I am reading Hidden In Plain Sight 10: How To Program A Quantum Computer (https://www.amazon.com/Hidden-Plain-Sight-10-Computer-ebook/dp/B07GPRBYVC).
There is something I don't understand.
In chapter 5, the author presents the projection of a complex vector on a coordinate system in Hilbert space (I hope I expressed it correctly...).
As I understand, the author claims that the vector 0.707 + 0.707i has a magnitude of 1.0, because, supposedly, 0.707^2 + (0.707i)^2 = 1.0.
But, 0.707^2 + (0.707i)^2 equals 0 (according to my understanding, and according to https://www.google.com/search?q=0.707%5E2%2B%280.707i%29%5E2).
So there must be something I misunderstand in the explanation. The author's explanation includes a diagram, so I'm adding it here as an image:
What is my mistake?
A complex number $a + bi$ has a real part that is $a$. $a$ is a real number. And a complex number has an imaginary part that is $b$. The imaginary part is also a real number. It does not include the $i$. It is the "coefficient" of the $i$.
When he says we square the real part and square the imaginary part and add them together you are squaring $a$ and $b$ and getting $a^2 + b^2$. You are not squaring $a$ and $bi$.
So in this case $|0.707 + 0.707i| = \sqrt{ 0.707^2 +0.707^2} =\sqrt{\frac 12 + \frac 12} = 1$.