POST- M::1 - Are Complex Numbers too complex ?

Welcome to the first post under the label "Mathematics Strategy". In the posts under this label, I will be discussing the strategies for all the Mathematics chapters that are included in JEE syllabus.
Mainly I am going to discuss four things,

• The Importance of the chapter from JEE point of view,
• The prerequisites of this chapter,
• The important concepts inside the chapter &
• The problem-solving techniques (The most important one).

The Importance Of The Chapter : -

A complex number is not as complex as it's applications are. Whenever it is required to consider a different axis other than the real number axes, everyone prefers the imaginary axis to do transformations and calculations. If you are going to have a look over the history of complex numbers, you may find numerous researches which say that complex numbers are not another form of numbers, in fact, they can be derived from the real number theory itself. Well, that is some higher level of mathematics and our concentration is towards JEE.

Complex numbers are used a lot in Engineering Sciences. If you are aiming to study Engineering then you are going to find these species over Control Theory (Laplace Transformation etc), Signal Analysis, Game Development and Software Development.

If you are aiming to study Physics then you will find complex equations in Quantum Mechanics which uses imaginary number and imaginary axis.

This obviously makes the topic an important part of your syllabus.

The prerequisites of this chapter:-

Before heading towards the chapter, make sure that you have a good command over the following chapters and concepts,

• Real Number System
• Trigonometry( Trigonometry ratio, Trigonometry equations)
• Finding Domain and Range
• Problem-solving by graph
• 2D Geometry
• Vector(Class-11 physics knowledge is sufficient to understand theory)

Important concepts inside the chapter:-

These are the important concepts inside this chapter, that you must cover in an efficient way to attempt and solve a JEE advanced problem related to complex number & its application,

  1 - Conjugate, Modulus, Amplitude of a complex number and properties like - Triangle Inequality, Amplitude deciding and geometrical representation (both cartesian & polar).

  2 - Application od 2D concepts to solve problems using complex number theory (like section formula, centroid, orthocentre, Circumcenter)   

  3 - Transforming 2D equations of Straight Line, Circle, Conic Section from the real number to complex number presentation (or you may say "General Locii on the complex plane")

   4 - Concept of "Vector" in complex number plane and rotation of complex number.

   5 - "De'moivre's Theorem" and it's application to solve complex looking proofs & equations.

   6 - Cube root of Unity and nth Root of Unity, their applications.

In the next section, I am going to discuss an overview of each concept and the ways to master them.

The problem-solving techniques:-

        1 - This section is considered as the favourite section for JEE main especially, because of the standard of question that can be formed by mixing these concepts together. Follow the ways below,

• Complete NCERT mathematics first.
• This concept is explained very well in TMH(Tata McGraw Hill) objective mathematics and you may follow TMH at the initial stage of practice.
• When you will get the comfort in the problems from TMH, switch to Arihant Algebra and complete the JEE advanced part(both theory and problems).
• For a final touch and self-test of complex number problems, complete all the previous year questions once. If you are going to face difficulty then note the exact part and practise 10 questions of that type from the book you are following.
• If you are a dropper of "Type-A", i.e you are taking coaching from some institute, then instead of TMH, complete the theory and basic question part from your material. Don't jump to brainstorming (JEE advanced) part initially. Complete Arihant and then jump into the material again.

           2 - This section scares aspirants because if the wrong technique is followed, a lengthy unsolve equation will confuse you. Hence here is the way you can complete this chapter,

• Try to learn the technique which allows you to relate complex plane and argand plane. You can find this technique in a good book like TMH/Arihant or else you may use coaching materials also if you are taking coaching.
• Before jumping into the subject, practise and understand the 2D chapter well because if you don't know how to find centroid in argand plane than you can not solve a problem which requires knowledge of centroid/circumcentre/orthocentre. Aspirants make a mistake here, they try to learn and remember the formula, but I suggest you, practise the derivations of 2D geometry manually. Arihant Coordinate geometry (By Amit M. Agarwal) or Cengage 2D geometry is the best book. That helps is JEE advanced.
• Now follow the same procedure that I have explained in the previous point. First TMH( or material) then Arihant and then previous year papers.
• One more thing for this concept, practise as more numerical as possible as this concept can be intermixed with many other concepts to make hard JEE advanced questions.

            3 - This concept is easy, you don't need to practise 100 questions from this concept. Learn how the transformation of real to complex equation and the vice-versa occurs. As I have said in the prerequisite section, you need to learn 2D equations manually first or else this section can make you lose confidence.

"Don't fear,nothing is impossible to achive if that is possible to imagine"

         4 -  Rotation of a complex number, is a fantastic tool for JEE questions. Questions which seem hard can be easily solved by rotating the vector represented in a complex plane and by using arguments. Here is the proper approach to master the questions,

• Learn vector and it's rotation in the Argand plane, I mean to say learn exactly which thing remains constant and which thing varies when a vector is rotated.
• Open Arihant Algebra and read theory under this section. Then practice the related questions.
• Once that is over, switch to previous year papers. Don't worry if you find the application of this concept very hard and time taking initially, I had also faced the same but with constant practise you can predict one day, which question can be solved by general technique and which requires the rotation technique for an easy approach. 

       5 - Honestly TMH mathematics is sufficient alone for "De' Moivre's theorem", but as a choice, you may use Arihant Mathematics for a better grip. Well, this is the easiest concept, so no extra technique is needed.        

      6 - Here comes a versatile topic. This is like the salt in our food, as this topic can be intermixed with Matrix, Quadratic Equation, Sequence and series and almost all algebra topics in JEE syllabus. This topic is not hard, just you need to have a clear idea of how things are getting solved by applying roots of unity. You may follow the following procedure,    

• Read theory of nth root of unity first because that is the main concept. Cengage Algebra by G.Tewani has the best set of theory and numerical to study the nth root of unity.
• After the initial practice of this section, turn your pages towards the cube root of unity. often questions are asked from cube root of unity and their algebra. TMH or Cengage has the best questions for this matter. Complete them.
• If you are preparing for JEE like a serious aspirant then you are going to find a lot of questions from this concept almost daily, because this concept is omnipresent. So you don't need to practise 100 questions from this chapter.

Well, after following my strategy, reading this much theory and solving this much of numerical you may be thinking yourself as a master of this chapter, but you are wrong my friend.

The most important and dangerous thing is yet to be practised, that is "Time". This 4-word thing is so dangerous that it can kick your rank below by 1000s. Almost everyone knows the same theory and techniques but only those aspirants succeed who knows how to solve a question in proper time. There is no special way to know the use of time except "Regular Practise".

To use the "Time" as a weapon, you need to attend mock tests. If you are in coaching then you have an upper-hand for this. If you are not in coaching then I suggest you, get enrolled in DLP series of some institute so that you may get the question papers of their mock tests. Also, nowadays many paid and free mock tests are available on the internet. Always prefer the online mode(or CBT) as things are little different in online mode ( We are going to discuss them in another post).

Final words for this post:-

This chapter requires constant practice. The more you practise, the less complex it seems. Not only for JEE but also in future, this chapter is going to be applied very often.

"Study well, practise daily, do smart work and succeed in your aim, that is what I believe."

All the best! 

P.S - In case of any doubt you can join the discussion forum made for the users - DISCUSSION. I will try to reply as many users as possible.

By, 

Abhijit Tripathy

Admin, Dropper's Den.