Thursday, December 11, 2014

Unizor - Trigonometry - Representation of Complex Numbers

We strongly advise students to refresh the knowledge of Complex Numbers. The lectures dedicated to these numbers are presented in the Algebra part of this course. Special attention should be dedicated to graphical representation of complex numbers.

Here is the main concept of the graphical representation of complex numbers.
We can always consider a complex number z = a + b·i as a pair of two real numbers (a, b) and each such pair (i.e. each complex number) we can put into a correspondence with a point on a coordinate plane with Cartesian coordinates (a, b).

But every point on a plane can be identified not only by its Cartesian coordinates, but a pair of polar coordinates - the distance from the origin r and a polar angle φ.
The conversion from the polar coordinates into Cartesian is a simple trigonometric identities:
a = r·cos(φ)
b = r·sin(φ)
The conversion from Cartesian coordinates to polar is also fully defined by the following equalities:
r = √(a^2+b^2)
cos(φ) = a/√(a^2+b^2)
sin(φ) = b/√(a^2+b^2)

Hence, each complex number a+b·i can be represented in the polar form as
r·cos(φ)+r·sin(φ)·i = r·[cos(φ)+i·sin(φ)]
In this form the distance from the origin r is called magnitude or modulus, or absolute value of a complex number, while the polar angle φ is called an argument or a phase.

Incidentally, real numbers can be considered as a subset of complex numbers with an imaginary component (that is, coefficient at i) equaled to zero. In polar form it means that the argument equals to 0 for positive real numbers or π for negative real numbers. In both cases the imaginary part equals to 0 since sin(0)=sin(π)=0.

While it's easy to add two complex numbers in their traditional form, their product looks much more complicated.

Here is a sum:
(a1+b1·i) + (a2+b2·i) =
= (a1+a2) + (b1+b2)·i

And here is a product:
(a1+b1·i) · (a2+b2·i) =
= (a1·a2−b1·b2)+(a1·b2+a2·b1)·i
because i^2 = −1

The situation with product is much simpler in the polar form of representation of complex numbers:
r1·[cos(φ1)+i·sin(φ1)] ·
r2·[cos(φ2)+i·sin(φ2)] =
= r1·r2·[cos(φ1)·cos(φ2) +
+ i·cos(φ1)·sin(φ2) +
+ i·sin(φ1)·cos(φ2) +
+ i^2·sin(φ1)·sin(φ2)] =
= r1·r2·{[cos(φ1)·cos(φ2) − sin(φ1)·sin(φ2)] +
+ i·[cos(φ1)·sin(φ2) + sin(φ1)·cos(φ2)]} =
= r1·r2·[cos(φ1+φ2)+sin(φ1+φ2)]
So, the magnitudes are multiplied but arguments are added. The final formula is relatively simple.

There is a very clear geometric meaning of multiplication of complex numbers in polar form.
Consider only complex numbers that have a magnitude of 1, that is all numbers of a form
They all lie on a unit circle around the origin of coordinates.

When one such number with an argument φ1 is multiplied by another with an argument φ2, the result will be a new complex number with a magnitude of 1, still on the same unit circle, and an argument equal to φ1+φ2.
It means that algebraic operation of multiplication is, geometrically, a rotation.
Remarkable, is not it!

If we consider multiplication by a complex number with a magnitude not equal to 1, geometrically, we still deal with a rotation, but also deal with a stretching by a factor equal to a magnitude of a multiplier.
Multiplication by a positive real number is only a stretching since positive real numbers, considered as a subset of complex numbers, have argument equaled to zero in polar form and, therefore, there is no rotation.
Multiplication by a negative real number is a stretching with a change in the direction to an opposite since negative real numbers, considered as a subset of complex numbers, have argument equaled to π in polar form, which, if added, changes the direction to an opposite.

Multiplication by i is a rotation by π/2=90° because, in polar form,
i = 0+1·i = cos(π/2)+i·sin(π/2),
that is, i is a complex number with magnitude 1 and argument π/2 in polar form.
Therefore, multiplication by i is a rotation by an angle π/2.
Incidentally, this is obvious in a traditional representation of complex numbers since
(a+b·i)·i = −b+a·i
and a segment connecting the origin of coordinates and a point (a,b) should be rotated by an angle π/2 around the origin of coordinates to coincide with a segment from the origin of coordinates to a point (−b,a).

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