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### Complex Numbers- Intro, Examples, Problems, MCQs - Argand Plane, Roots of Unity

Here's an outline and a summary of what's introduced in this tutorial
Target Audience: High School Students, College Freshmen and Sophomores, high school students in Europe and North America, Class 11/12 Students in India preparing for ISC/CBSE and Entrance Examinations like the IIT-JEE

, and anyone else who needs this Tutorial as a reference!
After studying the topic you might also be interested in attempting the following 2 Quizzes related to Complex Numbers.

Complex Numbers and The Argand Plane

## Basics of Complex Numbers, Real and Imaginary Parts, Iota

• Complex Numbers is the largest and the complete set of numbers, consisting of both real and unreal numbers.
• A complex number is of the form
i2=-1.
• A complex number is usually denoted by the letter ‘z’.
• ‘a’ is called the real part, and ‘b’ is called the imaginary part of the complex number.
• The notion of complex numbers increased the solutions to a lot of problems. For instance, had complex numbers been not there, the equation x2+x+1=0 had had no solutions.

Because of ‘i', we can now extend the square root to be defined for negative numbers as well. If y is a negative number, then y can be written as -|y| where |y| is the absolute value of y. Then, the square root of y is defined to be equal to be

y=i√|y|

## Algebra of Complex Numbers, Conjugate, Modulus and Argument, Euler's Form

• Complex Numbers have different algebraic laws as well. Let z1=a+ib,z2=c+id.
• z1+z2=(a+c)+i(b+d)
• z1-z2=(a-c)+i(b-d)
• 1/z1=(a-ib)/(a2-b2)
• Conjugate : Conjugate of a complex number z=a+ib is defined as

z=a-ib.

• Modulus of z is written as |z| and is defined to be equal to √(a2+b2)
• Thus, z.Conjugate(z)=|z|2.
• Commutative, Associative, and Distributive Laws of algebra apply to Complex Algebra as well.
• Complex Numbers can be represented pictorially on a plane known as Argand Plane.
• The plane consists of a Real Axis, and an Imaginary Axis.
• The angle that z makes with the real axis (counterclockwise) is called the argument of z, and the length of line segment joining z and the origin is equal to |z|, also sometimes written as ‘r’.
• Different representations of z,

[Euler’s Form]
• Observe that, e=r(cosθ + i sinθ)
• Multiplication and Division using using Euler’s Form:
1=r1eiθ1, z2=r2eiθ2
1z2=r1r2ei(θ1+θ2)

z1/z2=(r1/r2)ei(θ1-θ2)

• Conversion between various forms.

-1(|b/a|); then θ=α,π-α,α-π,-θ depending upon z being in first, second, third or fourth quadrant respectively. This is called the principal argument.
• Approaches to solving questions:
• Expand every complex number as a real and imaginary part.
• Use Euler’s form for every Complex Number
• Use Algebraic Identities, and solve in general, keeping in mind that complex numbers have a real and an imaginary part.

## Complex Numbers as Free Vectors

• Another approach to dealing with Complex Numbers is to treat them like free vectors. Free Vectors are vectors that can be moved around without changing direction, in other words, that do not really matter upon the origin. The approach that was used so far was treating complex number as a point.
• Any comple number such as 2+3i can be any one of the infinite number of vectors
• e is multiplied with the vector which is to be rotated, and rotates the vector by “θ” anticlockwise. Clockwise rotation is obtained by multiplying with e-iθ
• m, n are unit vectors; and we have ne=m, m e-iθ=n

## Here's a quick look at some of the questions we will solve

• What is the value of (1+i)(1+i2)(1+i3)(1+i4)?
• If (1+i)(1+2i)(1+3i). . .(1+ni)=a+ib, then what is  2X5X10X…X(1+n2)?
• If (x+iy)1/3=a+ib; then x/a+y/b=?
• Find the sum of i+i2+i3+…upto 1000 terms.
• If |z|=2, find the area of the triangle formed by z,iz,z+iz.
• If x=-5+2√-4, find the value of x4+9x3+35x2-x+4.
• Find the square root of 7-24i.
• Given that cosα+cosβ+cosγ=0 and sinα+sinβ+sinγ=0, prove that  sin3α+sin3β+sin3γ=3sin(α+β+γ)
• If cosα+2cosβ+3cosγ=0 and sinα+2sinβ+3sinγ=0. Show cos3α+8cos3β+27cos3γ=3sin(α+β+γ).
• Given that 1,α1,α2,…,αn-1 are nth roots of unity, Show that (1- α1)(1- α2)…(1- αn-1)=n.
• Show that sin(π/n)sin(2π/n)…sin((n-1)π/n)=n/2n-1

## MCQ Quiz #1

Companion MCQ Quiz/Worksheet #1 for Complex Numbers (The Basics) - test how much you know about the topic. Your score will be e-mailed to you at the address you provide.

## MCQ Quiz/Worksheet #2: More on Complex Numbers

Read the questions in the document below, and fill up the answers in the answer submission form below it.
Your score will be emailed to you

#### Questions: More on Complex Numbers

Answer Submission Form for MCQ Quiz #2

#### Answer Submission Form for MCQ Quiz #2: More on Complex Numbers

You might like to take a look at our other algebra tutorials:

 Introduction to Complex NumbersIntroduction to Complex Numbers and iota. Argand plane and iota. Complex numbers as free vectors. N-th roots of a complex number. Notes, formulas and solved problems related to these sub-topics. The Principle of Mathematical InductionIntroductory problems related to Mathematical Induction. Quadratic EquationsIntroducing various techniques by which quadratic equations can be solved - factorization, direct formula. Relationship between roots of a quadratic equation.  Cubic and higher order equations - relationship between roots and coefficients for these. Graphs and plots of quadratic equations. Quadratic Inequalities Quadratic inequalities. Using factorization and visualization based methods. Series and ProgressionsArithmetic, Geometric, Harmonic and mixed progressions. Notes, formulas and solved problems. Sum of the first N terms. Arithmetic, Geometric and Harmonic means and the relationship between them.

After reading this tutorial you might want to check out some of our other Mathematics Quizzes as well.
 Quizzes on ProgressionsMCQ #1: Arithmetic Progression MCQ #2: Geometric ProgressionMCQ #3 : More on Geometric Progressions.MCQ #4 : Harmonic Progressions. MCQ #5: More on Harmonic ProgressionMCQ #6: Mixed ProgressionsComplex NumbersMCQ #1MCQ #2: More on Complex NumbersQuadratic EquationsMCQ Quadratic EquationsQuadratic In-equationsMCQ Quadratic In-equations Coordinate Geometry - Straight LinesMCQ #1: Cartesian Planes, Straight Line BasicsMCQ #2 on Straight LinesMCQ #3 on Straight LinesMCQ #4 on Straight LinesCircles1 MCQ #1 on Circles. 2 MCQ #2 on Circles. 3 MCQ #3 on Circles. Conic Sections- Parabola, Hyperbola, Ellipse1 MCQ- The Basics of Conic Sections2 MCQ on Parabola..3 MCQ on Hyperbola4 MCQ on Ellipses. ProbabilityMCQ #1 on Basic ProbabilityMCQ #2: More Challenging Problems on ProbabilityMCQ #3- Conditional Probability and Bayes Th