(No. At least, if you take "geometric way" to mean "means to construct using a compass and straight-edge", then no. Transcendental numbers can't be constructed with a ruler and compass. I imagine that you could construct a tool to let you do this, in much the same way that you can construct a tool that lets you square a circle...)
Right. I was hoping that was the case, or my understanding of transcendental (heck, irrational) numbers was way off.
On the other hand, you can demonstrate pi, just not draw a line of length pi, with a compass (i.e., point to the circumference of a circle). My mind is trying to create some infinite-regress of seashell spirals to demonstrate e similarly...
You *can* draw many irrational numbers, just not transcendental numbers. You can, for example, construct the square root of two (the diagonal of a square).
Algebraic numbers are the numbers that are roots of some polynomial equation (with integer coefficients not all equal to zero). Transcendental numbers are the ones that aren't algebraic.
I don't really have a good enough intuitive understanding of e to have an idea how to demonstrate it geometrically.
You could rig up springs with the right coefficients, but that's cheating. You could demonstrate Pi and then derive e from it, but that's unsatisfying. You could could iterate triangles based on the ratio expansion, but that's an approximation.
Of course, in the end a geometric method can still only define it as precisely as the precision of the tools available, in our world.
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(Hmm, good question, is there a geometric way to derive e precisely?)
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On the other hand, you can demonstrate pi, just not draw a line of length pi, with a compass (i.e., point to the circumference of a circle). My mind is trying to create some infinite-regress of seashell spirals to demonstrate e similarly...
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Algebraic numbers are the numbers that are roots of some polynomial equation (with integer coefficients not all equal to zero). Transcendental numbers are the ones that aren't algebraic.
I don't really have a good enough intuitive understanding of e to have an idea how to demonstrate it geometrically.
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You could rig up springs with the right coefficients, but that's cheating. You could demonstrate Pi and then derive e from it, but that's unsatisfying. You could could iterate triangles based on the ratio expansion, but that's an approximation.
Of course, in the end a geometric method can still only define it as precisely as the precision of the tools available, in our world.
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