Complex numbers are composed mathematical objects and by calculating with these you avoid problems with square roots of -1, and more easily do calculations with rich information.
Algebra with ordinary plain numbers do not work in some instances, for example when a part of a function happens to express the sqare root of -1. By using composed mathematical objects called complex numbers and composted operations with these, instead of plain numbers and operations, one avoids this problem and the arithmetics get more practical for several applications.
THE TRADITIONAL DEFINITION OF COMPLEX NUMBERS AND THEIR ALGEBRA
A complex number is a composed mathematical object that can be written in the form a +/- bi, where a and b are real or ordinary numbers.
The unit i is defined as the square root of -1, that is i*i=-1. The a and bi parts of the object is usually called the real and imaginary part.
When you calculate with complex numbers, you calculate with whit them as if they were polynomials, but each time you get i*i you replace this with -1. Here is an example:
(4+3i)(2+5i) = 8+20i+6i-15=-9-15+20i+6i=-6+26i
In real life applications, you will typically use complex number for phenomenons that have two connected parts and where values of these parts behave like the real and imaginary part of this kind of mathematics, which appen to be faily many, especially in the field of physics and engineering. Then you can reduce the number of equations or functions you would have to deal with otherwise..
A MORE FUNDAMENTAL WAY OF DEFINING COMPLEX NUMBERS AND COMPLES-VALUED CALCULATIONS
You can also define complex numbers and calculations with complex numbers in another way. This definition is like the following.
A complex number is and ordered pair of two real numbers (a,b)
Adding two complec numbers (a,b) and (c,d) is done this way: (a,b)+(c,d)`= (a+c, b+d)
Multiplying two complex numbers (a,b) and (c,d) is done this way: (a,b)*(c,d)`= (ac-bd, ad+bc)
Using the same example this way we get: (4,3)*(2,5)=(4*2-3*5,4*5+3*2)=(8-15,20+6)=(-6,26), and this gives the same real part and imaginary part in the answar.
DERIVING THE SECOND DEFINITION FROM THE FIRST
If one performs a symbolic multiplication of two complec numbers with the first method, one gets:
(a+bi)*(c+di)=ac+adi+bci-bd=ac-bd+adi+bci
Since there is no way of adding togeather the term with and without i, the plus sign between them can be replaced by a comma while taking away the i, and one gets: (a,b)*(c,d)`= (ac-bd, ad+bc)
You can esily perform a similar transformation using addition.
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