The user-friendly version of this content is available here.
The following content is copyright (c) 2009-2013 by Goods of the Mind, LLC.
This essential trains for: SAT-II, AMC-10, AMC-12, AIME.
Synthetic definition of complex numbers:
The real numbers x and y are called the real part and the imaginary part of the complex number z.
The imaginary part is a real number - therefore, it is allowed to be equal to zero. In this case, the complex number z is real. It follows that the set of real numbers is a subset of the set of complex numbers.
Since there is a one-to-one correspondence between complex numbers and pairs of real numbers, the Cartesian plane is isomorphic with the set of complex numbers. Therefore, we can associate a point in the plane with any complex number in a unique way. The mapping can be, for example, that of setting the x-coordinate to be equal to the real part and the y-coordinate to be equal to the imaginary part. Such a set of points is called the complex plane or the Argand plane.
The distance between any complex number z and the origin of the Argand plane is called the absolute value of z:
The set of real numbers is the x-axis and the set of purely imaginary numbers is the y-axis. One can see that the intersection of the set of real numbers and the set of imaginary numbers is not empty, but the set that contains the element zero, since:
and, therefore, the intersection of the set of real numbers and the set of purely imaginary numbers is not empty, but the set that contains the element zero.
The angle formed by the segment from the origin to z with the x-axis is called the argument of z:
This angle is called the principal argument if it satisfies:
Equality of two complex numbers happens when their real parts are identical and their imaginary parts are identical. Therefore, an equation with complex numbers:
is, in fact, a system of two equations with real numbers:
Each complex number has a unique algebraic form:
A non-unique trigonometric form can be used, by introducing polar coordinates:
The procedure for obtaining the trigonometric form is:
Since the trigonometric functions are periodic, it is obvious that a rotation of 360° will not change the number z although it will change the argument of the trigonometric form. Therefore, this form is not a unique writing of z.
which is also obvious from the graphical representation:
Operations with complex numbers can be done in either form and must be equivalent. Moreover, they must coincide with regular operations for real numbers whenever the imaginary part is zero (the real numbers are a subset of the complex numbers).
Addition and subtraction: add/subtract the real parts and add/subtract the imaginary parts
Multiplication, division and raising to a power:
Observe how the result must be written as a complex number, i.e. there must be a real part plus an imaginary part.
In trigonometric form:
We can use trigonometric identities for the sum of arcs:
Multiplication of two complex numbers in trigonometric form results in multiplying the absolute values and adding the arguments.
Similarly, division results in dividing the absolute values and subtracting the arguments:
Raising a complex number to an integer power can be done by repeatedly multiplying it with itself:
We obtain de Moivre's formula:
which applies to roots as well, by taking the reciprocal of the exponent:
Conjugation of a complex number is an operation that reflects it with respect to the Re(z) axis. As a result, the imaginary part changes sign. Since the real part is unchanged, this operation does not affect real numbers. For the complex number defined as:
the conjugate complex number is:
and, we have the obvious property:
Why do we need the trig form? Changing the representation of a complex number from algebraic to trigonometric is not always easy without a calculator. Why do we need it?
The trigonometric form is important because de Moivre's formula works only on it. And what de Moivre does, is to help us raise a number to any power much more easily than if we used the algebraic form.
The above property allows us to make denominators real, if they have non-zero imaginary parts:
Moreover, we have been able to write the given fraction in proper algebraic form, by separating the real and imaginary parts.
Since any sum of two squares is zero only if both squares are zero, any complex number that has an absolute value of zero is identical to zero:
then z is a real number. The only complex numbers that are equal to their conjugates are the real numbers. This may be an indirect way for a problem to say 'this number is real'.
The powers of i form a repeating sequence: