Lesson Complex plane

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This Lesson (Complex plane) was created by by ikleyn(4) About Me : View Source, Show
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Complex plane


You have learned that real numbers may be represented in the straight number line (see, for example, the lesson "What is number line" in this site).

Number line is a graphical representation of the set of real numbers. It is a straight line (see Figure 1 below), with a point zero marked, where positive numbers are located to the right of zero and negative numbers to the left of zero. In Figure 1 numbers 4.5, 7.2 and -4 are depicted by the red, green and blue points respectively. The location of a number is defined by its distance from zero, which is equal to the value of the number. Each real number is depicted by some point in the number line; each point in the number line represents some real number.

number_line%28+600%2C+-10%2C+10%2C+4.5%2C+7.2%2C+-4+%29
Figure 1. Number Line

Complex plane in rectangular coordinates


Complex numbers may be depicted in the number plane. To do this, let us consider the plane with rectangular coordinate system (see Figure 2) with the same scale in both axes. The complex number a%2Bb%2Ared%28i%29 is depicted by the point (x,y) in the plane with abscissa (x-coordinate) equal to the real part of the complex number x=a and ordinate (y-coordinate) equal to the imaginary part of the complex number y=b.

This number plane is also called the complex plane.
In this way, every complex number uniquely corresponds to the point in the complex plane, and, in opposite, every point in the complex plane uniquely represents some complex number.

Examples
In Figure 2 point A with abscissa x=3 and ordinate y=2 represents the complex number 3%2B2%2Ared%28i%29.
Point B with abscissa x=-2 and ordinate y=4 represents the complex number -2%2B4%2Ared%28i%29.

Real numbers (in complex form they look like a%2B0%2Ared%28i%29) are depicted by points on the X-axis.
Pure imaginary numbers (in complex form they look like 0%2Bb%2Ared%28i%29) are depicted by points on the Y-axis.

drawing%28+300%2C+250%2C+-5%2C+5%2C+-5%2C+5%2C+%0D%0A++++++++++++grid%281%29%2C%0D%0A%0D%0A++++++++++++circle%283%2C+2%2C+0.1%29%2C%0D%0A++++++++++++locate%283%2C+2%2C+A+%29%2C%0D%0A++++++++++++line++%280%2C+0%2C+3%2C+2%29%2C%0D%0A%0D%0A++++++++++++circle%28-2%2C+++4%2C+0.1%29%2C%0D%0A++++++++++++locate%28-2.3%2C+4%2C+B+%29%2C%0D%0A++++++++++++line++%28+0%2C+++0%2C+-2%2C+4%29%2C%0D%0A%0D%0A++++++++++++circle%284%2C+0%2C+++0.1%29%2C%0D%0A++++++++++++locate%284%2C+0.7%2C+C+%29%2C%0D%0A++++++++++++line++%280%2C+0%2C+++4%2C+0%29%2C%0D%0A%0D%0A++++++++++++circle%28-3.0%2C+0%2C+++0.1%29%2C%0D%0A++++++++++++locate%28-3.3%2C+0.7%2C+D+%29%2C%0D%0A++++++++++++line++%280%2C++++0%2C+++-3%2C+0%29%2C%0D%0A%0D%0A++++++++++++circle%280%2C+++3%2C+0.1%29%2C%0D%0A++++++++++++locate%280.1%2C+3%2C+E+%29%2C%0D%0A++++++++++++line++%280%2C+++0%2C+3%2C+0%29%2C%0D%0A%0D%0A++++++++++++circle%280%2C+++-3%2C+0.1%29%2C%0D%0A++++++++++++locate%280.1%2C+-3%2C+F+%29%2C%0D%0A++++++++++++line++%280%2C++++0%2C+-3%2C+0%29%2C%0D%0A%0D%0A++++++++++++circle%28-2%2C+++-4%2C+0.1%29%2C%0D%0A++++++++++++locate%28-2.3%2C+-4%2C+G+%29%2C%0D%0A++++++++++++line++%28+0%2C+++0%2C+-2%2C+-4%29%0D%0A%29
Figure 2. Complex Plane

Examples
In Figure 2 point C with abscissa x=4 and ordinate y=0 represents the complex number 4%2B0%2Ared%28i%29, or, what is the same, the real number 4.
Point D with abscissa x=-3 and ordinate y=0 represents the complex number -3%2B0%2Ared%28i%29, or, what is the same, the real number -3.
Point E with abscissa x=0 and ordinate y=3 represents the complex number 0%2B3%2Ared%28i%29, or, what is the same, the pure imaginary number 3%2Ared%28i%29.
Point F with abscissa x=0 and ordinate y=-3 represents the complex number 0-3%2Ared%28i%29, or, what is the same, the pure imaginary number -3%2Ared%28i%29.

