17 May 17
Originally posted by sonhouseThat is all correct.
So one calculator I do i^2 = -1 ok. I do i^3 and it goes 'math error'. But another one (Calctastic app for android) it goes i^3= -i. i^4=1, i^5=i. Is that correct mathematically speaking?
Multipling by i on the complex plane is equivalent a rotation of -90degrees.
17 May 17
2 edits
Originally posted by wormerMy cell app gives me i^7 = -i. and i^8=1. Are those crazy numbers? i^9=i according to
also if you do i^7 or i^8 it will give a crazy number but you must realize that calculators use formulas when calculation this
CalcTastic app for my android I did i^9.9 and got -0.987688340595 and +0.156434465040i
What about that one?
17 May 17
Originally posted by sonhouseIf your calculator accepts complex inputs then the answer should either be correct, should show an error to say it can't do it, or you should throw out your calculator. Once you start questioning its wisdom there is no point having it.
My cell app gives me i^7 = -i. and i^8=1. Are those crazy numbers? i^9=i according to
CalcTastic app for my android I did i^9.9 and got -0.987688340595 and +0.156434465040i
What about that one?
17 May 17
Originally posted by twhiteheadWhich is why I am asking. How can you prove that one, i^9.9 = all that stuff?
If your calculator accepts complex inputs then the answer should either be correct, should show an error to say it can't do it, or you should throw out your calculator. Once you start questioning its wisdom there is no point having it.
17 May 17
1 edit
Originally posted by sonhouseEither with another calculator 🙂
Which is why I am asking. How can you prove that one, i^9.9 = all that stuff?
or log tables.
Have fun.
Actually, I believe its polar coordinates you need:
https://en.wikipedia.org/wiki/De_Moivre%27s_formula#Failure_for_non-integer_powers.2C_and_generalization
17 May 17
1 edit
Originally posted by sonhouseIt's the exponential of the log. Suppose x is a real number, and z = i ^ x, then:
Which is why I am asking. How can you prove that one, i^9.9 = all that stuff?
z = exp(x log(i))
We also have that i = exp(iπ/2), so
z = exp (iπx/2)
But this is only the principal value, the argument of the exponential can have any integer factor of 2πi added to it, so the full expression is:
i^x = exp(iπx(2n + 1/2))
where n is any integer.
17 May 17
2 edits
Originally posted by wormerAnother anamoly in calculators: Doing non-integer factorials.
graphing calculators will give estimations
I did 4.5! and it pumped out 59 and change.
But 4.5! = 4.5*3.5*2.5*1.5*0.5=29.53125
But by that same logic, 4.1*3.1*2.1*1.1*0.1 = 2.93601 Which also is odd. What am I doing wrong?
That seemed odd. On my Casio FX-115ES it goes 'math error' even though you can crank through it manually and come out with 29.53125.
17 May 17
Originally posted by sonhouseThey use something called the Gamma function, which is identical to factorial for the integers but defined for the set of real numbers.
Another anamoly in calculators: Doing non-integer factorials.
I did 4.5! and it pumped out 59 and change.
But 4.5! = 4.5*3.5*2.5*1.5*0.5=29.53125
But by that same logic, 4.1*3.1*2.1*1.1*0.1 = 2.93601 Which also is odd. What am I doing wrong?
That seemed odd. On my Casio FX-115ES it goes 'math error' even though you can crank through it manually and come out with 29.53125.
Gamma(n) = (n - 1)!
Gamma(1/2) = sqrt(π/2)
The second value is from memory and I might have it wrong, try putting 3/2 into your calculator, press the factorial button and then square the result.
It's defined by an integral and discussed at length in its own Wikipedia page.
17 May 17
Originally posted by DeepThoughtSo if I input 5.5 it should give me 4.5! right? But it goes 287.88 and change. I wish I knew just what it was computing. If I do 5! it gives 120.
They use something called the Gamma function, which is identical to factorial for the integers but defined for the set of real numbers.
Gamma(n) = (n - 1)!
Gamma(1/2) = sqrt(π/2)
The second value is from memory and I might have it wrong, try putting 3/2 into your calculator, press the factorial button and then square the result.
It's defined by an integral and discussed at length in its own Wikipedia page.
18 May 17
Originally posted by sonhouseNo, that rule only applies for positive integers.
So if I input 5.5 it should give me 4.5! right?
Imagine you put all the integer factorials on a graph then connected them up with a smooth line.
You would expect 4.5! to be between 5! (120) and 4! (24) but without graphing it, you won't know exactly where. Its not as simple as your subtract 1 calculation.
https://en.wikipedia.org/wiki/Gamma_function
18 May 17
Originally posted by sonhouseYou know that (-1)^2 = 1, (-1)^3 = -1, (-1)^4 = 1, and so on. But you also know that i^2 = -1. Hence i^2 = -1, i^4 = 1, i^6 = -1, and so on. Now take i^2 = -1 and multiply both sides by i, which gives i^3 = -i. Multiply both sides again by i^2 to get i^5 = -i^3 = i. And so on and so forth, giving all the integer powers of i needing to know only that i^2 = -1.
So one calculator I do i^2 = -1 ok. I do i^3 and it goes 'math error'. But another one (Calctastic app for android) it goes i^3= -i. i^4=1, i^5=i. Is that correct mathematically speaking?
It all becomes pretty intuitive when you learn how complex polar coordinates work.