Introduction to complex numbers
The complex numbers system
The number i
i is a number such that :
i^2=-1
Or equivalently:
i=\sqrt { -1 }
i is one of only a few numbers that has a name. Other named numbers are \pi and e. It creates the basis for the complex system and gives rise to solutions for polynomial equations.
i is a solution to the quadratic equation x^2=-1.
Complex Numbers
Let a and b be two real numbers. A complex number is written:
z=a+bi
The following numbers are complex numbers :
- z_1=1+2i
- z_2=3i-4
- z_3=5i
z=a+bi is called the algebric form of a complex number.
Real part and imaginary part
Let a and b be two real numbers and z a complex number such that:
z=a+ib
- a is called the real part of z.
- b is called the imaginary part of z.
Let z be the complex number such that z=2+4i :
- The real part of z is 2
- The imaginary part of z is 4
Let z be the complex number such that z=-2i+5 :
- The real part of z is 5
- The imaginary part of z is -2
The number i is never a component of the imaginary part.
- The imaginary part of 5i is 5.
- The imaginary part of 2-3i is -3.
Equality of Complex Numbers
Two complex numbers are equal if and only if the real and imaginary parts are equal.
- 2+8i \text{ is equal to }8i+2 \text{ because the real and imaginary parts are equal. }
- 7+3i \text{ is not equal to } 7-3i \text{ because the imaginary parts are not equal. }
Operations with complex numbers
Let a_1, a_2, b_1 and b_2 be real numbers and let z_1 and z_2 be complex numbers with :
- z_1=a_1+b_1i
- z_2=a_2+b_2i
Then:
z_1+z_2=\left(a_1+b_1i\right)+\left(a_2+b_2i\right)=\left(a_1+a_2\right)+\left(b_1+b_2\right)i
If :
- z_1=5+2i
- z_2=1-3i
z_1+z_2=\left(5+2i\right) +\left(1-3i\right)=\left(5+1\right)+\left(2-3\right)i=6-i
Let a_1, a_2, b_1 and b_2 be real numbers and let z_1 and z_2 be complex numbers with :
- z_1=a_1+b_1i
- z_2=a_2+b_2i
Then:
z_1-z_2=\left(a_1+b_1i\right)-\left(a_2+b_2i\right)=\left(a_1-a_2\right)+\left(b_1-b_2\right)i
If :
- z_1=8+3i
- z_2=1+2i
z_1-z_2=\left(8+3i\right)-\left(1+2i\right)=\left(8-1\right)+\left(3-2\right)i=7+i
Complex Conjugate
Let a and b be two real numbers and z a complex number such that:
z=a+bi
The conjugate of z is the following complex number:
\overline{z}=a-bi
Consider the complex number z=2+5i. The conjugate is :
\overline{z}=2-5i
If z is written as z=ib+a, then the conjugate of z is:
\overline{z}=-ib+a=a-bi
Considering the complex number z=i-3. The conjugate of z is :
\overline{z}=-i-3=-3-i
Let a_1, a_2, b_1 and b_2 be real numbers and let z_1 and z_2 be complex numbers with :
- z_1=a_1+b_1i
- z_2=a_2+b_2i
Then:
z_1.z_2=\left(a + bi\right) \times \left(c + di\right) = \left(ac-bd\right) + i\left(bc + ad\right)
If z_1=2+3i and z_2=3-3i, then:
\begin{aligned}z_1.z_2&=\left(2+3i\right) \times \left(3-3i\right) \\ &= \left(6-\left(-9\right)\right)+\left(-6+9\right)i \\&=15+3i \end{aligned}
Let z_1 and z_2 be two complex numbers such that:
- z_1=a+bi
- z_2=c+di
a, b, c, and d being four real numbers with z_2\neq 0.
The division of z1 by z2 is given by :
\begin{aligned}\dfrac{z_1}{z_2} &= \dfrac {\left(a + bi\right)} {\left(c + di\right)} = \dfrac {\left(a + bi\right)} {\left(c + di\right)} \times \dfrac {\left(c - di\right)} {\left(c - di\right)}= \dfrac {\left(ac + bd\right) + \left(bc-ad\right) i} {\left(c ^ 2 + d ^ 2\right)} \end{aligned}
If :
- z_1=1+2i
- z_2=2+3i
\begin{aligned}\dfrac{z_1}{z_2}&= \dfrac{\left(1 + 2i\right)\left(2-3i\right)}{\left(2 + 3i\right)\left(2-3i\right)}\\&= \dfrac {\left(2 + 6\right) + \left(4-3\right) i} {2 ^ 2 + 3 ^ 2} \\&= \dfrac {8 + i} {13} \\&= \dfrac{8}{13}+\dfrac{i}{13}\end{aligned}
Graphic représentation of complex numbers
Complex numbers can be represented on a two-dimensional graph:
- The horizontal axis represents the real part.
- The vertical axis represents the imaginary part.
Consider the following complex numbers:
- z_1=-1+4i
- z_2=3+2i
z_1 and z_2 can be represented by A_1\left(-1{,}4\right) and A_2\left(3{,}2\right) :
Let a,b be real numbers and consider the complex number z=a+bi. The reflection of z across the real axis is where the complex conjugate \overline{z}=a-bi is located on the complex plane.
The following graphic contains the location of 2+3i and its complex conjugate 2-3i.
Observe that 2-3i is the reflection of the point 2+3i across the real axis.
