Does Complex Numbers / Imaginary number i exists??
People studying Higher Mathematics are often confused with the existence of Complex Numbers and its Imaginary number “i”.
What are Complex Numbers:
Complex Numbers are numbers whose 1st part is Real while 2nd part is Imaginary and is expressed as a+bi
a –> Real Part
bi –> Imaginary Part and
i –> √ -1
wait √ -1 …. Is it POSSIBLE ??
Reason of Confusion:
- In schools and colleges everyone has learnt that we cannot have square root of Negative numbers i.e. √ negative number doesnot exists. Then how it is possible to have √ -1 .
- Moreover “i” is called as Imaginary Number hence i is often considered as it does not exists and was used just for theory but if it is really imaginary then how it is used in Real Life Mathematics? How “i” and Complex Numbers can be used in Real Life Exmaples?
Quite Correct !!! But then my question is does our day-to-day used numbers like 0-9, π, 1/3, √ 2 , etc. exists?
All the numbers are Human Creation for making life of us Simpler and all the numbers took 100’s & 1000’s of years for approval and to be consistently used.
Brief History of Numbers:
Initially Romans used numbers only from 1-9 i.e. without any ‘0’.
Zero: But then Indians introduced ‘0’ and when this new number went to Europe it was not accepted for 100’s of years. Reason was simple Europeans were very comfortable with 1-9 for their transactions but as years progressed they realized the importance of ‘0’ which makes solving certain equations in simple manner and started accepting it.
Fractions: Today fractions are widely used across all fields. But can fractions exists for all items?? NO… we cannot have 11⁄2 men … right ?? but we can definitely have 21⁄3 apples. Also to solve equations like 2x = 1, fractions were required and hence for convenience fractions gradually became part of our number system.
Negative Numbers: Like fractions even negative numbers for certain items doesn’t exists. We cannot have -20 trees or -2 people. But to solve equation like x+20 =0, we need negative numbers. x= -20 means debt of 20 items. Even negative numbers were not included in number system before but gradually people started understanding of them and started using them and also included them in our number system.
Irrational Numbers: There are certain numbers like √ 2 , π which cannot be expressed as fractions and
hence cannot fit anywhere in above number system.
But certain equations like x2=2 generates certain numbers,
called as Irrational Numbers, which are required to placed somewhere in our
Number System else it wont be possible to have solutions for x2=2.
So as seen above we used 0, negative numbers, Irrational, Vinculum numbers, etc. all these numbers are Human Creation and were invented to have solution to every problem/equation and not to forget to make life of human being simpler.
Same way is Mathematics — Invention, Modification and Addition.
Arrival of Complex Numbers: Similarly to make certain equations solvable like x2 = -1, Complex Numbers were introduced. To solve certain polynomial equations we often come across square root of negative numbers and without the concept of Complex Numbers we would end up in Dead End. Same thing had happened to Greek Mathematician “Heron“. While calculating volume of some frustum of a cone he came across √ 81 – 144 but unaware what to be done with √ negative number ,
he left the calculation there itself.
The imaginary word in complex numbers is the reason for confusion but these complex numbers does exists in same manner as other regular numbers exists.
So as seen above our Number System is indeed a 2 dimensional system and there might be every possibility that some another number might get add up to our Number System making it to a 3 – Dimensional Number System. Any idea which can be that number ???