i = [math]\sqrt{-1}[/math]

i[math]^{4}[/math] = [math]\sqrt{-1}[/math][math]^{4}[/math] = 1

[math]\sqrt[4]{1}[/math] = i = 1

 

Therefore [math]\sqrt{-1}[/math] = 1, but that would mean that 1*1 = -1 Which is impossible.

 

How is this? Did I make a miscalculation? Or is this really a paradox?

The sqrt(-1) is i my friend, as you stated in the first part of your question. Your logic is somewhat off base, but it's good that you are thinking. Keep it up.

No paradox. You just did many things that are wrong. You ignored that [math]x^2=1[/math] has two solutions and [math]x^4=1[/math] has four and you assumed [math](ab)^c=a^c\,b^c[/math], which is valid only for real c and positive real a and b.

I don't understand. All I did was take i, put it to the 4th power, and took that number and found its 4th root.

 

Shouldn't that, in the end, equal the same?

I don't understand. All I did was take i, put it to the 4th power, and took that number and found its 4th root.

 

Shouldn't that, in the end, equal the same?

 

No. D H said what you did wrong in your calculation, which is why you got a clearly wrong answer. To start with, there is more than one root. 3^2 is 9, but the square root of 9 is either 3 or -3. Etc.

So? The 4th root of 1 is -1 or 1. Neither would work.

So? The 4th root of 1 is -1 or 1. Neither would work.

 

No, it's either 1, -1, i, or -i.

Oh, I understand now. THANK YOU.

 

Phew, it was bugging me.

 

Let me see if I got this straight... the square root of -1, to the 4th power = 1. BUT taking the fourth root of 1 = 1, -1, i, or -i NOT both. So i doesn't necessarily = 1 or -1

 

What else could, to the fourth power, equal 1 other than 1 and -1? i seems to be both positive AND negative. (Which very well may be the point).

1^4 = 1

-1^4 = 1

i^4 = 1

-i^4 = 1

 

They're all equally fourth roots of one. Similarly,

 

1^2 = 1

-1^2 = 1

 

and

 

i^2 = -1

-i^2 = -1

I now see your second point as well, that even though they have the same 4th power, the 2nd power is variant, and therefore not equal to 1.

 

Thanks again.

Generally, xⁿ=1 for natural n has exactly n sollutions: typically called the Roots of unity.