>It would be helpful if you could show us an example of this cube root of 2,
says Kate, one of your eighth grade math students.
>Well, consider a cube with a volume of 2 cubic units. Its side length is the cube root of 2.
>Isn't it circular reasoning to prove the number exists that way?
asks David.
>We don't know if a cube with that volume is possible until we've found a cube root of 2.
>Let's try another approach. Make a graph of , then draw a line at . The point where the line meets the curve has x coordinate .
>Why is it okay to just plot a few points and then just guess where the curve goes?
asks Kate.
>You would never accept this sort of reasoning in a geometry proof, but when graphing it's apparently okay.
David raises his hand.
>I was wondering how the graphing calculator does it, so I asked my uncle who programs computers for a living. He said that it just plots a large number of points and connects consecutive points with the closest thing to a line it can draw. That's not really a smooth curve, it just looks like one.
Can you show a better example of ? What about other numbers David and Kate will learn about, like ln(2), or erf(1), or the sum of 1/2 to the power of every integer which is the ASCII encoding of a valid halting Brainfuck program? How far can you go?
says Kate, one of your eighth grade math students.
>Well, consider a cube with a volume of 2 cubic units. Its side length is the cube root of 2.
>Isn't it circular reasoning to prove the number exists that way?
asks David.
>We don't know if a cube with that volume is possible until we've found a cube root of 2.
>Let's try another approach. Make a graph of , then draw a line at . The point where the line meets the curve has x coordinate .
>Why is it okay to just plot a few points and then just guess where the curve goes?
asks Kate.
>You would never accept this sort of reasoning in a geometry proof, but when graphing it's apparently okay.
David raises his hand.
>I was wondering how the graphing calculator does it, so I asked my uncle who programs computers for a living. He said that it just plots a large number of points and connects consecutive points with the closest thing to a line it can draw. That's not really a smooth curve, it just looks like one.
Can you show a better example of ? What about other numbers David and Kate will learn about, like ln(2), or erf(1), or the sum of 1/2 to the power of every integer which is the ASCII encoding of a valid halting Brainfuck program? How far can you go?
