Does anyone know an answer to either/both of the following?
A) Does the term, “tangent” (from trigonometry) have any relation to a tangent line? Since trig functions are based on the unit circle, does the tangent of an angle have some relation to some tangent line to the unit circle? :dubious:
B) Does the conic section “hyperbola” have any relation to 4-D space where things are deemed hyperspace, hypersurfaces, etc? Why, then, is a hyperbola “hyper”? :dubious:
“Hyper” is just the Greek word for “over”. People also like to over (pun intended) use “super”, which is the same thing but in Latin, and “uber”, which, of course, is from German. They are all cognates with English “over”.
Hold on a moment while I power up my ubersuperhyperoverdrive.
Interesting, and consistent with conventional language, especially medical terms. Despite the etymology, I often see “hyper” to coin words that mean “even more than super”. Computer operating systems have a supervisor, and virtual machine models have what’s called a hypervisor. I guess because it is kind of on a higher dimension than a supervisor.
I was certain you were wrong about this, so I did a little research, and it turns out that you were entirely correct.
I would say that “hyper-” as a prefix in English implies excess of something, rather than just “more than super”. Hyperactive, hypercritical, hypersensitive - they all have negative connotations implying too much of the quality in question.
A hyperbola is a conic section that opens up forever. A parabola also opens infinitely - but the crossection of a cone that defines a parabola is parallel to the side of the cone, so the solution to the equation only opens one way, while a parabola has two symmetric opposite curves.
A cut into the cone less than parallel to the edge of the cone produces an ellipse, a single closed figure. A circle is a special case where the major and minor axes of the ellipse are equal - a cross-section perpendicular to the axis of the cone.
(I suppose there’s one more case, where the cross section plane is tangent to the cone surface, producing a straight line.)
Not sure what the exact logic for the choice of the word hyperbola is, but para- for parallel, hyper- for beyond the parallel…
And as for mathematical terms with dual meanings: I noted two seemingly sort-of-distinct meanings for “relation”, but as I gathered a bit more mathematical understanding, I came to understand that they actually were the same, or at least quite similar. I think I’ve noticed this also with other math terms that seem to have two meanings.
Relation:
(1) A set of ordered pairs. This could be anything as simple as just a few enumerated pairs like
{ (3, 5), (7, 18), (37, -6) }
or, what is typically of greater interest, the locus of some equation in two variables.
(2) “Relation” also refers to a statement about the relative sizes of two quantities, e.g.,
3 = 3
5 > 3
6 < 35
Of greater interest, a statement about the relative sizes of two quantities involving two variables:
3x = 7y
5x > 3y[sup]2[/sup]
3x[sup]2[/sup] < 4y[sup]2[/sup]
It took me quite a while into my math education before I realized that this usage of the term “relation” is, in fact, the same usage as in (1) above.
[QUOTE=Wikipedia]
The word “hyperbola” derives from the Greek ὑπερβολή, meaning “over-thrown” or “excessive”, from which the English term hyperbole also derives.
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Interesting that all three major types of conic sections share names with literary terms. “Parabola” comes from the same Greek word as “parable”
and “ellipse” comes from the same word as “ellipsis.”
[QUOTE=Wikipedia]
Ellipsis (plural ellipses; from the Ancient Greek: ἔλλειψις, élleipsis, “omission” or “falling short”) is a series of dots that usually indicates an intentional omission of a word, sentence, or whole section from a text without altering its original meaning.
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[QUOTE=Wikipedia]
In linguistics, ellipsis (from the Greek: ἔλλειψις, élleipsis, “omission”) or elliptical construction refers to the omission from a clause of one or more words that are nevertheless understood in the context of the remaining elements.
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