Main post link -> http://blog.brilliant.org/2013/02/13/complex-numbers/

Read the full explanation on the blog. Feel free to try out these test yourself questions here. If you show your work there is more to discuss.

**Test Yourself**

Determine the 4 roots of the equation \(x^4 + 1 = 0\), and find their real and imaginary parts.

Verify that the norm distributes over multiplication and division. Specifically, show that \(N( z \times w ) = N( z ) \times N( w )\) and that \(N\left( \frac {z}{w} \right) = \frac { N( z )} { N( w )}\). Give examples to show that the norm does NOT distribute over addition and subtraction.

If \(\frac { (1+2i)(2+3i)}{8+i} = a + bi\), what is \(a^2 +b^2 \)? Hint: There is no need to determine the exact values of \( a \) and \( b \).

Determine the square root of \(1-i\).

Note: We can show that the square root of \(z = a + bi\) is equal to \( \pm \left( \sqrt{ \frac {a + \sqrt{a^2+b^2}}{2}} + sgn(b) \sqrt{ \frac {-a + \sqrt{a^2 + b^2}}{2}} \right)\). Currently, our only way to show this is through brute force multiplication. We will be learning how to approach this problem otherwise.

## Comments

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TopNewestComplex roots include real roots, right? – Zi Song Yeoh · 3 years, 11 months ago

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– Harshit Kapur · 3 years, 11 months ago

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