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Uses of imaginary numbers

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What are imaginary numbers?

Calvin & Hobbes Imaginary

(A) Ask your math teacher 😉

(B) See Ask Dr. Math: What is an imaginary number? What is i?

Better Explained: A Visual, Intuitive Guide to Imaginary Numbers

The Number System Complex Imaginary Rational

Are they “real” in some sense?

In what sense are imaginary numbers just as real as “real” numbers? We should start by being aware that people used to say the same thing about fractions! People argued that either something is a number or it isn’t – how can one possibly have part of a number?

Later, people said the same thing about irrational numbers. And for quite a long time, people said the same thing about the number zero – people argued that there couldn’t possible be a number without value.

Yet today everyone agrees that fractions, irrational numbers, and zero are all “real.” How it possible that people didn’t “believe in” those numbers before, but they do now? Because today, from the very beginning of public school, we introduce students to these numbers and show how they all work together in a well-defined and useful system (“math”.)

The same could be true for imaginary numbers – what if we showed how imaginary numbers filled in a gap in our math system? When we see how imaginary numbers fill in this gap, then we realize that they work together in the same well-defined, useful system (“math”.)

Here’s one way of doing this: Consider this function 𝑓(𝑥) = 𝑥2 + 1

Here is this function’s plot in the real x-y plane:

plotting parabola on axis

Now according to the Fundamental Theorem of Algebra we should have n-roots for n-th degree polynomial. Yet when we consider the graph for this function it doesn’t appear to intersect the x-axis right.

Well, the thing is, we are not seeing it correctly and have not included a fundamental set of numbers : Complex Numbers which have both real and imaginary part but don’t get confused yet as both the parts are quite real.

The below GIF plots the the function in the complex plane The vertical axis that comes out of the paper is the imaginary axis, NOT the Z-axis.

from math.stackexchange

Plotting imaginary numbers GIF 2

from “Imaginary Numbers are Real,” Welch labs

How can one show that imaginary numbers exist? In the same way that one would show that fractions exist. First, let’s first show that fractions exist.

Of course, that’s something you know already, but the point is that exactly the same argument shows that imaginary numbers existHow can one show that imaginary numbers really do exist? Univ. of Toronto, Philip Spencer

Here’s a great video showing how imaginary numbers can be thought of as just as real as other numbers: Imaginary numbers are not some wild invention, they are the deep and natural result of extending our number system. Welch Labs .

How are imaginary numbers used?

I. Alternating current circuits

AC generator Wire through magnet

“The handling of the impedance of an AC circuit with multiple components quickly becomes unmanageable if sines and cosines are used to represent the voltages and currents.”

“A mathematical construct which eases the difficulty is the use of complex exponential functions. ”

.

II. Engineering – damped oscillators

Many objects have simple harmonic motion, aka oscillation. Objects move back and forth, and the “pull back” force is related to how far the object is pulled from the center.

This motion doesn’t last forever. Due to friction, the motion slowly dampens, or dies away, over time. This is called damped oscillation.

There are mechanical vibrations in any structures, such as bridges, overpasses, tunnel walls, and floors of shopping malls and buildings. Here’s a practical example of a problem that requires imaginary numbers in math to produce an engineering solution:

“An existing mid-rise office building included a gymnasium on the second floor. Floors above the gym level were occupied as offices by different tenants. Vibration complaints were reported by the tenants on the fourth floor at two different locations.

In essence, vibrations generated at second floor were traveling up through the columns and producing unacceptable vibrations at the fourth floor. The task was to verify the reported vibration complaints analytically, and then propose vibration mitigation measures.”

GIF Floor Vibration damped oscillation

Vertical vibration transmission from a gym, Floor Vibration Expert, Boston, MA

Here is an (exaggerated) analysis of how oscillation in bridge structures.

London's Millennium Bridge resonance

The same is true for studying a plucked violin or guitar string,

Violin string vibration

And of course the same kind of analysis is used for studying damped oscillations in car shock absorbers, pendulums, bungee jumping, etc.

The engineering of any of these involves equations that use imaginary numbers.

See Real World Example: Oscillating Springs (Math Warehouse)

III. In Economics

Economics calculator

Image from St. Lawrence University, Mathematics-Economics Combined Major

“Complex numbers and complex analysis do show up in Economic research. For example, many models imply some difference-equation in state variables such as capital, and solving these for stationary states can require complex analysis.”

and

“The application of complex numbers had been attempyed in the past by various economists, especially for explaining economic dynamics and business fluctuations in economic system In facr, the cue was taken from electrical systems. Ossicilations in economic activity level gets represented by sinosidual curves The concept of Keynesian multiplier and the concept of accelerator were combined in models to trace the path of economic variables like income, employment etc over time. This is where complex numbers come in.”
{By sensekonomikx, Yahoo Answers, Complex numbers in Economics?}

IV. Why use imaginary math for real numbers?

Electrical engineers and economists study real world objects and get real world answers, yet they use complex functions with imaginary numbers. Couldn’t we just use “regular” math?

Welch Labs plotting imaginary

Image from Imaginary Numbers Are Real, Welch Labs

Answer:
Imaginary numbers transform complex equations in the real X-Y axis into simpler functions in the “imaginary” plane.

 This lets us transform complicated problems into simpler ones.

Here is an explanation from “Ask Dr. Math” ( Math forum at National Council of Teachers of Mathematics.)

Complex imaginary 1

complex imaginary 2

Also

We sometimes just use imaginary numbers because they can be easier to use:  Engineers and physicists use the complex exponential 𝑒𝑗𝜔𝑡 instead of sines and cosines.

