5. Modulus

L3CALC

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‘Modulus’ (plural ‘moduli’) is the distance of the point $z(x+iy)$ from the origin of the argand diagram.

Modulus is written as $|z|$. Absolute value is added, since the negative/positive signs don’t really matter, since it’s just the distance from the origin.

To calculate distance, we can use the distance formula from L2 CoordGeo:

$DISTANCE=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$

But since we don’t have $x_2$, we just use the following Pythagorean theorem:

$|z|=\sqrt{x^2+y^2}$

For instance, for the complex number $z=2+4i$, the modulus $|z|$ can be calculated by the following:

$|z|=\sqrt{2^2+4^2}\approx 4.472135955$

Again, this represents the distance of the line (from origin to complex number) from the origin.

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We also know that:

$|z|=|\overline{z}|$

Why? Because that the modulus is just the distance from the origin. As mentioned in Note 4, the conjugate of $z$ ($\overline{z}$), is just mirrored by the real axis.

Since it is just mirrored by the real axis, it will have the same modulus.