Euler Identity in Home

[Manolo Sobrino]

Joe, I don't know what to do when people just don't read the posts in the thread.

As jebem told us, the TIs 86 and 89 work with pi=3.1415926535898, and all the results I quote are for pi=3.1415926535898. When you press Pi it shows a greek Pi, and if you evaluate that you get 3.14159265359, that's because the display mode of floats in those calculators just shows 12 significant digits.

If you evaluate pi-3.14159265359, you get -2E-13. The values those TIs calculate for sin(pi) are accurate and consistent for a 14 significant digit calculator:

Type sin(pi) or

sin(3.1415926535898)

And you get zero

And the values those same TIs calculate for sin(3.14159265359) are accurate as well: type sin(3.14159265359) and... do you know what you get?

sin(3.14159265359) = -2E-13

That's right. And why is that good here? That's good because these machines work with 14 digit precision, they can give you that.

And why sin(3.14159265359)=-2.06761537357E-13 in the OP's calculator is "wrong" (not really wrong, but rounding pending) and it should show 0E12 instead? Because it comes from a machine with 12 digit precision. It shouldn't know that, because in order to know that you need more than 12 significant digits, actually 25 significant digits. That's my whole point. The "awful consequences" are that you believe that as the result shows 12 significant digits in floating point notation it all makes sense, but that's really a joke that shows the absurd of mixing things up.

And if you can work with 25 significant digits (at least), how come a TI can give you: sin(3.14159265359*(1+1E-11))=-3.16E-11, which is accurate and it's all this thing can get, while the OP's calculator gives you -3.0206761537357*10^-11? Because of rounding 3.14159265359*(1+1E-11)->3.14159265362.

How did you calculate Sin(3.14159265359*10^-12)? Maybe the implementation I'm working with is wrong, but I get (not in a TI) just that, and it should be that because in the Taylor expansion of sin(x), the second term for x=3.14159265359*10^-12 is -(1/6)*3.10063*10^-35... give me a break.

DavidM: Just read the posts in the thread. And yeah, Wolfram Alpha gives you that for "Decimal approximation" and 0E-12 for "Result". Or at least that was what it gave me the day I posted it on this thread.

(edit)



And this is what Pi+ArcSin looks like (it should cross the y axis a little bit upwards, but you get the idea, I don't have time to get the perfect ticks right now). If you round Pi to 3.14159265359 and every other number to 12, what does it mean having a value of Sin of -2.06761537357E-13? Well, according to the significant digits you are considering, no more than you just can say that sin(3.14159265359) is 0E-12.

Maybe it's so obvious for me that I'm just incapable of explaining it to you. At least I hope that you understand now why Wolfram Alpha gives you that 0.*10^-12 "Result".