Imaginary Numbers Historyand Practical Applications Practical Applications

Definition of Imaginary Numbers: b of or having to do with the even root of a negative number or any expression involving such a root.

HISTORY b There has always been a natural progression of numbers: natural, negative, rational, irrational, imaginary. b Ancient Greeks were disturbed by the thought of irrational numbers (the hypotenuse of isosceles right triangles) b Mathematicians for a long time were unwilling to accept that solutions to equations could be a number less than zero. b Many times we have had to change our beliefs.

The term imaginary b All numbers in math are imaginary in the sense that they are only in our minds. b The word “imaginary” is unfortunate. b Imaginary numbers do correspond to reality, but not in the simple, intuitive sense that whole numbers did.

Why the word “imaginary”? b 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. b They eventually realized that the term “I” was a good idea, and the term imaginary never was changed. b 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? b Mathematicians could not find a solution to x^2 + 1 = 0 b 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 b The earliest record of the square root of negative numbers appears in Stereometrica by Heron of Alexandria. (AD50) sqrt (81-144) b In India in 850, Mahavira wrote “As in nature of things, a negative is not a square, it has no square root.” b Until 1500’s, mathematicians were puzzled by the square root of a negative number.

Girolamo Cardano b Cardano was one of the first to work with imaginary numbers. b He wrote a book about them in 1545 called “Ars Magna”. b He was the first to actually use imaginary numbers to solve a problem. b At first, he called complex numbers “fictitious”.

Important People b Leonard Euler (1748) introduced the letter i into the world of complex numbers. b Casper Wessel (1799) came up with the graphical representation of complex numbers. b Rene Descartes invented the terms “real” and “imaginary” b Carl Friedrich Gauss (1832) introduced complex numbers. It was through his influence that they became universally accepted.

Uses of imaginary numbers: b The most common purpose of I.N. is the representation of roots of polynomial equations in one variable. b In analysis, it is much quicker and easier if you use imaginary numbers in trig form (polar form). b I.N. opens up vast fields of study from Abstract Algebra to Complex Analysis.

Electrical Engineering b I.N. are used to keep track of amplitude and phase of electrical oscillation. (audio signal, electric voltage, and current that powers electrical appliances. b The state of a circuit element is much better if it is described by one complex number than two real numbers.

More Electrical Engineering b They use complex numbers in analyzing stress/strains on beams of buildings and bridges. b I. N. must be used when electricity flows through devices where no real current can go. b More imaginary numbers than real numbers are used in electrical problems.

Electromagnetic Field b There are electric and magnetic components. b There is a real number describing the intensity of each component. b It is much simpler to use a complex number versus a pair of real numbers.

Quantum Mechanics b A field of physics b Helps form the description of electronic states (fluorescent lights) b Electronic devices (magnetic disk drives) b Chemistry (covalent bonding between atoms) b To calculate where a particle is in space, you must use complex numbers.

More uses of imaginary numbers b Telecommunications (cellular phones) b Radar (assists navigation of planes) b Biology (analysis of firing events from neurons in the brain) b Differential Equations (wavelike functions) b To describe the behavior of electrons. b Physics of electric circuits. b Modeling the flow of fluids around various obstacles.

A few final points: b It is helpful in many real life situations, to be able to get a solution for every polynomial equation. b 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 b Mathematics is done by posing problems, creating new notation, and expanding our current number system. b Mathematics is creative, making the impossible, possible!

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