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Complex numbers

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Unraveling the Mysteries of Complex Numbers: A Comprehensive Guide

Comprehensive Definition, Description, Examples & Rules 

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Introduction

Complex Numbers are an element of the number system in mathematics that deals with square and cubic roots of negative numbers. With the help of complex numbers, mathematicians solve polynomial equations which do not have real roots. 

Complex Numbers paved the way for better understanding and progress of multiple fields. Today, we see the application of complex numbers in many fields like Electronics, Electromagnetism, Signal Processing, Quantum Mechanics, and many more. 

Understanding Complex Numbers

Definition of Complex Numbers

A complex number is a number that consists of a combination of Real and Imaginary numbers. It can be written in the format ‘a+ib’ where ‘a’ is the Real Number (Re), ‘ib’ is the Imaginary Number (Im), and ‘i’ is called iota. The value of i is (√-1). 

For example, in the Complex Number (z)= 9+7i, the Real Number is 9, and the Imaginary Number is 7i. 

Real vs. Imaginary Numbers

A Real Number is a number that exists in the number system as positive, negative, 0, an integer, rational, irrational number, fraction, etc. 

For example, -3,-9/190, 0, √9, 12, 89/96, etc. are real numbers.

Any number that is not real is an Imaginary Number. When we try to square an Imaginary Number, it will generate a negative result. 

For example, the square of √-17 will be -17. Numbers like √-99, √-25, √-3, etc. are Imaginary Numbers.

Complex Numbers were discovered to solve equations that do not have real roots. Thus, when we solve any equation using complex numbers, the result will be imaginary roots.

To better understand the format of complex numbers, please go through the following table: 

Complex Number (z)

Real Number (a)

Imaginary Number (ib)

8+9i

8

9i

-5+6i

-5

6i

-3-2i

-3

-2i

4

4

0i

Representation on the Complex Plane

Since the complex numbers consist of the real part (Re) and imaginary part (Im), they can be plotted on an Euclidean plane, where [(Re), (Im)] is considered as a pair just like (x,y). Concerning complex numbers, this Euclidean plane is called the Complex Plane or Argand Plane. The x-axis depicts the real part whereas the y-axis represents the imaginary part.  

Representation on the Complex Plane

The process of plotting complex numbers on a graph is identical to the process of plotting points on a (x,y) graph.

Exploring Imaginary Numbers

With real numbers, you will always get a positive result after squaring them, for example, 42= 16 and (-4)2= 16. However, the case with imaginary numbers is different on squaring, the result is always negative.

For example, (√-4)2= -4 or (√-9)2= -9. Thus, imaginary numbers are defined as numbers that produce a negative result on squaring. 

The imaginary part, depicted by ‘i’, is called iota. The value of ‘i’ is always (√-1). Therefore, you write 9i as [9*(√-1)] or (√-9). Imaginary numbers do not exist, but since we want them, we imagine them. 

Real-World Examples of Imaginary Numbers

Imaginary numbers play a crucial role in our daily lives. Many machines and processes run on these imaginary numbers effortlessly. The applications of Imaginary Numbers are visible in the following fields: 

  1. Signal Processing: Imaginary numbers are applied to cellular and wireless technology, radars, and even biology to measure brain waves. Since the waves are of sine and cosine, it is easy to use imaginary numbers. 
  2. AC Current Analysis: Alternating Current or AC continuously changes between the positive and negative in a sine waveform. The use of imaginary numbers makes the calculations easy to avoid energy loss and cases of electrocution. 

There are other uses of imaginary numbers as well, but two uses are directly applicable to our daily lives. 

The Exponential Form of Complex Numbers

Complex numbers can be categorised into three forms, Standard or Rectangular, Polar, and Exponential Form. Out of these three, the standard/rectangular form is the easiest and least complicated method of representing complex numbers. However, we can convert a complex number in general or rectangular form to polar or Exponential Form. 

The three forms of complex numbers are as follows:

  1. Rectangular/Standard Form: a+ib
  2. Polar Form: r(cosθ+isinθ)
  3. Exponential Form: reiθ

Conversion from Standard to Exponential Form

To convert a complex number from general form to polar or exponential form, we need the values of r and θ. Here, r is the modulus of z; |z|= √(a2 +b2) and θ is the amplitude or argument of z, and eiθ = (cosθ+isinθ) that is known as Euler’s Formula.  

First, we convert the general form of a complex number to polar form with the help of an example:

We will take the complex number 1+ √3i.