Conjugate complex numbers are depicted in the complex plane as a pair of points symmetric to the abscissa axis. For example, points B and G in Figure 2 represent conjugate complex numbers -2%2B4%2Ared%28i%29 and -2-4%2Ared%28i%29. Points E and F also represent conjugate complex numbers 3%2Ared%28i%29 and -3%2Ared%28i%29.

Complex numbers can also be displayed by line segments (vectors) issued from the system coordinate origin (0,0) and terminating at the appropriate point of the complex plane. In Figure 2 such vectors are portrayed for the complex number 3%2B2%2Ared%28i%29 (point A), complex number -2%2B4%2Ared%28i%29 (point B), and complex number -2-4%2Ared%28i%29 (point G).

Both of two ways for representing complex numbers in the complex plane (depicting points and displaying vectors) are applicable. Let me remind you that vectors are the straight line segments having the length and the direction. The later means that the starting point and the ending point are defined for vectors.

Complex plane in coordinates "radius - polar angle"


In this lesson above we discussed a coordinate presentation of complex numbers in the complex plane with the rectangular coordinate system. Now we are going to introduce another presentation of complex numbers in the complex plane - so called trigonometric presentation.

The modulus of the complex number a%2Bb%2Ared%28i%29 is the length of the vector corresponding to this complex number. The modulus of the complex number a%2Bb%2Ared%28i%29 denotes as abs%28a%2Bbi%29 and as r. From the drawing in the Figure 3 we have
abs%28a%2Bbi%29+=+r+=+sqrt%28a%5E2%2Bb%5E2%29.
The modulus of the real number is the same as its absolute value.

Examples
The modulus of the complex number 3%2B4%2Ared%28i%29 is equal to r=sqrt%283%5E2%2B4%5E2%29=sqrt%2825%29=5.
The modulus of the complex number 1%2Bred%28i%29 is equal to r=sqrt%281%5E2%2B1%5E2%29=sqrt%282%29=1.41 (approximately).
The modulus of the complex (actually, real) number 5%2B0%2Ared%28i%29 is equal to r=sqrt%285%5E2%2B0%5E2%29=sqrt%2825%29=5.
The modulus of the complex (actually, imaginary) number 0%2B3%2Ared%28i%29 is equal to r=sqrt%280%5E2%2B3%5E2%29=sqrt%289%29=3.

drawing%28+210%2C+175%2C+-2%2C+5%2C+-2%2C+5%2C+%0D%0A++++++++++++grid%281%29%2C%0D%0A%0D%0A++++++++++++circle%283%2C+4%2C+0.1%29%2C%0D%0A++++++++++++locate%283.1%2C+4%2C+A=a%2Bbi+%29%2C%0D%0A++++++++++++line++%280%2C+0%2C+3%2C+4%29%2C%0D%0A++++++++++++line++%280%2C+0%2C+3%2C+0%29%2C%0D%0A++++++++++++line++%283%2C+0%2C+3%2C+4%29%2C%0D%0A++++++++++++locate+%281.5%2C+0.7%2C+a%29%2C%0D%0A++++++++++++locate+%283.1%2C+2.2%2C+b%29%2C%0D%0A++++++++++++locate+%281.5%2C+2.9%2C+r%29%2C%0D%0A++++++++++++arc%280%2C+0%2C+2%2C+2%2C+300%2C+360%29%0D%0A%29
Figure 3. The modulus and
the argument of the complex number

The angle alpha between the axis of abscissas and the vector depicting the complex number a%2Bb%2Ared%28i%29 is called an argument of the complex number a%2Bb%2Ared%28i%29.
In Figure 3 the vector OA is pointing to the complex number 3%2B4%2Ared%28i%29. The angle XOA is the argument of the complex number 3%2B4%2Ared%28i%29.

For the complex number 0 (zero) an argument is not defined.
For any non-zero complex number the argument has infinitely many values. These values differ by an integral number of complete rotations (i.e., by 360°*k, where k is any integer). Thus, for example, the argument of the complex number 1%2Bred%28i%29 is 45°, 45°+360°=405°, 45°-360°=-315°, and, generally, 45°+/-360°*k.

