Imaginary Numbers

Historyand

Practical Applications

Definition of Imaginary Numbers:

of or having to do with the even root of a negative number or any expression involving such a root.

HISTORY There has always been a natural

progression of numbers: natural, negative, rational, irrational, imaginary.

Ancient Greeks were disturbed by the thought of irrational numbers (the hypotenuse of isosceles right triangles)

Mathematicians for a long time were unwilling to accept that solutions to equations could be a number less than zero.

Many times we have had to change our beliefs.

The term imaginary

All numbers in math are imaginary in the sense that they are only in our minds.

The word “imaginary” is unfortunate.

Imaginary numbers do correspond to reality, but not in the simple, intuitive sense that whole numbers did.

Why the word “imaginary”?

The reason the mathematicians choose “I” as the new name was because they still were unsure as to the validity of this number and if it really was a number.

They eventually realized that the term “I” was a good idea, and the term imaginary never was changed.

It isn’t that imaginary numbers aren’t real, but they reveal new aspects of reality that were not immediately clear to us.

Why did “i” come about?

Mathematicians could not find a solution to x^2 + 1 = 0

People wanted to be able to take the square root of a negative number and you can’t if you limit yourself to the reals.

The Beginnings

The earliest record of the square root of negative numbers appears in Stereometrica by Heron of Alexandria. (AD50) sqrt (81-144)

In India in 850, Mahavira wrote “As in nature of things, a negative is not a square, it has no square root.”

Until 1500’s, mathematicians were puzzled by the square root of a negative number.

Girolamo Cardano

Cardano was one of the first to work with imaginary numbers.

He wrote a book about them in 1545 called “Ars Magna” .

He was the first to actually use imaginary numbers to solve a problem.

At first, he called complex numbers “fictitious”.

Important People

Leonard Euler (1748) introduced the letter i into the world of complex numbers.

Casper Wessel (1799) came up with the graphical representation of complex numbers.

Rene Descartes invented the terms “real” and “imaginary”

Carl Friedrich Gauss (1832) introduced complex numbers. It was through his influence that they became universally accepted.

Uses of imaginary numbers:

The most common purpose of I.N. is the representation of roots of polynomial equations in one variable.

In analysis, it is much quicker and easier if you use imaginary numbers in trig form (polar form).

I.N. opens up vast fields of study from Abstract Algebra to Complex Analysis.

Electrical Engineering

I.N. are used to keep track of amplitude and phase of electrical oscillation. (audio signal, electric voltage, and current that powers electrical appliances.

The state of a circuit element is much better if it is described by one complex number than two real numbers.

More Electrical Engineering

They use complex numbers in analyzing stress/strains on beams of buildings and bridges.

I. N. must be used when electricity flows through devices where no real current can go.

More imaginary numbers than real numbers are used in electrical problems.

Electromagnetic Field

There are electric and magnetic components.

There is a real number describing the intensity of each component.

It is much simpler to use a complex number versus a pair of real numbers.

Quantum Mechanics

A field of physics Helps form the description of

electronic states (fluorescent lights)

Electronic devices (magnetic disk drives)

Chemistry (covalent bonding between atoms)

To calculate where a particle is in space, you must use complex numbers.

More uses of imaginary numbers

Telecommunications (cellular phones)

Radar (assists navigation of planes)

Biology (analysis of firing events from neurons in the brain)

Differential Equations (wavelike functions)

To describe the behavior of electrons.

Physics of electric circuits. Modeling the flow of fluids around

various obstacles.

A few final points:

It is helpful in many real life situations, to be able to get a solution for every polynomial equation.

If we are willing to think about what happens in the set of complex numbers, then it will help us draw conclusions about real world situations.

MATHEMATICS Mathematics is

done by posing problems, creating new notation, and expanding our current number system.

Mathematics is creative, making the impossible, possible!

THE END