Complex numbers and equations
Discriminant
The discriminant of the quadratic equation ax^2+bx+c=0 is :
b^2-4ac
Consider the equation:
x^2-5x+2=0
The discriminant is:
\left(-5\right)^2-4\left(1\right)\left(2\right)=17
Quadratic Formula
Let a, b, and c be three real numbers. Given a quadratic equation of the form ax^2+bx+c=0 , the solutions are given by:
x=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}
Consider the equation 2x^2-5x+4=0. The solutions are given by:
x=\dfrac {- \left(- 5\right) \pm \sqrt {\left(-5\right) ^ 2-4 \left(4\right)}} {2 \left(4\right)} \\\\=\dfrac {- \left(- 5\right) \pm \sqrt {25-16}} {2 \left(4\right)} \\ = \dfrac {5 \pm \sqrt {9}} { 8} \\ = \dfrac {5 \pm 3} 8
Therefore, the equation has two solutions :
- x_1=\dfrac{5-3}{8}=\dfrac{2}{8}=\dfrac{1}{4}
- x_2=\dfrac{5+3}{8}=\dfrac{8}{8}=1
In the complex system, square roots of negative numbers can be found. Hence, quadratic equations with negative discriminant will also have solutions.
Consider the equation :
2x^2+x+1=0
The square root of the discriminant is :
\sqrt{1^2-4\left(1\right)\left(1\right)}=\sqrt{-3}=\sqrt{3}i
The quadratic formula provides the solution to a quadratic equation of the form ax^2+bx+c=0 :
- A positive discriminant will provide two real solutions to the given equation.
- A discriminant will be zero when there is only one real solution.
- A negative discriminant will provide two complex solutions that are conjugates of each other (the imaginary part will be non-zero).
Consider the equation x^2+4x+5=0. The discriminant is equal to:
4^2-4\left(1\right)\left(5\right)=16-20=-4.
The discriminant is negative and therefore the quadratic has two complex roots :
- z_1=\dfrac{-4+\sqrt{-4}}{2}=\dfrac{-4+2i}{2}=-2+i
- z_2=\dfrac{-4-\sqrt{-4}}{2}=\dfrac{-4-2i}{2}=-2-i
Conjugate Theorem
If a polynomial with real coefficients has a root of the form z=a+bi, then its complex conjugate \overline{z}=a-bi is also a root of the polynomial.
Consider the following equation:
2x^2-5x+7=0
Assume that \dfrac {5 + i \sqrt {31}} {4} is a solution to the equation. Then the other solution is:
\overline{\dfrac {5 + i \sqrt {31}} {4}}=\dfrac {5 - i \sqrt {31}} {4}
Fundamental Theorem of Algebra
Every polynomial having real coefficients and a degree greater than or equal to one has at least one complex root.
2x+1=0 has a root when x=\dfrac {-1} {2}.
x^2+2x+1=0 has a root when x=-1.
2x^2-5x+7=0 has roots when x=\dfrac {5 \pm i \sqrt {31}} {4}.
Trigonometric form, exponential form and polar form of complex numbers
A complex number is determined by its magnitude and the angle that the complex number makes with the positive real axis. Therefore, in addition to the algebraic form, a given complex number can be written in forms that use those two elements.
Consider the complex number 2+2i.
The magnitude of 2+2i is \sqrt{2^2+2^2}=2\sqrt{2}, and the angle that 2+2i makes with the positive real axis is \theta=\dfrac{\pi}{4}.
Trigonometric form (and exponential form) of a complex number
Absolute value of z
Let z=a+bi. \left| z \right| is called the absolute value of z and equals:
|z|=\sqrt{a^2+b^2}
|4-3i|=\sqrt{4^2+3^2}=\sqrt{25}=5
|2+3i|=\sqrt{2^2+3^2}=\sqrt{13}
Let z be a complex number. Then:
|z|=\sqrt{z\cdot \bar{z}}
Consider the complex number 2+i. Then:
|2+i|=\sqrt{\left(2+i\right)\left(2-i\right)}=\sqrt{4-2i+2i+1}=\sqrt{5}
Trigonometric form of a complex number
Suppose a,b are real numbers and z=a+bi. Let \theta be the angle which z makes with the real axis as in the following graphic:
The trigonometric form of z is:
z=|z|\left(\cos\left(\theta\right)+i\sin\left(\theta\right)\right)
Consider the complex number z=2+2i. Then:
- |z|=\sqrt{2^2+2^2}=2\sqrt{2}
- The angle z makes with the real axis is \dfrac{\pi}{4} radians.
Therefore:
2+2i=|2+2i|\left(\cos\left(\dfrac{\pi}{4}\right)+\sin\left(\dfrac{\pi}{4}\right)\right)
2+2i=\sqrt{2}\left(\cos\left(\dfrac{\pi}{4}\right)+\sin\left(\dfrac{\pi}{4}\right)\right)
Exponential form of a complex number
Let a,b be real numbers and z=a+bi. Let \theta be the angle that z makes with the real axis as in the following graphic:
The exponential form of z is:
z=|z|e^{i\theta}
Consider the complex number z=-1. Then:
- |z|=|-1|=1
- The angle z=-1 with the positive real axis is \pi.
Therefore:
z=e^{i\pi}
Polar coordinates and polar form
Polar form
Let z be a complex number such that \theta is the angle that z makes with the positive real axis as in the following graphic:
The polar form of z is:
\left(|z|,\theta\right)
The numbers |z| and \theta are the polar coordinates of z.
Consider the complex number z=3-3i. Then:
- |z|=\sqrt{3^2+\left(-3\right)^2}=3\sqrt{2}.
- z makes an angle of \dfrac{7\pi}{4} with the positive real axis.
Therefore, the polar form of z is \left(3\sqrt{2},\dfrac{7\pi}{4}\right).