# Multiply Complex number to form a real value

In python, I have seen the following:

result = np.conjugate(result) * result


Which I presume conjugates the complex number and multiplies by the result (of the conjugate) and thus forms a real value from this.

I want to replicate this, but, I am stuck on the formula involved:

z = (-0.0106392, -0.0106392i)


Now if I conjugate these, for example, I use the following formula:

(-1 * -0.0106392)


The complex number now becomes: - z = (-0.0106392, 0.0106392i)

Now I would like to multiply these two numbers, in order to form a real value.

Could someone please give me an example? I know I could use FOIL.

Thanks

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Are you wondering about the identity $(a+bi)(a-bi)=a^2+b^2$? – Lubin Jan 18 '14 at 18:53
@Lubin Hey! Just the calculation to transform the complex number into a an actual value through the use of multiplication. In the "python" code, they conjugate the complex number, and then multiply to form an actual (double) value, rather than a complex one – Phorce Jan 18 '14 at 19:02
Right. That’s how the identity arises. Throw “FOIL” away and just multiply out. – Lubin Jan 18 '14 at 21:28

The identity Lubin gave above, which multiplies a complex number $z=a+bi$ (where $a$ and $b$ are real numbers) by its conjugate, $\bar{z}=a-bi$, yields a real number, $a^2 + b^2$, the square root of which is equal to $z$'s distance from the origin when we plot $(a,b)$ in the plane. (This is called the absolute value or norm or modulus of $z$.)
Alternatively, if you were interested only in the real part (that is, $a$), in Python you can use the code var_name_here.real(). Note also that the imaginary part, var_name_here.imag(), is a real number. These two numbers can be viewed as the legs on a right triangle, the hypotenuse of which has length equal to the absolute value. The legs of the triangle, then, are how far over (the real part) or up (the imaginary) $z$ is from the origin.
As for an example, Lubin's comment includes one. Just remember that $i^2=-1$, and you should be able to fill in the intermediate steps.