Square Root of i

[BigGlenntheHeavy]

\(\displaystyle Is \ the \ expression \ \sqrt i \ defined? \ If \ so, \ find \ a \ possible \ value \ for \ it.\)

\(\displaystyle Hint: \ Euler's \ formula\)

 

[pitoten]

http://en.wikipedia.org/wiki/Imaginary_unit

 

[Mrspi]

\(\displaystyle Is \ the \ expression \ \sqrt i \ defined? \ If \ so, \ find \ a \ possible \ value \ for \ it.\)

\(\displaystyle Hint: \ Euler's \ formula\)

You're kidding, right? There are thousands of websites which explain this, and any student who has been through pre-calculus should be able to explain it as well. And you don't need Euler's formula, either.

let a + bi = sqrt(i)

Square both sides:

(a + bi)[sup:llqnuywt]2[/sup:llqnuywt] = [sqrt(i)][sup:llqnuywt]2[/sup:llqnuywt]

a[sup:llqnuywt]2[/sup:llqnuywt] + 2 a b i + bi[sup:llqnuywt]2[/sup:llqnuywt] = i

a[sup:llqnuywt]2[/sup:llqnuywt] + 2 a b i - b[sup:llqnuywt]2[/sup:llqnuywt] = 0 + 1 i

Equate the real parts, and equate the "imaginary" parts....

a[sup:llqnuywt]2[/sup:llqnuywt] - b[sup:llqnuywt]2[/sup:llqnuywt] = 0

and

2 a b = 1

solve for a and b....