to reading out the symbols:

The derivative of a power of a variable with respect to that variable is equal to the product of the index of that power and the previous power of the variable.

If you read that enough times you might be able to unpick its meaning, but it doesn’t have the same ring to it as ‘the derivative of a sum of functions is equal to the sum of their derivatives.’ I certainly wouldn’t call it ‘natural’.

Reading Euclid’s Elements, we see a lot of complicated prose amongst the geometric figures:

Proposition 26 from Sir Thomas Heath’s translation of ‘The Elements of Euclid’, 2nd ed.

It used to be that all mathematics was written in prose, and not just the statements of the theorems but their proofs too. The story of how the introduction of symbolic notation changed this and revolutionised mathematics has been told before. As Tobias Danzig put it, ‘the letter liberated algebra from the slavery of the word.’

I believe that both Proposition 26 and the way to differentiate a power of a variable are best understood with the help of symbolic notation, diagrams, and examples and non-examples. Once we come to apply them, however, we just need a way to tap into this understanding.

Rather than proving Proposition 26 again whenever it’s needed in a geometric proof, we can simply give its statement — all 5+ lines of it — and reference its proof. Going further, we could just say ‘using Proposition 26’ and forgo its statement as well. However, we have a more descriptive name for ‘Proposition 26’ which not only allows its statement and proof to be looked up but also hints at the statement; Proposition 26 tends to be known as the ‘Angle-Side-Angle congruence theorem/postulate’. When it is applied to justify a step of working, this full name can be abbreviated down as far as ‘ASA’.

Back to the problem of what to say when applying the following:

As with ‘Proposition 26’, rather than proving this rule from scratch or giving its statement in full, we can simply say we are applying ‘the power rule’.

The power of terminology is not only in being able to precisely talk about minutiae, but to give succinct and descriptive names to common concepts and techniques. There is a balance to be struck between defining too many terms and too few. The former sends the reader to the dictionary more often than is helpful; the latter leads to complex sentences of basic words where it’s easy to miss the forest for the trees.

Terminology is not the only tool we have for communicating mathematics; it complements, and is complemented by, diagrams and symbolic notation. It is, however, an essential tool in a way that, arguably, the other two aren’t. As Wittgenstein wrote, ‘The limits of my language mean the limits of my world.’