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**Complex Numbers**

Complex numbers are an extension of real numbers (everyday numbers, including those with decimal figures and square roots). They can be expressed as the sum of a real number and an imaginary number. A typical complex number would be:

z = 3 + 4i

The squares of complex numbers may be positive, negative or complex. For example, in the case above z squared would be:

z · z = (3 + 4i) · (3 + 4i) = 3 · 3 + 3 · 4i + 4i · 3 + 4i · 4i = 9 + 24i – 16 = -7 + 24i

Complex numbers may be identified as numbers in two dimensions, where the real part is the horizontal component and the imaginary part is the vertical one. In order to obtain an idea of the length of the number, squaring it won’t work, since we may not get a positive number. Instead, we multiply by its conjugate, which is the same number with the opposite sign in the imaginary part. Using the example below, z conjugate would be:

z* = 3 – 4i

It is easy to see that z · z* is positive:

z · z* = (3 + 4i) · (3 – 4i) = 3 · 3 – 3 · 4i + 3 · 4i – 4i · 4i = 9 – (-16) = 25

That is why, in Quantum Mechanics, probability amplitudes (which are complex numbers) are multiplied by their conjugates in order to obtain the probability.

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