# Iota square?

1. Oct 26, 2008

### mkbh_10

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

2. Oct 26, 2008

### Hurkyl

Staff Emeritus
Huh?

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

3. Oct 26, 2008

### mkbh_10

iota which belongs to the complex number domain

4. Oct 26, 2008

### Hurkyl

Staff Emeritus
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'.

5. Oct 26, 2008

### HallsofIvy

Staff Emeritus
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 realy 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.

Last edited: Oct 26, 2008