## I. What are imaginary numbers?

### (A) Ask your math teacher 😉 That’s a major part of high school math.

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

### Better Explained: A Visual, Intuitive Guide to Imaginary Numbers

## II. Are they “imaginary” or are they real in some sense?

### How can one show that imaginary numbers really do 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 exist*: How 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 .

## III. How are imaginary numbers used?

### I. Alternating current circuits

### “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. In Economics

### “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?

### 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” ( The Math Forum at, National Council of Teachers of Mathematics.)

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 entire universe runs 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, you might want to 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.

## 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 argued[2] 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 about a related, very weird topic, negative probabilities.

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