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Complex Number

Basic Concepts of Complex Numbers

A number which is written in the form of x + iy, is known as the complex number. Here x and y are real numbers and i=√-1, x is the real part of the complex number denoted by Re(z) = x and y is the imaginary part of the complex number denoted by Im(z) = y. Example for a complex number: 5 + i3 and 4 + i(-1/11).

**Power of i:**

i=√-1, i^{2}= -1, i^{3}= - i and i^{4}=1.

**Square roots of a negative real number:**

i^{2} = -1 and (-i)^{2} = i^{2} = -1, square roots of -1 are i, -i.

If z = x + iy is a complex number then conjugate of this complex number can be defined as z¯= x - iy. Conjugate is a mathematical identity having a reciprocal relation.

Conditions : If a complex number x + iy = 0, here x = y = 0.

If a complex number x + iy = c + id , here x = c and y = d.

**Cube Roots of Unity:**

(1)^{1/3 }=1, -1+i√3 / 2, -1-i√3 / 2 =1,x, x^{2 } where x = -1+i√3 / 2

**Properties of Cube Roots of Unity**

? Cube Roots of Unity are in Geometric Progression.

? Complex cube root of unity is the square of the other complex cube root of unity i.e x = -1+√3i / 2, x^{2 }= -1-√3i / 2

? 1 + x + x^{2 }= 0

? Product of all cube roots of unity is 1 i.e. x^{3} = 1

? 1 / x = x^{2} or 1 / x^{2} = x

**Addition of two Complex Numbers**:

We have two complex numbers a = b + ic and z = x + iy. The sum of these two complex numbers is also a complex number, given by a + z = (b + x) + i(c + y)

**Properties of Addition of Complex Numbers:**

1. The Closure Law : Sum of two complex numbers results into another complex number i,e. a+z is a complex number for all complex numbers a and z.

2. The Commutative Law : a + z = z + a where z and a are complex numbers

3. The Associative Law : (a + z) + d = a + (z+d) where a, z and d are complex numbers

4. Zero Complex Number = 0 + i0

5. Additive Inverse of Complex Number : when z = a + ib, then we also have a complex number - z = -a + i(-b) called as additive inverse of Complex Number.

**Difference of Complex Numbers;**

We have two complex numbers a = b + ic and z = x + iy. The difference of these two complex numbers is also a complex number, given by a - z = (b - x) + i(c + y)

**Multiplication of Complex Numbers:**

We have two complex numbers a = b + ic and z = x + iy. The product of these two complex numbers is also a complex number, given by az = (bx - cy) + i(by + cx)

**Properties of Multiplication of Complex Numbers:**

6. The Closure Law: Product of two complex numbers results into another complex number i,e. az is a complex number for all complex numbers a and z.

7. The Commutative Law: az = za where z and a are complex numbers

8. The Associative Law: (az)d = a(zd) where a, z and d are complex numbers

9. The Distributive Law: a( z + d) = az + ad

(a + z)d = ad + zd where a, z and d are complex numbers

10. Existence of Multiplicative Identity = 1 + i0, such that z.1 = z for every complex number z.

**Division of Complex Numbers;**

For complex numbers z_{1} and z_{2} where z_{2} is not equal to zero, the quotient z_{1}/ z_{2} is defined by

z_{1}1/z_{2} .

**Modulus of a Complex Number:**

If a+ib is a complex number, then the non-negative real number √a^{2}+b^{2 }is the modulus (or absolute value or magnitude) of the complex number. It is denoted by | a+ib|=√a^{2}+b^{2}.

**Note: a+ib is the Rectangular(Cartesian) form of complex numbers.**

**Note: Polar form of a complex number z = a+ib is r(cosθ+isinθ), where r = √a**^{2}**+b ^{2}, where cosθ = a / r and sinθ = b/r, θ is argument of z.**

**Note: The Complex number can be expressed in the exponential form as z=r(cosθ+isinθ). By Euler's Formula, we have e ^{iθ}=cosθ+isinθ, generally we select a smallest positive value for θ.**

**Importance of Complex Numbers:**

There are two specific districts that use the possibility of complex numbers, in light of present circumstances: Certified sums that are ordinarily depicted by complex numbers instead of real numbers. Honest to goodness sums which, nonetheless, they're delineated by genuine numbers, are regardless best grasped through the investigation of complex numbers.The issue is that most by far are looking for instances of the principle kind, which are truly extraordinary, however, instances of the second kind happen continually.

Here are a couple instances of the essential kind. In equipment, the state of a circuit part is depicted by two honest to goodness numbers (the voltage V crosswise over it and the present I coursing through it). A circuit segment moreover may have a capacitance C and an inductance L that (in limited terms) delineate its slant to oppose changes in voltage and current independently. These are endlessly enhanced depicted by complex numbers. Rather than the circuit part's state being depicted by two assorted genuine numbers V and I, it can be portrayed by a solitary complex number z = V + iI. So likewise, inductance and capacitance can be considered as the genuine and nonexistent parts of another single complex number w = C + i L. The laws of power can be conveyed using complex expansion and augmentation. Another case is electromagnetism. Rather than endeavouring to delineate an electromagnetic field by two genuine amount (electric field quality and attractive field quality), it is best depicted as a solitary complex number, of which the electric and attractive segments are fundamentally the genuine and fanciful parts.

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