Table of Contents

# Geometry Of Complex Numbers

Hans Schwerdtfeger wrote this book, first published in 1962 as Volume 13 of the University of Toronto Press’s Mathematical Expositions series. In 1979, Dover Publications produced a corrected edition in its Dover Books on Advanced Mathematics series. The Mathematical Association’s Basic Library List Committee has recommended that it be included in undergraduate mathematics libraries.

The geometrical representation of a complex number is one of the fundamental laws of algebra. A complex number, such as z = + I can be represented as a point P(,) in the Argand plane, where is the real part and is the imaginary part. I = 1 is the value of i. Students will learn about the Argand plane representation of Z modulus, section formula, polar form, and many other essential topics of complex number geometry.

**The term “complex number” is defined as “a number that is**

A complex number is one that has both real and imaginary elements. The formula for a complex number is a+ib, where an is a real number and ib is an imaginary number. Furthermore, a,b are real numbers, and I = -1.

As a result, a complex number is simply the sum of two integers, namely a real and an imaginary number. One side is completely real, while the other is completely made up.

**Real figures**

Real numbers are any numbers that exist in a number system, such as positive, negative, zero, integer, rational, irrational, fractions, and so on. Re is the symbol for it (). 12, -45, 0, 1/7, 2.8, 5, and other real numbers are examples.

**Imaginary numbers**

Imaginary numbers are numbers that are not real. We get a negative return when we square an imaginary number. Im is the symbol for it (). For example, the numbers -2, -7, and 11 are all fictional.

To answer the equation x2+1 = 0, complex numbers were introduced. The equation’s roots are of the form x = -1, and there are no true roots. As a result of the addition of complex numbers, we now have imaginary roots.

The symbol I which stands for Iota, is used to represent -1. (Imaginary number).

**Complex Numbers Algebraic Operations**

There are four different types of algebraic operations that can be performed on complex numbers, as listed below. More information about these algebraic operations, as well as solved examples, can be found in the linked article. The four operations on complex numbers are as follows:

Addition

Subtraction

Multiplication

Division

**Complex Numbers’ Roots**

When we solve a quadratic equation of the form ax2 +bx+c = 0, we can discover the roots of the equations in three ways:

There are two types of real roots: distinct real roots and similar real roots.

**Formulas for Complex Numbers**

Combine comparable phrases while executing complicated number arithmetic operations like addition and subtraction. It means that you should add real numbers to real numbers and imaginary numbers to imaginary numbers.

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

Subtract (a + ib) from (c + id) to get (a – c) + i(b – d).

**Multiplication**

When two complex integers are multiplied by each other, the technique should be identical to that of binomial multiplication. It denotes the application of the FOIL method (Distributive multiplication procedure).

(ac – bd) + i(ad + bc) = (c + id)

**Division**

The FOIL Method can be used to divide two complex numbers by multiplying the numerator and denominator by the denominator’s conjugate value and then applying the FOIL Method.

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

**A complex number’s geometrical representation.**

One of the fundamental laws of algebra is the geometrical representation of a complex number. A complex number z = + I can be represented as a point P(,) in the Argand plane, were and are the real and imaginary parts, respectively.

**The following are the properties of complex numbers:**

When you put two conjugate complex numbers together, you get a real number.

Multiplying two conjugate complex numbers produces a real number.

If x and y are real numbers and x+yi = 0, then x and y must be equal.

If p, q, r, and s are real numbers, then p+qi = r+si, p = r, and q=s.

The commutative law of addition and multiplication apply to complex numbers.

**z1 + z2 = z1 + z2**

**z1 x z2 = z2 x z1 x z2 x z1 x z2**

The associative law of addition and multiplication apply to complex numbers.

**(z1+z2) +z3 = (z2+z3) + z1**

**(z1.z2).z3 = (z1.z2).z3 = (z1.z2).z3 = (z1.z2).**

**(z2.z3)**

The distributive law z1.(z2+z3) = z1.z2 + z1.z3 applies to complex numbers.

If the sum of two complex numbers is real, and the product of two complex numbers is also real, these numbers are conjugate.

|z1+z2|

The result of multiplying two complex numbers and their conjugate value should be another complex number with a positive value.

Also read: **What Is Mutually Exclusive Events In Probability?**

**FAQs**

##### What is a complex number's geometric representation?

One of the fundamental laws of algebra is the geometrical representation of a complex number. A complex number z = + I can be represented by the point P (,) in the Argand plane, where is the real part and is the imaginary part. I = sqrt -1

##### What is the best way to express 0 as a complex number?

0 is a real number, as we all know. Real numbers are also included in the category of complex numbers. As a result, 0 is a complex number and can be written as 0+0i. Examine the axes in the graph below to see how complex numbers are represented. The x-axis represents the real part, while the y-axis represents the imaginary part.