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# Uses of imaginary numbers

## Are they “real” in some sense?

### 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

from “Imaginary Numbers are Real,” Welch labs

## How are imaginary numbers used?

### “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. ”

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### 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.”

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

### III. In Economics

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

and

## 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?

Image from Imaginary Numbers Are Real, Welch Labs

## V. The universe seems to be based on complex numbers

### 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.

### 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

## VI. Negative Probabilities

### 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