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When watching some videos on various bits of mathematics, I observed the word "argument" being used to describe what is in layman's terms 'the inside of the bracket'.

I can understand the use of the word 'argument' in the context of complex numbers; attributed to Jean Argand's work with complex numbers.

So for example, for the complex number $z= \cos \theta + i \sin \theta$, we say that $\theta$ is the argument of $z$. In this context, the use of the word "argument" makes sense because it is the input of the function as well as having some relation to the Argand diagram.

However, why is it used in a more general context of $f(x)$, where $x$ is then said to be the 'argument' of $f$?

Trogdor
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    "In this context, the use of the word "argument" makes sense because it is the input of the function as well as having some relation to the Argand diagram." - Why does the use of "argument" make sense in this context? I see no obvious preference for argument in this context vs. the more general context other than the common prefix "arg" of "argument" and "Argand" which is quite likely just a coincidence. – Colm Bhandal Aug 18 '15 at 18:01

2 Answers2

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Tracing the etymology of a word can be a difficult task. For what it is worth, the usage of the word argument as "the angle, arc, or other mathematical quantity, from which another required quantity may be deduced, or on which its calculation depends" dates back to 1405AD. Cite: the Oxford English Dictionary.

Umberto P.
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It is a term with two different meanings. In the case of functions, which have inputs and an output, the inputs are commonly called arguments or parameters. In the case of the complex plane (also called the Argand diagram) any complex number is uniquely determined if you know both its distance from the origin and the anticlockwise angle at which the vector from the origin makes with the real axis, and that angle is called the argument of the complex number, while that distance is called the modulus. (Yes "modulus" is another term with a different meaning in number theory.)

user21820
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    This doesn't answer the question. The OP is asking about why the variables of a function are called "arguments". – Alex M. Aug 18 '15 at 17:00
  • @AlexM.: I did answer the question! "$x$" in "$f(x)$" is the input given to $f$ in that expression and hence called the argument given to $f$. – user21820 Aug 18 '15 at 17:04
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    And the OP asks whys is the argument called "argument". Why do we use this word? You didn't answer this. More clearly, in $f(x) = x^2 + 1$ why do we call $x$ argument, given that it has no connection with the argument as in "complex numbers argument". – Alex M. Aug 18 '15 at 17:05
  • @AlexM.: It wasn't clear that the asker was asking why we use that particular word for that concept. Rather it was clear that the asker was wondering why we call "$x$" argument when he knew about arguments in the complex plane. – user21820 Aug 18 '15 at 17:06
  • @AlexM.: The answer is that it is just used that way. There is no better answer. There is no really good reason why we call groups groups, rings rings, fields fields, modules modules, ... – user21820 Aug 18 '15 at 17:10
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    Oh yes, there is, it is called "etymology" and "history of mathematics". Why were they called so "in the beginning"? This is what the OP asked, and I'm pretty curious myself to find the answer. @Umberto P. gets much closer to what the answer should be. – Alex M. Aug 18 '15 at 17:21
  • @AlexM.: I totally disagree that Umberto gets much closer. The resource he cites clearly merely summarizes the usage observed, not the reason for the usage. – user21820 Aug 18 '15 at 17:41