Overview on Complex Numbers

Complex numbers are those that can be written in the form a+ib, where a and b are real numbers and i is an imaginary number denoted by the letter iota. The value of I is equal to (-1) For instance, the expression “2 plus 3i” is a complex number, in which “2” is a real number (Re) and “3i” is an imaginary number (Im).

A complex number is the result of the combination of a real number and an imaginary number.

Definition

A real number and an imaginary number make up a complex number, which is essentially the combination of the two. The formula for the complex number is a+ib, where a represents a real number and ib represents an imaginary number. Additionally, a and b are both real numbers, and the value of i is a negative one.

A complex number is an easy way to represent the addition of two numbers, namely a real number and an imaginary number. This is because a complex number is simply the addition of a real number and an imaginary number. It can be broken down into two categories: the purely real and the purely imagined parts.

Real numbers

Real numbers are any numbers that are present in a number system, whether they be positive, negative, zero, integer, rational, irrational, fractions, or any other type of number. It is denoted by the symbol Re (). Real numbers include the following: 12, -45, 0; 1/7; 2.8; -5; and so on and so forth.

Imaginary Numbers

Imaginary numbers are any numbers that do not exist in the real world. When we square an imaginary number, the result that we get is a negative value. It is denoted by the symbol Im (). As an illustration, the numbers -2, -7, and 11 are all examples of imaginary numbers.

In order to solve the equation x² + 1 = 0, the complex numbers were brought into play. The equation can be solved to find roots of the form x = -1, but there are no real roots. Imaginary roots are a result of the development of complex numbers, which led to their introduction.

We represent the value -1 with the symbol i which stands for the letter iota (Imaginary number).

Notation

A complex number is defined as the solution to an equation of the form z=a+ib, in which both a and b are real numbers. The real component is represented by the symbol Re z = a, whereas the imaginary component is represented by the symbol Im z = ib.

Z = a +  ib

Absolute Value

The number itself is considered to be a real number’s absolute value because it is the number itself. The value of x in its absolute form is denoted by the modulus, which is written as |x|. Because of this, the modulus of any value always gives a positive value, as a result of which;

|3| = 3

|-3| = 3

Now, in the event that we have complex numbers, we will use a different method to find the modulus.

Let’s say that the equation z = x+iy is a complex number. Then, the modulus of z will be as follows:

|z| = √(x² + y²)

The Pythagorean theorem can be applied in a complex plane, which allows us to derive this expression. As a result, the modulo of the complex number z is extended from 0 to z, and the modulos of the real numbers x and y are extended from 0 to x and y respectively. Now, these values shape a right triangle, and the vertex of the acute angle is represented by the number 0. Now, using the Pythagorean theorem, we can see that:

|z|² = |x|² + |y|²

|z|² = x² + y²

|z| = √(x² + y²)

Formulas for Complicated Numbers

When working with complex numbers in arithmetic operations like addition and subtraction, it is helpful to combine terms that are similar to one another. It indicates that the real number should be added to the real number, while the imaginary number should be added to the imaginary number.

•Addition (a + ib) + (c + id) = (a + c) + i(b + d)

•Subtraction (a + ib) – (c + id) = (a – c) + i(b – d)

•Multiplication

The process of multiplying two complex numbers by each other should follow a pattern that is analogous to the pattern that is followed when multiplying two binomials. It indicates that the Distributive multiplication process, also known as the FOIL method, is being used.

(a + ib). (c + id) = (ac – bd) + i(ad + bc)

•Division

Multiplying the numerator and denominator by the complex number that corresponds to the conjugate value of the denominator is the first step in performing the division of two complex numbers. The next step is to use the FOIL method.

(a + ib) / (c + id) = (ac+bd)/ (c² + d²) + i(bc – ad) / (c² + d²)

Conclusion

Complex numbers are those that can be written in the form a+ib, where a and b are real numbers and i is an imaginary number denoted by the letter iota. The value of I is equal to (-1) For instance, the expression “2 plus 3i” is a complex number, in which “2” is a real number (Re) and “3i” is an imaginary number (Im). A complex number is the result of the combination of a real number and an imaginary number. A real number and an imaginary number make up a complex number, which is essentially the combination of the two. The formula for the complex number is a+ib, where a represents a real number and ib represents an imaginary number. Additionally, a and b are both real numbers, and the value of i is a negative one. A complex number is defined as the solution to an equation of the form z=a+ib, in which both a and b are real numbers.