The polar form of complex numbers takes the pure mathematical calculation one step further and starts to incorporate the theory of polar co-ordinates. Complex numbers themselves were created to solve the problem of how to come up with a realistic solution to the negative root of a negative number. Until the time of the reformation, the result of the square root of a negative number was always positive. This way around this problem was to use an imaginary number that became known as i.
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Complex numbers are created from two components a “real” number and an “imaginary” number. This is not as complicated as it sounds and understanding these concepts will help when finding the polar form of a complex number. You should get some tried-and-true math tricks to help with your computational abilities. Moreover, the basic presentation of an imaginary number expressed mathematically is:
i=√-1 which becomes i2=(√-1)2 = -1
This shows that i can symbolize the negative of a square root but the expression cannot be taken further to;
i2=(√-1)2=√(-1)2=√1 = 1
as “i” cannot be both 1 and -1. By using the imaginary number, you have managed to circumvent the tradition that the square root of a negative number is always positive.
Incorporating the imaginary number into a standard equation gives you a complex number such as x+yi where x is a real number and y and an imaginary one.
8–2i = x+yi
The answer is very straightforward as it highlights the main concept of complex numbers that they can only be of equal value if the imaginary and real number are the same. Therefore, the solution to this problem is that 8=x and -2=y. The end result of this is that the real and imaginary parts of the equation have to be combined like for like, the same as when dealing with binomials.
The polar coordinate system is a two dimensional system that uses a distance “radial coordinate” and angle or “angular coordinate” from a fixed point called the “pole” This concept has been around since the 1st millennium and has been developed and refined over hundreds of years with the first recorded calculations appearing around the middle of the 17th century.
A good way to visualize how polar coordinates work is to think of a clock. If the hour hand on a clock is pointed at 3 then it is in the horizontal position and can be thought of as the horizontal axis on a graph. Now if the time showing on the clock is 3:10 then the minute hand will be pointing to 2. To get the distance or “r” we simply measure the length of the minute hand which for this example is 5 inches.
Therefore r=5. A clock face is a circle to represents 360 degrees, and there are twelve numbers on a clock face so 360 degrees divided by 12 gives 30 degrees. The minute hand is pointed at 2 so it is 2 times 30 degrees or 60 degrees. There is one number between 2 and 3 so the angle Θ is 30 degrees. As a result, the polar coordinates will be (5,30). You certainly want to understand these in order to help you Beat the SAT Math.
Polar Form of a Complex Number
It is possible to consider a complex number as a vector and the polar form to be represented using the polar co-ordinates described above. The purpose of the polar form of complex numbers is to display a complex number in terms of the distance and direction from the positive horizontal axis. This is just the start to gaining The Secrets of Mental Math.
If you take the two parts of a complex number, the real number becomes the horizontal axis and the imaginary number the vertical axis. You can then get to the real complex parts as in polar coordinates using r and Θ. As you can see from the diagram r is the angular length as in the minute hand of the clock and Θ the degree from the horizontal axis.
Starting with Pythagorean theory of r2=x2+y2 and then incorporating basic trigonometric theory you end up with:
tanΘ = y/x
cosΘ = x / r
sinΘ = y /r
Break out by multiplying each side by to get rcos Θ=x and rsin=y
The rectangular form of the complex numbers you can see in the diagram is written as:
z = x+yi
= rcosΘ + (rsinΘ)i = r(cosΘ + isinΘ)
The most important part to remember in this is that r is the absolute value and Θ is the argument of the complex number.
We can now display the polar for of the complex number as:
x + yi = r(cosΘ + i sinΘ)
For example to find the polar form of 5–2i you will need to follow these steps
r=√x2 + y2
= √ 52 + (-2)2
= √29 or 5.385
Now you need to calculate Θ using tan
We know that Θ=360 degrees and α=21.8
So the final step is
5–2i = 5.385(cos338.2 + isin338.2)
or 5–2i = 5.385∠338.2
This can then be simply plotted on a graph to give the graphical representation of the polar form of a complex number by drawing a line with the length of 5.385 drawn at an angle of 338.2 degrees.
Why do you need to grasp complex numbers? To start, it is because real numbers are a subset of all complex numbers. There are few aspects of mathematics that do not make use of complex numbers. In the past, you were told you could not take the square root of a negative number. But, when dealing with complex numbers, you can. Sure, the i is imaginary, but is that not advanced mathematics in general? Most of the equations and numbers are made up, however, they cause you to improve your problem solving skills in addition to understanding the scientific laws of the universe.
The name complex numbers in itself denotes that finding the polar form, can be, well, complex. However, it doesn’t have to be. Math is all about practice. What appears difficult, at first, gets easier with more and more application and exercises. In addition, learning these skills do have practical applications, as well. Fields in statistics, economics, biology, electrical engineering, chemistry and biology all require an understanding of complex numbers. Therefore, understanding how to find the polar form of complex numbers will be beneficial as you move forward in your career. To learn more, read Multiplying Complex Numbers: Methods and Strategies.