If two complex numbers are equal, then they have the same modulus, but their argument values may differ by an integral number of complete rotations (i.e., by 360°*k, where k is any integer).
In opposite, if two complex numbers have the same modulus and their argument values differ by an integral number of complete rotations (i.e., by 360°*k, where k is any integer), then these complex numbers are equal.

The argument of the complex number a%2Bb%2Ared%28i%29 is expressed via the coordinates a and b by the following formulas:
tg%28alpha%29+=+b%2Fa, sin%28alpha%29+=+b%2Fsqrt%28a%5E2%2Bb%5E2%29, cos%28alpha%29+=+a%2Fsqrt%28a%5E2%2Bb%5E2%29.
Note that each formula alone is not enough to determine (to restore) the argument in unique way, even in the range from 0° to 360°. In order to uniquely identify the argument alpha in this range, you have to take into account the quadrant in the complex plane where the given complex number is located. You can do it using values of coordinates a and b.

Examples

Find the argument of the complex number 1%2Bred%28i%29.
tg%28alpha%29+=+1%2F1+=+1, arctg%281%29 = 45°. Since 1%2Bred%28i%29 belongs to the 1-st quadrant, the argument alpha is equal to 45° + k*360°, k is any integer.

Find the argument of the complex number -1%2Bred%28i%29.
tg%28alpha%29+=+1%2F%28-1%29+=+-1, arctg%28-1%29 = -45°. Since -1%2Bred%28i%29 belongs to the 2-nd quadrant, the argument alpha is equal to -45° + 180° + k*360°, k is any integer.

Find the argument of the complex number -1-red%28i%29.
tg%28alpha%29+=+-1%2F%28-1%29+=+1, arctg%281%29 = 45°. Since -1-red%28i%29 belongs to the 3-rd quadrant, the argument alpha is equal to 45° + 180° + k*360°, k is any integer.

Find the argument of the complex number 1-red%28i%29.
tg%28alpha%29+=+-1%2F1+=+-1, arctg%28-1%29 = -45°. Since 1-red%28i%29 belongs to the 4-th quadrant, the argument alpha is equal to -45° + k*360°, k is any integer.

Trigonometric presentation of complex numbers


Abscissa a and ordinate b of the complex number a%2Bb%2Ared%28i%29 can be expressed in terms of the modulus r and the argument alpha by the formulas
a+=+r%2Acos%28alpha%29, b+=+r%2Asin%28alpha%29.
Therefore, every complex number can be represent in the form r%2A%28cos%28alpha%29%2Bi%2Asin%28alpha%29%29, where r%3E=0.
This expression is called the trigonometric form of the complex number.

Examples

Represent complex number -1-red%28i%29 in the trigonometric form.
We have 1%2Bi+=+sqrt%28%28-1%29%5E2%2B%28-1%29%5E2%29%2A%28cos%285%2Api%2F4%29%2Bi%2Asin%285%2Api%2F4%29%29+=+sqrt%282%29%2A%28cos%285%2Api%2F4%29%2Bi%2Asin%285%2Api%2F4%29%29.

The trigonometric form of the complex number 3%2B0%2Ared%28i%29 is 3%2A%28cos%280%29%2Bi%2Asin%280%29%29.

The trigonometric form of the complex number -3%2B0%2Ared%28i%29 is 3%2A%28cos%28pi%29%2Bi%2Asin%28pi%29%29.

The trigonometric form of the complex number 0%2B3%2Ared%28i%29 is 3%2A%28cos%28pi%2F2%29%2Bi%2Asin%28pi%2F2%29%29.

The trigonometric form of the complex number 0-3%2Ared%28i%29 is 3%2A%28cos%283%2Api%2F4%29%2Bi%2Asin%283%2Api%2F4%29%29.



For your convenience, below is the list of my relevant lessons on complex numbers in this site in the logical order. They all are under the current topic Complex numbers in the section Algebra II.
Complex numbers and arithmetical operations over them
Complex plane                                                                                                    (this lesson)
Addition and subtraction of complex numbers in complex plane
Multiplication and division of complex numbers in complex plane
Raising a complex number to an integer power
How to take a root of a complex number
Solution of the quadratic equation with real coefficients on complex domain
How to take a square root of a complex number
Solution of the quadratic equation with complex coefficients on complex domain



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