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so a straight line can also be a curve
Theoretically yes but actually no. A curve is a line of which no part is straight and which shows diagrammatically a continuous variation of quantity or force etc. It can also be a surface of which no part is plane.
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[bean]
so a straight line can also be a curve
[/bean]
[rubrick]
Theoretically yes but actually no. A curve is a line of which no part is straight and which shows diagrammatically a continuous variation of quantity or force etc. It can also be a surface of which no part is plane.
[/rubrick]
Like so much of what I think I almost know, the meaning of curve is something I have osmoted and not something I have ever looked up.
There is a difference between the common understanding of a term and the way the term is used in a specialized case like mathematics. (Forgive me for stating the obvious.) Ask a race car driver how to accelerate a car -- you'll get an odd stare or glare and, if you get any response at all, a reply in tones clearly indicating that the question was asked by an idiot, that "you step on the ACCELERATOR." Ask an eager algebra student the same question and he'll immediately recognize that a deceleration is a negative acceleration and he could reasonably say "you can step on the gas or the break." An engineer realizes that if a direction changes, even if the speed is constant, that this is an acceleration and so he might respond "step on gas or brake, or turn the steering wheel." Some people might give clever responses like running into a wall or shifting weight or what have you. But everyone answered correctly in the scenario. (I use this example all the time, but don't recall if I've used it in this forum.)
But back to the point, I think in the common language that clearly curved lines are not straight lines. In the technical language of the mathematician, I do not know the correct definition, but I do know that I have heard mathematicians talk about curves (generically, as the loci of functions) that could, in some cases, be straight lines. I don't know that these guys used the terminology correctly, but I do know they used the words that way.
Trivial example:
Find the area under the curve y=2x between x=2 and x=5.
(Clearly this is a straight line.)
I'm quite sure I've heard this said in class, and I 'think', but am not sure that I've seen this language used in text books. Again, just because mathematicians have used the terms like this doesn't mean they used it correctly.
k
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Joined: Jan 2001
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old hand
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old hand
Joined: Jan 2001
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Sorry, I wasn't very clear. Mostly we use curve to mean the plot of a function, like the calculus usage that TFF gave above. (Especially when the context is extrapolation/interpolation, where we are most certainly talking about a plot of data and a best-fit curve.) So the definition might be something like: curve, all the points which satisfy some equation. (If they all happen to lie on a straight line we still refer to it as a curve, and I apologize.)
Again, just because mathematicians have used the terms like this doesn't mean they used it correctly.
They've used the word correctly in the given context, in terms of what all mathematicians understand "curve" to mean. In any language, one word may have multiple "correct" definitions - it's all good as long as everyone using the word has the same idea of which definition is being employed. You'll find the terminology used consistently in math books written in English, so this is by definition, correct usage in in this context, since everyone in the field has tacitly agreed to that usage. If they were inconsistent, it might be a mistake, but that terminology is quite consistent.
I mean, much as we may rail against changes or irritating neologisms, usage is what makes language "correct" or "incorrect". If the majority of people accept and use a word or construction, then it's correct. So, the majority of mathematicians consider any locus of points* satisfying an equation a "curve", and it is thus correct.
Edit: *Should clarify that I mean in 2-D, specifically.
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