Question 22159: In the real world, where might these so-called Imaginary numbers be used?
When using a formula, we often know the value of one variable to a greater degree of accuracy than we know that others. What affect, if any, does it makes on our use of a formula if we know that value of one variable to a greater degree of accracy than another?
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! COMPLEX NUMBERS IN REAL WORLD.
Let us see 2 of the several popular approaches to problem solving which are at the opposite ends of a spectrum .
1.Some times we start with several small examples ,try to look for a pattern, make a bigger picture and solve the general problem.ex..we may note incidenence of a particular phenomena (say a decease)in a place, locate other cases of similar nature and then try to generalize .
2.On the other hand some times it pays to prove a general pattern first and then deduce from it several smaller particular instances of it easily. examples are many in maths and other science/real life stuations.
Study of complex numbers and its application to real world falls in the second category
A complex number consist of 2 parts one real part and another imaginary part. so real world is a part of the complex world. That is if we solve a problem in the more general field of complex numbers, it can be easily applied to prove theorems in real numbers as they are only particular cases of complex numbers as elaborated bove.
If we look at the evolution of the numbers, they all came up on necessity.
first we started with positive integers, then subtraction necessitated negative integers
followed by division necessitating fractions….like that we had rational and then irrational numbers….subsequently, powers and roots led to concept of imaginary numbers.
Thus complex numbers are a part of expansion of the real world.
Another factor that needs mention in this context is the necessity to simplify the communication and operation. For example when asked to give the performance of a school we can go on narrating that last year, Jennifer got 70% ,alice got 60% etc.. listing several facts, but after a while the listener fails to comprehend the information and loses interest. On the other hand , I we give a summary saying an average of 90 % in the school got A grade etc...it registers forcefully and conveys the meaning effectively. so concepts like average etc. simplifies communication . So is the case with complex numbers. The operations of addition, subtraction, multiplication , division, powers, roots etc.of complex numbers provide a compact tool to convey and perform complicated operations in real world. In maths itself ,they provide easy tools to solve various problems in trigonometry, algebra etc..(like use of Demoviers theorem ) .In engineering applications they provide a very useful tool to solve differential equations, definite integration problems etc. Many such applications abound in electricity,magnetism etc.
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