This is a question in introduction to pure mathematics. I am pretty sure I am close to the answer but I can't quite decide why this proves that there is no complex numbers:
$$|z| = |z + i√5| = 1$$
$$\sqrt{cos²Θ + (isinΘ + i\sqrt5)²} = \sqrt1$$
$$cos²Θ – sin²Θ – 2\sqrt5sinΘ – 5 = 1$$
$$cos²Θ – (1 – cos²Θ) – 2\sqrt5sinΘ – 5 = 1$$
$$2cos²Θ – 2√5sinΘ – 5 = 0$$
I can see are that there is no imaginary component left in the equation and there is no way to equate it to $cos\theta + isin\theta$.
Best Answer
Consider a simple geometric interpretation: if $z$ is a point on the unit circle in the complex plane, and we require $z+i\sqrt{5}$ to also be on this circle, this is obviously impossible since the diameter of the circle is $2$: for any line drawn through the circle, the points of intersection of that line with the circle cannot possibly be more than a distance of $2$ apart, yet $|i \sqrt{5}| = \sqrt{5} > 2$.