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| Integers, Pi, and Number Lines. |
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| Peter Taylor:
:-DD Love u all. |
| xrunner:
--- Quote from: Peter Taylor on July 05, 2022, 04:53:39 pm ---A circle can be drawn to an exact (integer, or number) circumference. Cut a piece of string to that circumference, and make a circle. A circle can be drawn to an exact (integer, or number) radius. Cut a piece of string to that radius, and make a circle. What can't be done is find how long to cut a piece of string to that radius, given the length of string to make the circumference, because the ratio of the circumference of a circle to its radius is not a number, but a question. --- End quote --- But I can draw a line from the circumference to the center of the circle I just made (I must know where the center is or I could not have made the circle with an exact piece of string. So all I have to do is measure the line I just drew, and that's the diameter; not a question anymore but an answer. How can that happen? |
| IanB:
--- Quote from: Peter Taylor on July 05, 2022, 04:53:39 pm ---What can't be done is find how long to cut a piece of string to that radius, given the length of string to make the circumference, because the ratio of the circumference of a circle to its radius is not a number, but a question. --- End quote --- Absolutely it can be done. Turn a cylinder on the lathe to an exact diameter of 1 unit. Wrap a piece of cord snugly around the cylinder, and cut it where it overlaps, so that the two ends meet. You now have piece of cord exactly pi units long. |
| Vtile:
--- Quote from: Nominal Animal on June 30, 2022, 06:55:54 pm --- That is, we never, ever try to divide anything by zero there. Instead, we can think of the process as looking at how the ratio changes as \$h\$ gets closer to zero, and extrapolate the value the ratio would be if we could calculate it at \$h = 0\$. --- End quote --- IIRC Newton had this concept of using something almost a zero (taste, feels and quaks like zero) as divisor when forming the concept and ideas of modern calculus. Much more intuitive, but unfortunately not mathematically rigorous. That was fixed and reformed much later with idea of limits. ....In sixties the duck did born again with hyper-reals. Quack quack. |
| Vtile:
--- Quote from: IanB on July 05, 2022, 07:05:58 pm --- --- Quote from: Peter Taylor on July 05, 2022, 04:53:39 pm ---What can't be done is find how long to cut a piece of string to that radius, given the length of string to make the circumference, because the ratio of the circumference of a circle to its radius is not a number, but a question. --- End quote --- Absolutely it can be done. Turn a cylinder on the lathe to an exact diameter of 1 unit. Wrap a piece of cord snugly around the cylinder, and cut it where it overlaps, so that the two ends meet. You now have piece of cord exactly pi units long. --- End quote --- Only if the string has diameter of zero. |
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