# Qubit, one or two complex numbers?

I'm currently reading up on quantum computing and it seems like I have found some contradiction about how to represent qubits.

It is often stated that a qubit is represented as $a|0\rangle + b|1\rangle = (a, b)$ with both a and b being complex numbers.

However, it is stated just as often, that there is only one complex number needed, namely b, since to ignore the global phase shift means, that a becomes real while b stays complex. See for example: this and this.

What is it now? What don't I get here?

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In the representation $|\psi\rangle = a|0\rangle + b|1\rangle$ we must have $|a|^2+|b|^2=1$, so that gives us one constraint. The second is that an overall global phase doesn't make any difference. We can use these two freedoms to chose $a$ and $b$ in a specific way. Traditionally we choose them such that $$|\psi\rangle = \cos \theta|0\rangle+\exp(i\phi)\sin \theta|1\rangle$$