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(This question was first asked here. I modified it and moved it here based on some suggestions.)

I have some difficulty in understanding the complex exponential function. So I decide to review the good old exponent function which I learned long ago.

I look up the word exponent, the online dictionary says this:

noun

  1. a person or thing that expounds, explains, or interprets: an exponent of modern theory in the arts.
  2. a person or thing that is a representative, advocate, type, or symbol of something: Lincoln is an exponent of American democracy.
  3. Mathematics. a symbol or number placed above and after another symbol or number to denote the power to which the latter is to be raised: The exponents of the quantities xn, 2m, y4 , and 35 are, respectively, n, m, 4, and 5.

I am not a native English speaker. To me, these 3 meanings are so distinct.

So when and why did powers come to be called 'exponents'?

ADD 1

According to here, the exponent part is also called index, or power. I think the latter ones index or power are more acceptable.

enter image description here

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    $\begingroup$ What I find mysterious about the terminology is, that besides being called exponent, index or power, the $b$ in $a^b$ is the logarithm to base $a$ of $a^b$. But no book I've seen calls it explicitly that way, or draws a picture like the one in your question and adds "or logarithm" below "or index, or power". $\endgroup$ – Michael Bächtold Jul 9 '18 at 10:24
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    $\begingroup$ @Michael: If logarithms were made to seem simple and approachable, students might be able to understand them, and we can't have that! $\endgroup$ – Kevin Jun 6 at 15:35
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The first use of the Latin exponentem seems to be [thnaks to @fdb's answer]:

with the latin source: expono - to display.

Leonhard Euler uses it into his:

For the "definition", see:

§ 172 English transl. by John Hewlett (1822) : To avoid this inconvenience [that of writing e.g.: $aaaaa$], a much more commodious method of expressing such powers has been devised, which, from its extensive use, deserves to be carefully explained. Thus, for example, to express the hundredth power, we simply write the number $100$ above the quantity, whose hundredth power we would express, and a little towards the right-hand; thus $a^{100}$ represents $a$ raised to the $100$th power, or the hundredth power of $a$. It must be observed, also, that the name exponent [German: der Exponent] is given to the number written above that whose power, or degree, it represents, which, in the present instance, is $100$.

A corresponding early English textbook uses instead "index" :

instead of $xx$ we write $x^2$. [...] These products are called powers of $x$; the figures representing the number of repetitions, are called the indexes of those powers [...].

Exponent and index are used into:

These figures which express the number of factors that produce powers are called their indices or exponents.

They are used as synonyms also later; see:

if the symbol $a$ be repeated any number of times ($n$) in the expression $aaa \ldots$ &c it is written $a^n$, and is called the $n^{th}$ power of $a$, and the number $n$ is called the index or exponent of the power.


Thus, I'm prone to think that Euler's "revival" of Stifel's term was the source of modern usage.

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This is a question about etymology and the answer should be sought in the standard etymological dictionaries. The DWDS says of „Exponent“ „als mathematischer Terminus 1544 bei M. Stifel nachweisbar“. That is a good two centuries before Euler. http://www.dwds.de/wb/Exponent

The logic is that the index numbers are “placed outside” the line of writing.

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  • $\begingroup$ This is such a vivid explanation. Very informative. Thanks! $\endgroup$ – smwikipedia Jul 9 '18 at 1:01
  • $\begingroup$ For those who don't read German, the idea is "ex"=outside, "ponere"=put. $\endgroup$ – Ben Crowell May 30 at 0:15

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