Why do we write $a^n$ instead of $^n\!a$ for exponentiation?

For subtraction I can understand why $2-3 = 2+(-3)$ since we read from left to right, but I don't see why this need apply to 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?

A similar question, with the focus on the historical reasons, has also been asked on History of Science and Math.

• @user201168 I'm aware that $^n\!a$ is used for tetration, but that's not my question. – Frank Vel Dec 17 '14 at 23:02
• You should browse Cajori's History of Mathematical Notations. – Mariano Suárez-Álvarez Dec 17 '14 at 23:04
• From a purely typographical point of view alone the common notation seems a lot more natural to me. – quid Dec 17 '14 at 23:06
• @par You can multiply three by itself a hundred times, you will still get nine every time. – bof Dec 17 '14 at 23:06
• @par I didn't exclude the other, but treating exponentiation with the symbol $^$ you often need to apply this is as right-assosiative. $a^{b^c} = a^{(b^c)}$ while $a-b-c = (a-b)-c$ and $a \div b \div c = (a\div b) \div c$, which is left-assosiative. So it breaks a pattern. – Frank Vel Dec 17 '14 at 23:10

Wikipedia says "The modern notation for exponentiation was introduced by René Descartes in his Géométrie of 1637", and has a link to a page from Descartes.

• This answer just answers a part of the question. I'll admit the question is poorly phrased, but I was hoping to get the answer to "What benefit is there to writing the base before the exponent?", and this doesn't. Nor does it say "why was $a^n$ chosen". Although both have been addressed and partly answered in the comment section of the question. – Frank Vel Dec 21 '14 at 17:34

I'm going to say that it's also because of the way we read.

$2^3$ is "two to the cube" or "two to the third" whereas $^23$ would be "to two, the three?".

Of course, this could be very bad reason since you can argue that the operation $3^2$ existed before we decided to read it...

(I wanted to write this as a comment better than a proper answer, but I can't write comments yet I'm afraid)

• a squared three? Of course the way we read these things were made up after the notation, so that's not a good argument. – Frank Vel Dec 17 '14 at 23:20
• @fvel this is actually debatable. Earlier math was written in a more verbal way; and even now often the ideas and talking about them comes before notations are decided on. – quid Dec 17 '14 at 23:25
• Yes, I know. That's why I wanted to write it as a comment. It didn't feel like an actual answer, but I felt it was worth pointing this out. – Gin Dec 17 '14 at 23:26
• @quid Good point, although that raises the question, why would you say "two to the third" instead of "a cubed two"... – Frank Vel Dec 17 '14 at 23:27
• "The cube of two" and "the third power of two" sound only slightly stilted to me. We very often talk about "powers of two", so it's not like it's too alien. – Dan Uznanski Dec 17 '14 at 23:28