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Why do we write $a^n$ instead of $^n\!a$ for exponentiation? What benefit is there to writing the base before the exponent? With addition and multiplication order doesn't matter since $a + b = b + a$, so why was $a^n$ chosen, and who popularised this notation?

To me the first is preferable since it's a logical choice, as you put the number in front when you add: $$a+a+a = 3a$$ Thus it makes sense to write: $$a\times a\times a = \,^3\!a$$

It would also retain left-associativity as is commonly used with subtraction and division: $$(a\text{^} b)\text{^}c = \,^\left(^ab\right)c$$

Historically the best choice isn't always the one which got popularised, but there must have been some reason why this choice were made and preferred in its period of invention.

A similar question was previously asked on Mathematics.

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Frank Vel
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  • fvel, please don't cross-post questions. @MauroALLEGRANZA, thanks for pointing that out. – HDE 226868 Dec 21 '14 at 15:21
  • @HDE226868 I was suggested by a user there to ask this question here. I did not know about cross-posting were unadvised, as the question is relevant on both sites. That said, I have rephrased the question on this site to focus on the historical details of why this notation was chosen, whereas the other asked of the advantages in general, with no connections to historical reasons being implied. – Frank Vel Dec 21 '14 at 16:58
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    Personally, I think that Descartes' choice for $a^3$ instead of $^3a$ was "more rational"; especially with "ancient" printing, it is quite easy to equivocate between $^3a$ and $3a$ ... – Mauro ALLEGRANZA Dec 21 '14 at 17:07
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    Note also that $^3a$ has some currency as tetration, i.e., $^3a=a^{a^a},$ providing also a modern reason to avoid that notation. (Personally, I dislike this notation, preferring Knuth's $\uparrow^2$, but that's neither here nor there.) – Charles Dec 21 '14 at 17:24
  • @Charles This is irrelevant I think, as there were no use of tetration at the time the notation for exponentiation was invented? I don't have any sources for this claim though, but I assume the notation for exponentiation were invented before tetration. – Frank Vel Dec 21 '14 at 17:27
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    @fvel: That's why I called it a "modern reason". – Charles Dec 21 '14 at 17:28
  • @MauroALLEGRANZA I have never mistaken $2^2$ for $22$, but I admit it's useful, as I have seen $2a2$ being written as $2\cdot a^2$. – Frank Vel Dec 21 '14 at 17:28
  • @fvel The question on Mathematics asks in part "who popularised this notation?" I'd say that this post is already covered in the original post. – HDE 226868 Dec 21 '14 at 20:33
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    @fvel I understand completely what you mean. I apologize if I came off as harsh; a couple weeks ago we had a cross-posting incident here from Mathematics where a question was cross-posted and an answer from Mathematics. There was another issue with a second answer (that ended up being deleted), a meta post, dialogue with a Math mod to solve it. . . The whole thing was a nightmare. I guess I was a little worried this might end up like that, so I sincerely apologize if I seemed hostile, rude or otherwise unpleasant. That certainly wasn't my intent! – HDE 226868 Dec 21 '14 at 22:03
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    @HDE226868 Part of the problem also stems from the fact that I had no knowledge of HSM prior to asking my question on Mathematics, so I naively phrased it to allow for historical answers. I was then advised that the question would be better on this site. For future reference, what do I do when this happens? The question and answer would certainly be better suited on the relevant site, but as it was only part of my question, what am I to do? Simply cross-post the part relevant for this site with linking? Remove the part relevant for the relevant site, and ask the answerer to move his answer? – Frank Vel Dec 21 '14 at 22:55
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    @fvel I've been thinking of writing a meta post on Mathematics about what to do, given that I've seen suggestions like the one made in the comment crop up a lot recently. I think (and this is just me saying this, not as a mod) that if you want it here, you should flag your post for migration and ask a mod on Mathematics to migrate it. If you like the answer, though, just keep it all on Mathematics. You'll most likely get good answers there, and it's certainly on-topic there. – HDE 226868 Dec 21 '14 at 23:05
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    The problem is exactly that: It's on-topic on both sites. What if a future user comes to this site and searches for the question, gets no result, and poses the same question here? The problem was also that I was asking multiple questions, which to me seemed to be related and would give a nuanced and detailed answer. Yet only some of those would fit here. This is an inherent flaw of SE, as it would be better if it allowed questions from different sites to be merged somehow. You should definitely make a meta post, because it's better to hear more people's view on this. – Frank Vel Dec 21 '14 at 23:55
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    @fvel note that merging of questions is an option we have here. – Danu Dec 22 '14 at 07:39
  • @Danu Ah, I was not aware. Should I cross-post and notify a moderator to merge then? I'm still unsure of how I should proceed when this happens... – Frank Vel Dec 22 '14 at 09:29

2 Answers2

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Browsing the "original" historiographical" source for the "power" symbolisms , i.e. :

can be very istructive, showing how Descartes' choice was the last step of a complex process :

  • Pietro Antonio Cataldi : $5$ $3(crossed)$ for $5x^3$

  • Joost Bürgi : with $vi$ on top of $8$ for $8x^6$ (obvious problem : how to handle two variables : $x$ and $y$ ?)

  • Romanus (Adriaan van Roomen) : $A(4)$ for $A^4$

  • Pierre Hérigone : $a3$ for $a^3$ and $2b4$ for $2b^4$

  • James Hume (1636) : $A^{iii}$ for $A^3$.

Thus, Cajori concludes that :

Hérigone and Hume almost hit upon the scheme of Descartes. [...] Where Hume would write $5a^{iv}$ and Hérigone would write $5a4$, Descartes wrote $5a^4$.

Mauro ALLEGRANZA
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See Earliest Use of Symbols for Operations for a discussion. Summary: Descartes 1637 is the first to use exponents for powers in the modern way: $a^3$.

Gerald Edgar
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    I think this answer could be significantly improved by expanding it a little bit; right now it's very short and not terribly enlightening. – Danu Dec 22 '14 at 07:36