Iota Squared: Mathematical Proof and Physical Meaning

[mkbh_10]

i want the mathematical proof of iota squre = -1 , also what does it mean physically .

 

[Hurkyl]

i want the mathematical proof of iota squre = -1 , also what does it mean physically .

Huh?

Well, what do you mean by "iota"? And what physical meaning are you ascribing to "iota"?

 

[mkbh_10]

iota which belongs to the complex number domain

 

[Hurkyl]

The imaginary unit is usually written as an 'i' or a 'j' -- not by the Greek letter iota, which is why I was confused.

That $i^2 = -1$ is just a basic arithmetic fact, on par with $1 + 1 = 2$.

The 'physical meaning' of i, as with any number, depends on how you use it. e.g. "2" doesn't have any 'physical meaning', although "2 apples", "2 volts" and "2 meters" all have (very different) 'physical meanings'.

 

[HallsofIvy]

In Saying that "i²= -1 is just a basic arithmetic fact", Hurkyl means, basically, that this is how "i" is defined.

But here is a little deeper way of looking at it. If we define the "complex" numbers to be the set of pairs of real numbers, (a, b) with addition and multiplication defined by (a, b)+ (c, d)= (a+ b, c+ d), (a, b)(c, d)= (ac- bd, ad+ bc) then it can be shown that
1) This system forms a "field"
2) The subset of all pairs of the form (a, 0) is a subfield and is isomorphic to the field of real numbers through the isomorphism (a, 0)---> a so we can "label" the pair (a, 0) simply by "a".
3) If we label the pair (0, 1) by "i" then (a, b)= (a, 0)+ (b, 0)(0,1)= a+ bi.
4) so "i²" means (0, 1)(0, 1)= (0(0)- 1(1), 0(1)+ 1(0))= (-1, 0)= -1.

The avoids the problem that saying "i²= -1" doesn't really define i since there are two complex numbers with that property.

And, as Hurkyl said, NO number has any intrinsic "physical meaning". It depends upon how you use them in a specific application.