18
$\begingroup$

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.

|improve this question
$\endgroup$
  • 3
    $\begingroup$ 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 $3a$ ... $\endgroup$ – Mauro ALLEGRANZA Dec 21 '14 at 17:07
  • 1
    $\begingroup$ 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.) $\endgroup$ – Charles Dec 21 '14 at 17:24
  • 1
    $\begingroup$ @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! $\endgroup$ – HDE 226868♦ Dec 21 '14 at 22:03
  • 1
    $\begingroup$ @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? $\endgroup$ – Frank Vel Dec 21 '14 at 22:55
  • 2
    $\begingroup$ @fvel note that merging of questions is an option we have here. $\endgroup$ – Danu♦ Dec 22 '14 at 7:39
16
$\begingroup$

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 $5a^4$.

|improve this answer
$\endgroup$
6
$\begingroup$

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: $a^3$.

|improve this answer
$\endgroup$
  • 2
    $\begingroup$ I think this answer could be significantly improved by expanding it a little bit; right now it's very short and not terribly enlightening. $\endgroup$ – Danu♦ Dec 22 '14 at 7:36

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.