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This might seem like an inappropriate question and even though its easy to retort using analytic geometry, I don't feel convinced that a tangent really does intersect a circle at only one point. Maybe it's better to ask why does it look like that the tangent intersects the circle at more than one point?

If, for example, we have a circle with the equation $x^2 + y^2 = 100$, and a tangent to this circle $y = 10$, then although substituting the value of $y$ in equation of the circle gives just one answer in the form of $x^2 = 0$, but when I look at the diagram of these equations and think about how if we place an extremely large straight bar on a (large enough) circular body, the bar won't really touch the circular body at just one point. I was also going to write about celestial bodies but I changed my decision because I realized that they aren't perfectly circular (or spherical).

How can I convince myself that a tangent intersects a circle at just one point?

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    $\begingroup$ What is your definition of a tangent? Also consider that a circle is a convex set. $\endgroup$ – Jack D'Aurizio Sep 5 '15 at 13:41
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    $\begingroup$ Straight bars and circular bodies are NOT lines and circles. The latter are abstract mathematical objects. You solved the equation and got correct answer and there is nothing more to be said here. $\endgroup$ – Blazej Sep 5 '15 at 13:55
  • $\begingroup$ If we know a tangent is perpendicular to the line from the centre to the point of tangency, then the result follows from the fact that the hypotenuse is always greater than a leg. $\endgroup$ – André Nicolas Sep 5 '15 at 13:56
  • $\begingroup$ So what other point besides (0,10) do you think your examples intersect at? $\endgroup$ – Deusovi Sep 5 '15 at 13:57
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    $\begingroup$ @modularnix: It DOESN'T in the real world though. There are no perfect circles or lines in the real world, only in math. Math is a world of perfect abstractions. $\endgroup$ – Deusovi Sep 5 '15 at 14:18
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The real world often doesn't have the same properties of math. Math is a world of perfect abstractions, and it almost never applies exactly because of the simple fact that perfect circles, lines, and shapes don't actually exist. A tangent line only touches a circle at one point; a real world approximate line touching an approximate circle will most likely be touching over an interval of space. (Or not at all, but then we get into a discussion about what it means to be touching at the atomic level.)

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No, tangent only touches the circle at one place.
Everything is not perfectly sphere or circular, so it seems that it is touching the circle at multiple points. "How do I convince myself?" Ok, first tell me how do you convince yourself that bacteria, which are not visible to the naked eye, exist?
And also, if in the case of plot, it is pretty easy to explain, see this,

If you zoom in, you will find that it is indeed touching at only one point.
A partially non rigorous way is to think about derivatives, derivative of a circle (suppose for the positive side) exists, and gives the slope of only one tangent at a point, so there is only one point.
We define the tangent like that only, so that it just "touches" the point.
Sorry uniquesolution, an example taken: Consider the example of a wheel. When we ride bicycle on mud, mud splashes in $90^{\circ}$. Not in any other angle. Even if it does, primarily mud goes at right angle, but then it diverges due to different factors. As said before, real life is not perfect.

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Many years ago my father showed me a book called "non standard analysis" by Abraham Robinson. When I asked what was that all about, he said "it is pretty clear that a wheel resting on the road touches it at more than one point." Of course, we all know it is so. He didn't really mean to suggest that a wheel touches the road at only one point, possibly as much as the poster of this question did not intend to be presented with detailed pictures as to why a tangent line touches the circle at one and only point. In fact, mathematical theory is of course an idealisation, and everyone ought to agree that the real world offers us no manifestation of a mathematical tangent line to a mathematical circle. But that's philosophy, and this is a mathematics forum.

The answer by Aditya Agarwal is of course correct, within our "usual" world of mathematical analysis. However, there is a discipline called Non-standard analysis, advocated by Abraham Robinson about 60 years ago in Princeton, where our standard construction of fundamental notions such as real numbers, limits, and continuity, is questioned, and an alternative theory is developed, where you can have a tangent line touch at more than one point. See here: http://plato.stanford.edu/entries/continuity/#9

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    $\begingroup$ "it is pretty clear that a wheel resting on the road touches it at more than one point". First of all, it is not perfectly circular. And also it is 3-dimensional. And, even if the first two conditions are satisfied, then because wheel has a weight, and it will get pressed on the road. So it will touch at multiple points. $\endgroup$ – Aditya Agarwal Sep 5 '15 at 14:32
  • $\begingroup$ @AdityaAgarwal -- Good job! $\endgroup$ – uniquesolution Sep 5 '15 at 14:33
  • $\begingroup$ Your link is quite good. (y) Interesting! $\endgroup$ – Aditya Agarwal Sep 5 '15 at 14:34

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