#### Science Forum

sonhouse
Science 16 May '17 16:55
1. sonhouse
Fast and Curious
16 May '17 16:551 edit
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?
2. 16 May '17 17:061 edit
It all looks correct to me. Its interesting as I had never thought about it that way.
3. wolfgang59
evolved
17 May '17 03:09
Originally posted by sonhouse
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?
That is all correct.
Multipling by i on the complex plane is equivalent a rotation of -90degrees.
4. 17 May '17 13:55
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
5. sonhouse
Fast and Curious
17 May '17 14:042 edits
Originally posted by wormer
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
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

6. 17 May '17 14:39
Originally posted by sonhouse
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

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.
7. sonhouse
Fast and Curious
17 May '17 14:50
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.
Which is why I am asking. How can you prove that one, i^9.9 = all that stuff?
8. 17 May '17 15:291 edit
Originally posted by sonhouse
Which is why I am asking. How can you prove that one, i^9.9 = all that stuff?
Either with another calculator ðŸ™‚
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
9. 17 May '17 15:39
graphing calculators will give estimations
10. DeepThought
17 May '17 21:211 edit
Originally posted by sonhouse
Which is why I am asking. How can you prove that one, i^9.9 = all that stuff?
It's the exponential of the log. Suppose x is a real number, and z = i ^ x, then:

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.
11. sonhouse
Fast and Curious
17 May '17 21:362 edits
Originally posted by wormer
graphing calculators will give estimations
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.
12. DeepThought
17 May '17 21:47
Originally posted by sonhouse
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.
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.
13. sonhouse
Fast and Curious
17 May '17 23:32
Originally posted by DeepThought
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.
So 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.
14. 18 May '17 10:45
Originally posted by sonhouse
So if I input 5.5 it should give me 4.5! right?
No, that rule only applies for positive integers.
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
15. 18 May '17 15:41
Originally posted by sonhouse
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?
You 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.

It all becomes pretty intuitive when you learn how complex polar coordinates work.