 1= r cosθ 

√3= r sinθ

 r= |z|= √(12+(√3)2)= 2

Therefore, cosθ= ½ and sinθ= √3/2

Thus, θ= π/3

Therefore, Polar form is z= 2[cos(π/3)+i*sin(π/3)]

To convert the complex number  1+ √3i into exponential form, we will directly put the values of r and θ, and we get:

z= 1+ √3i= 2ei(π/3)

Application and Advantages

The exponential form of complex numbers is used in engineering, mainly computer science, and civil and mechanical engineering. The advantage of using the exponential form is that it facilitates faster calculation when it comes to the multiplication and division of complex numbers as compared to the standard or rectangular form. 

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Key Takeaways

  1. ‘z’ represents Complex Numbers.
  2. Complex numbers are of the form a+ib.
  3. ‘i’ is known as Iota.
  4. The value of i is √-1.
  5. You can operate all basic arithmetic operations on Complex Numbers.
  6. There are three forms of complex numbers:
  • Standard/Rectangular Form
  • Polar Form
  • Exponential Form

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Frequently Asked Questions

It is an equation that consists of a Real number and an imaginary number that’s represented by ‘z’ and the form ‘a+ib’.

To convert a complex number into its exponential form, we first need the value of ‘r’. The value of r is equal to √(a2 +b2). With the help of r, we get the value of θ. When we have the values of r and θ, we put these values in Euler’s Formula, which will give us the value of e. Substituting the value of ein the formula, ‘re’ will result in the exponential form of a complex number. 

You can perform all basic arithmetic operations, i.e., Addition, Subtraction, Multiplication, and Division with Complex Numbers.

Real-world applications of complex numbers are Signal Processing, AC Current Analysis, Quantum Mechanics, 2D image representation, and design.

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2D Shapes2cosacosb Formula30-60-90 Formulas3D ShapesAbsolute Value FormulaAcute AngleAcute Angle triangleAdditionAlgebra FormulasAlgebra of MatricesAlgebraic EquationsAlgebraic ExpressionsAngle FormulaAnnulusAnova FormulaAnti-derivative FormulaAntiderivative FormulaApplication of DerivativesApplications of IntegrationArc Length FormulaArccot FormulaArctan FormulaArea Formula for QuadrilateralsArea FormulasArea Of A Sector Of A Circle FormulaArea Of An Octagon FormulaArea Of Isosceles TriangleArea Of ShapesArea Under the Curve FormulaArea of RectangleArea of Regular Polygon FormulaArea of TriangleArea of a Circle FormulaArea of a Pentagon FormulaArea of a Square FormulaArea of a Trapezoid FormulaArithmetic Mean FormulaArithmetic ProgressionsArithmetic Sequence Recursive FormulaArithmetic and Geometric ProgressionAscending OrderAssociative Property FormulaAsymptote FormulaAverage Deviation FormulaAverage Rate of Change FormulaAveragesAxioms Of ProbabilityAxis of Symmetry FormulaBasic Math FormulasBasics Of AlgebraBinary FormulaBinomial Probability FormulaBinomial distributionBodmas RuleBoolean AlgebraBusiness MathematicsCalculusCelsius FormulaCentral Angle of a Circle FormulaCentral Limit Theorem FormulaCentroid of a Trapezoid FormulaChain RuleChain Rule FormulaChange of Base FormulaChi Square FormulaCirclesCircumference FormulaCoefficient of Determination FormulaCoefficient of Variation FormulaCofactor FormulaComplex numbersCompound Interest FormulaConditional Probability FormulaConeConfidence Interval FormulaCongruence of TrianglesCorrelation Coefficient FormulaCos Double Angle FormulaCos Square theta FormulaCos Theta FormulaCosec Cot FormulaCosecant FormulaCosine FormulaCovariance FormulaCubeCylinderDiscover the world of MathsEllipseEquilateral triangleEuler’s formulaEven numbersExponentsFractionHeptagonHyperbolaIntegersIntegrationIntegration by partsLinesLocusMatricesNatural numbersNumber lineParallelogramPercentage formulaPerimeterPolygonPolynomialsPrismProbabilityPyramidPythagoras theoremScalene triangleSetsSlope formulaSolid shapesSphereSquareStandard deviation formulaSubtractionSymmetryTimeTrianglesTrigonometry formulaTypes of anglesValue of PiVectorVolume formulasVolume of a coneVolume of sphere formulaWhole numbers
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