1
$\begingroup$

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, y 4 , and 3 5 are, respectively, n, m, 4, and 5.

I am not a native English speaker. To me, these 3 meanings are so distinct. Why are they sharing the same word exponent? I think the subtle relations among these 3 meanings may help me better understand the human rationale/subconsciousness behind the mathematical concept.

ADD 1

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

enter image description here

$\endgroup$
  • 8
    $\begingroup$ This question belongs on another site in the Stack Exchange network - ell.stackexchange.com $\endgroup$ – Ian Miller Sep 14 '16 at 3:09
  • $\begingroup$ @IanMiller Thanks for the reminding. I did considered that site. But I am a bit afraid if it will get enough explanations in a mathematical way. $\endgroup$ – smwikipedia Sep 14 '16 at 3:12
  • $\begingroup$ While some math terms are related to everyday meanings, many are not (normal, field, ideal etc). I don't see the other meanings of exponent being related to the math one in any meaningful way, subtle or not. That said (and as suggested already) ell.stackexchange.com will likely provide more insights on the history and appropriation of exponent in math. $\endgroup$ – dxiv Sep 14 '16 at 3:19
  • 3
    $\begingroup$ If the question were re-framed as "When and why did powers come to be called 'exponents'?" it might be a good fit for hsm.stackexchange.com. $\endgroup$ – mweiss Sep 14 '16 at 3:20
  • $\begingroup$ @mweiss I didn't know that site until now. I will try it. Thanks! $\endgroup$ – smwikipedia Sep 14 '16 at 3:41
3
$\begingroup$

You are not the first to ask that question (perhaps in the stackexchange network you do).

Googling a bit: http://mathforum.org/library/drmath/view/65184.html

Looking up the context of this sentence in Smith, I find that it follows a table of powers of 2:

0   1   2   3   4   5   6   7   8 
1   2   4   8  16  32  64  128 256

He says (in Latin), "Just as, by addition, (in the upper row) 3 [added] to 5 makes 8, so (in the bottom row), by multiplication, 8 [multiplied] by 32 makes 256". [...] So the words "exponent" refers to > the number in the top row corresponding to a given number in the bottom row; this is the number of 2's multiplied to make the given number--what we today would call its base-2 logarithm, or the exponent of 2 that gives the number.

In short, it's impossible to know the exact meaning or reason of choosing exponent instead of any other word. I don't think you'll be able of deducing the intrinsics of Smith's mind from their explanations when choosing exponent, because he died five centuries ago.

| cite | improve this answer | |
$\endgroup$
  • 3
    $\begingroup$ Nice find. From the same page linked off that answer: "In mathematics, power refers to the number arrived at by raising a number to an exponent. In the mathematical expression 3^2=9, three is the base, two is the exponent, and nine is the power. Students often refer to the exponent as the power, but this is not historically correct, although it has become so common, even among many teachers, that some dictionaries refer to the power as the exponent.". $\endgroup$ – dxiv Sep 14 '16 at 3:45
  • $\begingroup$ @dxiv Thanks for pointing that out. I was one of those who refer to the exponent as the power... $\endgroup$ – smwikipedia Sep 14 '16 at 5:53
  • 1
    $\begingroup$ @smwikipedia Most of us use power and exponent interchangeably nowadays in contexts where there is no potential for confusion. FWIW I don't recall seeing the exponent called index in US-English, but that may be just my anecdotal ignorance ;-) $\endgroup$ – dxiv Sep 14 '16 at 6:04
  • 1
    $\begingroup$ While this link may answer the question, it is better to include the essential parts of the answer here and provide the link for reference. Link-only answers can become invalid if the linked page changes. - From Review $\endgroup$ – Scientifica Sep 14 '16 at 7:42
  • 1
    $\begingroup$ @Scientifica Done. $\endgroup$ – Peregring-lk Sep 14 '16 at 14:04
2
$\begingroup$

The common thread in the three definitions is that an exponent is something which increases or highlights a thing.   To make something larger, or more clear.

1.a person or thing that expounds, explains, or interprets: an exponent of modern theory in the arts.

They are people or things which seek to increase or highlight knowledge.

2.a person or thing that is a representative, advocate, type, or symbol of something: Lincoln is an exponent of American democracy.

They are people or things which seek to increase or highlight principles; by example or through leadership.

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 $x^n, 2^m, y^4$ , and $3^5$ are, respectively, $n, m, 4,$ and $5$.

They are things (mathematical operations) which increase the values.   Albeit, in mathematical usage an exponent may perhaps do so negatively, depending.

| cite | improve this answer | |
$\endgroup$
  • 1
    $\begingroup$ It may also have something to do with the fact that power was used since the Greeks, and there was a need to differentiate the exponential function from the power function . $\endgroup$ – dxiv Sep 14 '16 at 3:48

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