Complex Numbers and Quadratic Equations

Real number

A real number is one that can be placed on the real line. Rational and irrational numbers are included in real numbers. Rational numbers, such as positive and negative integers ( -8,0,11), fractions, and irrational numbers (3/4, 6/8, 4.5), are all examples of real numbers.

Imaginary Number

As 1 is the unit for real numbers, similarly unit for imaginary number is i, which is the square root of −1 i = √ -1 When we square i, we get −1 i² = −1 Examples of imaginary numbers: 5i, 5i\6

Complex Numbers

A complex number can be written in the form as: a+bi, where a denotes the real number while b denotes the imaginary number. An example of the complex number is: 42+6i

Types of Complex numbers

Any complex number can be divided into four main categories based on the composition of the real and imaginary parts:

• imaginary number
• zero complex number, where the imaginary part is zero
• purely imaginary number, where the real part is zero
• purely real numbers.

Quadratic Equation in the complex number system

A quadratic equation is a mathematical equation of squares of a variable in algebra. It’s also known as the  ‘2nd-degree equation’ (because of x²). Thus conventionally  complex numbers and quadratic equations can be represented as: ax² + bx + c = 0, where x is a variable, while a, b, and c are known values.

Mathematical Operations of Complex Numbers

Addition of a Complex number

While adding two complex numbers, we add real and imaginary numbers separately. (a+bi) + (c+di) = (a+c) + (b+d)i Let us consider solving the following problem: (4 + 5i ) + ( 3 + 8i ) = 4 + 5i + 3 + 8i (4 + 5i ) + ( 3 + 8i ) = 4 + 3 + 5i + 8i (4 + 5i ) + ( 3 + 8i ) = 7 + 13i

Subtraction of complex Numbers

Let z¹ =l + ia z²= m + i b Then z¹ – z²= z¹+ (-z²) z¹– z² = (l + ia ) + (- m + ib) z¹– z²= (l – m) + (a – b) [Distribute the negatives] z¹– z² = (l – m) + i( a – b) [ Combining the like terms]

Multiplication of Complex number

Complex number multiplication is similar to multiplying real numbers. We can assume i as a variable just like x. Multiplying complex numbers follows the distributive property. a(b + ci ) = ab + a(ci) 6(11 + 4i ) = 6(11) + 6 (4i) = 66 + 24 i

Conjugate of a complex number

Conjugate in simple words means the opposite. In complex numbers, it is merely formed by reversing the sign of mathematical operations. Conjugate of a + b = a – b, and vice-versa.

Division of Complex numbers

Division of Complex numbers can be carried out by multiplying both numerator and denominator by the conjugate of the denominator. For example, consider solving the following: (3+4i) (5 – 6i) Multiply numerator and denominator by the conjugate of (5 – 6i) = ( 5 + 6i) Step 1:  (3+4i)i ( 5 + 6i) = 3( 5 + 6i) + 4i(( 5 + 6i) = 15 + 18i + 20i + 24i2 = 15 + 38i + 24i2 Substitute the value of i2 = -1 (3+4i)i ( 5 + 6i) = 15 + 38i + 24(-1) = 15 + 38i -24 = -9 + 38i Step 2 : (5 – 6i) ( 5 + 6i) = 5( 5 + 6i) – 6i ( 5 + 6i) = 25 + 30i – 30i – 36i2 =25 – 36i2 Substitute the value of i2 = -1 (5 – 6i) ( 5 + 6i) = 25 – 36 (-1) = 25 +36 = 61 Step 3:  (3+4i) (5 – 6i) = (-9 + 38i) 61 Putting in a + bi form, we get (3+4i) (5 – 6i) = -9/61 + 38 i

Modulus of a complex number

It can be defined as the square root of the sum of the squares of real and imaginary parts of a complex number. Consider a complex number z = a + ib where a and b are real and i = √-1. Then the value of the non-negative square root of a2+ b2 is called the modulus or absolute value of z. Modulus is denoted by the mod(z) or |z|

Complex number representation on Argand plane

A point P(a,b) on a xy plane can represent the complex number z=a+ib. The actual part is represented by the x-axis, while the imaginary part is represented by the y-axis. Like any vector quantity, a complex number on the argand plane can be represented using modulus and arg(z), which is direction. As a result, in the argand plane, all complex numbers can be described as position vectors. The addition and subtraction of complex numbers can be done in the same way that vector addition and subtraction can be done.

Square Root of a complex number

The Polar form representation of the square root of a complex number is determined by using the nth root theorem for complex numbers. The nth Root Theorem asserts that the nth root of a complex number z = r(cos + I sin) is z1/n = r1/n [cos [(+ 2k)/n] + I sin [(+ 2k)/n], where k = 0, 1, 2, 3,…, n-1. We multiply by 2k to get the periodic roots of the complex number. Using the nth root formula, we can figure out how to obtain the square root of a complex integer in polar form. The formula is as follows: [cos [(+ 2k)/2] + I sin [(+ 2k)/2]] z1/2 = r1/2,  with k = 0 and 1

Conclusion

Thus complex numbers and quadratic equations involve a deep understanding of both linear and quadratic equations along with a grasp on complex numbers. The roots of quadratic equations of complex numbers are also worth mentioning.