What are Complex Numbers?
Before we can explain what complex numbers are, let’s first assume that √-1 = i or i2 = -1, which simply means that i can be assumed as the solution of the equation x2 + 1 = 0. In this case, i is called as Iota in the world of complex numbers.
We can also put it together as,
i2 = -1
i3 = i2 * i = -i
i4 = i2 * i2 = 1
Therefore, we can conclude now that i4n = 1, where n is any positive interger.
Also, keep in mind that i + i2 + i3 + i4 = 0 or in + i2n + i3n + i4n = 0
Any number that comes in the form of a + ib, where a and b which are real numbers can be defined as complex numbers. For instance: 4 + 5i, -3 +23i, -3 + i√5 are all called complex numbers.
Complex numbers are written by z or w (= a + ib), which has two parts: one part is known as the real part and another is the imaginary part. z= a (or b = 0) is called the purely real number whereas z = ib ( or a = 0) can be called the purely imaginary number. Also, you need to note that zero or 0 + i0 is both purely real and purely imaginary but not imaginary.
The Real part of z is a, which is denoted by Re(z) and the Imaginary part of Z is b, which is denoted by Imag(z). For z = 1+2i, re(z) = 1 and Imag (z) = 2
Any complex number can be safely known as the super-set of all the other possible numbers.
N⊂ W ⊂ I⊂Q ⊂R ⊂C
What are Quadratic Equations?
To begin with, any Algebraic Equation that has the degree n is called the Polynomial Equation. A quadratic equation can be defined as the special case of it where the degree is equal to 2. So, one can conclude that a polynomial with degree 2 or two roots (or solution) is the quadratic equation.
The most common form of any Quadratic equation is y = ax2+ bx + c, where a ≠ 0, b and c can be any real (or complex) number. Here, a, b and c are known as the Coefficients. And a is known as the Leading Coefficient.
The shape of all the quadratic equations is always parabolic in nature. E.g. when it comes to y = ax2+ bx + c, the parabola is open upwards if a is positive and open downwards if a is negative. Whereas for x = ay2+ by + c, the parabola is open rightwards if a is positive while open leftwards if a is negative.
A quadratic equation consists of two roots or solution due to its degree 2. Both these roots can be specified with the help of Quadratic Formula as:
Talking about the Roots of a given equation, it signifies the point on the x-axis where the curve for any function cuts the x-axis. The number of points where the curve cuts the x-axis reflects the number of roots or solution or degree of the given polynomial.
The graph of a quadratic equation is in Parabolic Shape. If a is positive, i.e. a > 0, then the parabola will be open-upwards and hence will only have the least value or minimum value of this function while for a to be negative, i.e. a < 0, the parabola will be open-downwards and will have a definite maximum value but not a minimum value.
What is Complex Root Or Complex Solution?
When it comes to the roots for an equation, they are the set of points where the graph for the given function intersects or touches the x–axis. The total number of roots can be directly related with the degree of a polynomial.
However, we assume as the complex root (or solution) when the graph of a polynomial does not intersect. It means that there is no real number and the complex number would only satisfy the functional equation that is given.
For solving the Quadratic equation, if b2 – 4ac < 0, then the roots will not be real roots and the roots, in this case, will be known as the Complex Roots.
Different Forms of Representation for Complex Numbers
Complex numbers can be mainly of four types.
Discriminant of a Quadratic Equation
In quadratic, the value of D = b2 – 4ac means discriminant and it plays a major part in finalizing the nature of the roots. It is also represented by delta or ∆.
If D = 0, means equal roots. Graphically, parabola touches the x-axis at a single point.
If D > 0, means real and distinct roots, Graphically, parabola intersects the x-axis at two distinct points.
And if D < 0, it signifies non-real or imaginary or complex roots. In this situation, the parabola never manages to touch the x-axis.
Different Ways to Solve Quadratic Equations
While solving a Quadratic equation, we have to find out the values of the variable that satisfy the given equation. Quadratic equations can be done in two simple ways. The first technique is called the Factorization Method and the second one is known as the Hindu Method or Sri dharachary Method, which is a formula-based approach.
Examples of Complex Numbers and Quadratic Equations
There is a wide range of applications for Complex Numbers, but most of them are used in the field of electrical engineering. In Electrical engineering, there are applications in areas such as the design of circuits using capacitor and inductors, oscillations, electromagnetism, and many more.
A quadratic equation is responsible for tracing the parabolic cubic curve. We all invariably come across a lot of real-life situations wherein the path or the equation follows the parabolic curve i.e. they can be easily linked with the help of quadratic equation. For instance, a ball that is thrown in the air or any person jumping from a building where the path traced by them in their journey follows the parabolic curve. We can understand the instantaneous locations during the journey while understanding quadratic equations. Quadratic equations are also helpful in finding out the profit of any business.