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Complex Numbers

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

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

  2. 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.

  3. 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 \( b \).

  4. 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.

Note by Peter Taylor
4 years, 7 months ago

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Complex roots include real roots, right? Zi Song Yeoh · 4 years, 7 months ago

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@Zi Song Yeoh Yes Harshit Kapur · 4 years, 7 months ago

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