Arithmetic of complex numbers in cartesian form, including conjugates, and the Argand diagram (complex plane).
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Practice exam-style cartesian form problems
The imaginary number i is the square root of −1:
In general, imaginary numbers are of the form bi,b∈R∖{0}. Notice that
Conclusion: the square of any imaginary number is negative.
A complex number
is the sum of a real number a and an imaginary number bi.
We call a the real part of z and b the imaginary part of z:
For example, z=2−3i is a complex number with a real part Re(z)=2 and imaginary part Im(z)=−3.
If two complex numbers are equal, then both their real and imaginary parts are equal:
The expression x+iy is often referred to as the Cartesian form of z.
Real numbers are a subset of complex numbers a+bi where b=0. Imaginary numbers are also a subset of complex numbers with a=0.
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The product of two complex numbers in Cartesian form is
Complex numbers can be visualized in the complex plane, also known as the Argand Diagram.
To plot a complex number, the real part determines the x-coordinate and the imaginary part determines the y-coordinate. Therefore the complex number a+bi has coordinates (a,b) on the plane.
It is conventional to use arrows from the origin to the point (a,b) to represent complex numbers.
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The conjugate of a complex number z is the complex number with the same real component and the opposite imaginary component:
Since the real components of z and z∗ are the same, and the imaginary components are opposite, on the complex plane z∗ is the reflection of z in the x-axis.
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The following properties hold for complex conjugates:
Fractions with complex denominator can be made real using a process analogous to rationalizing the denominator. For a fraction with a complex denominator c+di, we multiply both the numerator and the denominator by the conjugate c−di to get the fraction in a more workable form:
This allows us to split z into its real and imaginary components.
We can solve complex equations involving z and z∗ by using the fact that a complex number z takes the form a+bi. Recall that for complex numbers z1 and z2,
We use this fact to equate the real and imaginary parts of both sides, which creates a solvable a system of two equations in two unknowns from one given equation.
When a quadratic
has
it has no real roots since the square root in
is not a real number. Instead, the square root will give an imaginary number, making the roots complex.