Why? This  notation makes differential equations much easier to deal with. This relates to why we use imaginary numbers when studying electrical impedance. A more detailed explanation is here:

Why is impedance represented as a complex number rather than a vector?

Examples of real world uses

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/impcom.html

Careers That Use Complex Numbers, by Stephanie Dube Dwilson

Imaginary numbers in real life: Ask Dr. Math

Imaginary numbers, Myron Berg, Dickinson State Univ.

V. The universe seems to be based on complex numbers

If we look only at things in our everyday life – objects with masses larger than atoms, and moving at speeds far lower than the speed of light – then we can pretend that the entire word is made of solid objects (particles) following more or less “common sense” rules – the classical laws of physics.

But there’s so much more to our universe – and when we look carefully, we find that literally all of our classical laws of physics are only approximations of a more general, and often bizarre law – the laws of quantum mechanics. And QM laws follow a math that uses complex numbers!  When you have time, look at our intro to the development of QM and at deeper, high school level look at what QM really is .

Scott Aaronson writes about a central, hard to believe feature of quantum mechanics:

“Nature is described not by probabilities (which are always nonnegative), but by numbers called amplitudes that can be positive, negative, or even complex.”

He points out that this weird reality seems to be a basic feature of the universe itself
“This transformation is just a mirror reversal of the plane. That is, it takes a two-dimensional Flatland creature and flips it over like a pancake, sending its heart to the other side of its two-dimensional body.

But how do you apply half of a mirror reversal without leaving the plane? You can’t! If you want to flip a pancake by a continuous motion, then you need to go into … dum dum dum … THE THIRD DIMENSION.

More generally, if you want to flip over an N-dimensional object by a continuous motion, then you need to go into the (N+1)st dimension.

But what if you want every linear transformation to have a square root in the same number of dimensions? Well, in that case, you have to allow complex numbers. So that’s one reason God might have made the choice She did.”

– PHYS771 Quantum Computing Since Democritus, Lecture 9: Quantum. Aaronson is Professor of Computer Science at The University of Texas at Austin.

Imaginary Numbers May Be Essential for Describing Reality

A new thought experiment indicates that quantum mechanics doesn’t work without strange numbers that turn negative when squared.

Charlie Wood, Quanta Magazine , 3/3/2021

A group of quantum theorists designed an experiment whose outcome depends on whether nature has an imaginary side. Provided that quantum mechanics is correct — an assumption few would quibble with — the team’s argument essentially guarantees that complex numbers are an unavoidable part of our description of the physical universe.

“These complex numbers, usually they’re just a convenient tool, but here it turns out that they really have some physical meaning,” said Tamás Vértesi, a physicist at the Institute for Nuclear Research at the Hungarian Academy of Sciences who, years ago, argued the opposite. “The world is such that it really requires these complex” numbers, he said.

 

VI. Negative Probabilities

In 1942, Paul Dirac wrote a paper “The Physical Interpretation of Quantum Mechanics” where he introduced the concept of negative energies and negative probabilities: “Negative energies and probabilities should not be considered as nonsense. They are well-defined concepts mathematically, like a negative of money.”

The idea of negative probabilities later received increased attention in physics and particularly in quantum mechanics. Richard Feynman argue that no one objects to using negative numbers in calculations: although “minus three apples” is not a valid concept in real life, negative money is valid. Similarly he argued how negative probabilities as well as probabilities above unity possibly could be useful in probability calculations.

  • Wikipedia, Negative Probabilities, 3/18

John Baez ( mathematical physicist at U. C. Riverside in California) writes

The physicists Dirac and Feynman, both bold when it came to new mathematical ideas, both said we should think about negative probabilities. What would it mean to say something had a negative chance of happening?

I haven’t seen many attempts to make sense of this idea… or even work with this idea. Sometimes in math it’s good to temporarily put aside making sense of ideas and just see if you can develop rules to consistently work with them. For example: the square root of -1. People had to get good at using it before they understood what it really was: a rotation by a quarter turn in the plane. Here’s an interesting attempt to work with negative probabilities:

Gábor J. Székely, Half of a coin: negative probabilities, Wilmott Magazine (July 2005), p.66–68

He uses rigorous mathematics to study something that sounds absurd: half a coin. Suppose you make a bet with an ordinary fair coin, where you get 1 dollar if it comes up heads and 0 dollars if it comes up tails. Next, suppose you want this bet to be the same as making two bets involving two separate ‘half coins’. Then you can do it if a half coin has infinitely many sides numbered 0,1,2,3, etc., and you win n dollars when side number n comes up….

… and if the probability of side n coming up obeys a special formula…

and if this probability can be negative whenever n is even!

This seems very bizarre, but the math is solid, even if the problem of interpreting it may drive you insane.

By the way, it’s worth remembering that for a long time mathematicians believed that negative numbers made no sense. As late as 1758 the British mathematician Francis Maseres claimed that negative numbers “… darken the very whole doctrines of the equations and make dark of the things which are in their nature excessively obvious and simple.”

So opinions on these things can change. By the way: experts on probability theory will like Székely’s use of ‘probability generating functions’. Experts on generating functions and combinatorics will like how the probabilities for the different sides of the half-coin coming up involve the Catalan numbers.

Learning standards

Massachusetts Mathematics Curriculum Framework 2017

Number and Quantity Content Standards: The Complex Number System

A. Perform arithmetic operations with complex numbers.

B. Represent complex numbers and their operations on the complex plane.

C. Use complex numbers in polynomial identities and equations.

Common Core Mathematics

High School: Number and Quantity » The Complex Number System

Resources

https://www.mathwarehouse.com/algebra/complex-number/real-world-example-scillating-springs-explained.php

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