Why is i^2 equal to negative one and not positive one?

[srfriggen]

Someone please tell me what is wrong with this logic:

i = √-1

i²= √-1√-1 = √(-1)(-1) = √+1 = 1

But also i² = (√-1)^1/2= -1

 

[mfb]

The square root is not unique any more in the complex numbers. The laws that work for positive real numbers don't work for complex numbers in general.

 

[Mark44]

... and the property of radicals that you used -- $\sqrt{a}\sqrt{b} = \sqrt{ab}$ -- is applicable only if both a and b are nonnegative.

 

[fresh_42]

This is worth reading it in the context:
https://www.physicsforums.com/insights/things-can-go-wrong-complex-numbers/

 

[srfriggen]

Thank you all for the response, much appreciated!

 

[FactChecker]

The symbol, i, was defined as the imaginary number which, when squared, gives -1. That definition should not be violated. It worked out very well with several other things. It allows every polynomial to be factored completely. It provides a good way to represent rotations in the two-dimensional complex plane with multiplication by complex numbers.

 

[micromass]

https://en.wikipedia.org/wiki/Split-complex_number

 

[fresh_42]

https://en.wikipedia.org/wiki/Split-complex_number

How does it come you know all the exotic corners out there? I'm flabbergasted every single time.
But that's not what I wanted to say. I like to take the chance to thank you for your Insight on the subject. I've linked it now for the third or forth time (hoping it will be read). Very useful.