Tuesday, 7 June 2016

Complex numbers

complex number is a number that can be expressed in the form a + bi, where a and bare real numbers and i is the imaginary unit, that satisfies the equation i2 = −1.In this expression, a is the real part and b is the imaginarypart of the complex number.

  Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point (a, b) in the complex plane. A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the complex numbers contain the ordinary real numbers while extending them in order to solve problems that cannot be solved with real numbers alone.
As well as their use within mathematics, complex numbers have practical applications in many fields, including physics, chemistry, biology, economics,electrical engineering, and statistics. The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers. He called them "fictitious" during his attempts to find solutions to cubic equations in the 16th century.

Complex numbers allow solutions to certain equations that have no solutions in real numbers. For example, the equation
has no real solution, since the square of a real number cannot be negative. Complex numbers provide a solution to this problem. The idea is to extend the real numbers with the imaginary unit i where i2 = −1, so that solutions to equations like the preceding one can be found. In this case the solutions are−3 + 3i and −3 − 3i, as can be verified using the fact that i2 = −1:
According to the fundamental theorem of algebra, all polynomial equational with real or complex coefficients in a single variable have a solution in complex numbers.

A complex number is a number of the form abi, where a and b are real numbers and i is the imaginary unit, satisfying i2 = −1. For example, −3.5 + 2i is a complex number.
The real number a is called the real part of the complex number a + bi; the real numberb is called the imaginary part of a + bi. By this convention the imaginary part does not include the imaginary unit: hence b, not bi, is the imaginary part.The real part of a complex number z is denoted by Re(z) orℜ(z); the imaginary part of a complex number z is denoted by Im(z) or ℑ(z). For example,
Hence, in terms of its real and imaginary parts, a complex number z is equal to . This expression is sometimes known as the Cartesian form of z.
A real number a can be regarded as a complex number a + 0i whose imaginary part is 0. A purely imaginary number bi is a complex number 0 + bi whose real part is zero. It is common to write a for a + 0i and bifor 0 + bi. Moreover, when the imaginary part is negative, it is common to write a − bi withb > 0 instead of a + (−b)i, for example 3 − 4iinstead of 3 + (−4)i.
The set of all complex numbers is denoted by 
To add two complex numbers we add each part separately:
(a+bi) + (c+di) = (a+c) + (b+d)i
Example: (3 + 2i) + (1 + 7i) = (4 + 9i)


To multiply complex numbers:
Each part of the first complex number gets multiplied by
each part of the second complex number
Just use "FOIL", which stands for "Firsts, Outers,Inners, Lasts"

  • Firsts:a × c
  • Outers:a × di
  • Inners:bi × c
  • Lasts:bi × di
(a+bi)(c+di) = ac + adi + bci + bdi2
Like this:

Example: (3 + 2i)(1 + 7i)

(3 + 2i)(1 + 7i)= 3×1 + 3×7i + 2i×1+ 2i×7i
= 3 + 21i + 2i + 14i2
= 3 + 21i + 2i − 14(because i2 = −1)
= −11 + 23i

But There is a Quicker Way!

Use this rule:
(a+bi)(c+di) = (ac−bd) + (ad+bc)i
Example: (3 + 2i)(1 + 7i) = (3×1 − 2×7) + (3×7 + 2×1)i = −11 + 23i

1. Add: (7 + 5i) + (8 - 3i)

2. Add: (2 + 3i) + (-8 - 6i)

3. Express the sum of and
in the form .

4. Add and .

Example: i2

i can also be written with a real and imaginary part as 0 + i

Complex Plane

We can also put complex numbers on a Complex Plane.
  • The Real part goes left-right
  • The Imaginary part goes up-down
i2 = (0 + i)2= (0 + i)(0 + i)
= (0×0 − 1×1) + (0×1 + 1×0)i
= −1 + 0i

Prove that tex2html_wrap_inline287 for any integer n.
Corollary 1.2.15: DeMoivre's Formula
For any integer n and any real number twe have
(cos(t) + i sin(t))n = cos(nt) + i sin(nt)
DeMoivre's Formula is quite something. It says that if you take a number on the unit circle (i.e. with lenght 1) with initial argument (angle) t and multiply it by itself, it simply rotates around the unit circle by that angle t. Each time you multiply the number by itself, the vector rotates another tdegrees. In other words, in this case the power operator results in a simple rotation.

Powers of a vector z with |z|=1
Two interesting questions related to this rotation, taken from the field of Complex Dynamics, are: suppose z is a complex number with |z|=1. Then:
  • find conditions for Arg(z) such that zn = z for some integer n. Such a point, incidentally, is called periodic of order n.
  • if Arg(z)/ is irrational, what can you say about the sequence {z, z2, z3, z4, ...}? Does it, for example, converge? Such a sequence, incidentally, is called the orbit of z.
Polar coordinates can be especially helpful for finding roots, in particular for complex numbers of lenght 1.
Proposition 1.2.16: Finding Roots
For any positive integer n and any non-zero complex number a = r cis(t) the equation zn = a has exactly n distinct roots given by:
z =
where k = 0, 1, 2, ... n-1.
In a previous example we found the two square roots of i, which turned out to be a fair amount of work. The above proposition allows us to dispense with such a question quickly. For example, the two solutions for
z2 = i = cis(/2)
z1 = cis(/2/2) = cis(/4)
z2 = cis((/2 + 2)/2) = cis(5/4)
which you can quickly check using DeMoivre's Formula. Here is a geometric interpretation of this proposition: let's find, for example, the three third-roots of i, i.e. we want to find all solutions toz3 = i.
1: Draw the vector i2: Divide angle by 3 for first root3: Draw 3 equally spaced segments,
starting at the first root
Note, in particular, that the third root of i turned out to be -i, which indeed checks out:
(-i)(-i)(-i) = i*i*(-i) = (-1)*(-i) = i
This proposition is very satisfying: it says that at least every simple polynomial equation of degreen has n solutions. Later we will see that this is true in general: every n-th degree polynomial hasn roots, no "if's" and "but's". This, in fact, is a sign of things to come: many theorems in complex analysis will turn out to be very "satisfying" and nicely structured, which is one reason that the study of complex analysis is a lot of fun (I think -:). But first a few more 'profane' examples.
Example 1.2.17: Finding roots geometrically
  • Find the cube roots of 8 and iand draw them.
  • Find all 4 fourth-roots of -1 and draw them geometrically
  • Find all 5 fifth-roots of 1 and draw them geometrically
  • Find both square-roots of 3i-2 by (a) using polar coordinates and (b) using rectangular coordinates and aformula from the previous section. Confirm that both methods result in the same answers.
Let's conclude this chapter with a result that illustrates what a 'nicely structured' theorem in complex analysis can look like.
Proposition 1.2.18: Roots of Unity
The n n-th roots of unity are given bywnk, where k = 0, 1, 2, ... n-1 and
wn = cis(2/n)
They form the vertices of a regular polygon and add up to zero, i.e. they satisfy the equation:
1 + wn + wn2 + ... + wnn-1 = 0
This is neat: not only does the equation zn = 1have exactly n solutions (one of which is, of course, z=1), but the solutions have this really pretty geometric structure of forming a regular polygon, which implies algebraically that they add up to zero as vectors. Here are, for example, the eight roots of z8=1:
You can see their regular structure. When you add them all as vectors, you indeed get the zero vector.

(.Ref from Wikipedia and Mathisfun private.Shu.edu)