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Uses of imaginary numbers

I. What are imaginary numbers?

Calvin & Hobbes Imaginary

(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

The Number System Complex Imaginary Rational

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 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 .

III. 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. 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.”


“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

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.)

Complex imaginary 1

complex imaginary 2

Examples of real world uses:


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




Lectures on the history of physics

Galileo and Einstein: Lectures on the history of physics

Michael Fowler – University of Virginia Physics



Physics of Batman: The Dark Knight

Batman Angular

Let’s assume that the memory fiber used in “The Dark Knight” is real, and that it can be used to change the shape of a cape into gliding wings with the application of an electrical current.  (No such material yet exists, but materials scientists may be getting close.)

Why don’t people use some form of bat wings? Let’s analyze the forces your arms would have to exert in order to successfully use bat wings.


Adapted from “The Physics of Batman: The Dark Knight – High Dive”, Adam Weiner, 08.15.2008

Batman spreads his wings & moves into a circular path.
His motion goes from vertical to horizontal.
The force of air resistance increases dramatically when he opens his wings.
This force turns his linear path into a circular path.
This inward pointing force is a centripetal force.

Law of physics: No object travels in a circular path (Newton’s 1st law), unless some force continually pulls it radially inward.

The balance of inertia and a radially inward force can create circular motion.

Centripetal force depends on the radius of the curve (r) and the radial velocity (v)

F = mv2/r

When a glider – or a Batwing – is bent into the wind, one can use the force to deflect the glider, plane or Batman.

Red arrow to upper right = “lift” (due to the wind hitting the wings)

Red arrow down = weight

Horizontal green arrow is the horizontal component of lift (aka centripetal force)

Vertical green arrow is the vertical component of lift. (If itis big enough, then one can glide for long periods of time)

What about Newton’s 3rd law of motion?

To hold his arms out, Batman has to exert the same force back on the air. So while he moves in a circle, we can calculate the force that will be exerted on Batman’s arms.

circle radius = 20 meters

man + equipment mass = 80 kg

speed remains constant during this turn

Let’s estimate the force on Batman’s arms as he sweeps through the bottom of the arc.

F = weight + centripetal force

F = m g + m v2/r = m ( g + v2/r )

= 80 kg (9.8 m/s2 + [40 m/s]2 /20 m) = 7200 N

= about 1600 pounds

This means that Batman has to hold 800 pounds on each arm!  Imagine lying on your back, on a workout bench, holding your arms out and having 800 pounds of weights placed on each one!  This is probably impossible for someone to do without super-strength.

Perhaps there is a way out of this. Maybe there are some hinges that connect the wings to the Bat suit. If so, then these hinges could be doing some of the supporting, rather than Batman’s arms.

Cartoon Laws of Physics

Cartoon Law I

Any body suspended in space will remain in space until made aware of its situation.

Daffy Duck steps off a cliff, expecting further pastureland. He loiters in midair, soliloquizing flippantly, until he chances to look down. At this point, the familiar principle of 32 feet per second per second takes over.

Cartoon Law II

Any body in motion will tend to remain in motion until solid matter intervenes suddenly.

Whether shot from a cannon or in hot pursuit on foot, cartoon characters are so absolute in their momentum that only a telephone pole or an outsize boulder retards their forward motion absolutely. Sir Isaac Newton called this sudden termination of motion the stooge’s surcease.

Cartoon Law III

Any body passing through solid matter will leave a perforation conforming to its perimeter.

Also called the silhouette of passage, this phenomenon is the speciality of victims of directed-pressure explosions and of reckless cowards who are so eager to escape that they exit directly through the wall of a house, leaving a cookie-cutout-perfect hole. The threat of skunks or matrimony often catalyzes this reaction.

Cartoon Law IV

The time required for an object to fall twenty stories is greater than or equal to the time it takes for whoever knocked it off the ledge to spiral down twenty flights to attempt to capture it unbroken.

Such an object is inevitably priceless, the attempt to capture it inevitably unsuccessful.

Cartoon Law V

All principles of gravity are negated by fear.

Psychic forces are sufficient in most bodies for a shock to propel them directly away from the earth’s surface. A spooky noise or an adversary’s signature sound will induce motion upward, usually to the cradle of a chandelier, a treetop, or the crest of a flagpole. The feet of a character who is running or the wheels of a speeding auto need never touch the ground, especially when in flight.

Cartoon Law VI

As speed increases, objects can be in several places at once.

This is particularly true of tooth-and-claw fights, in which a character’s head may be glimpsed emerging from the cloud of altercation at several places simultaneously. This effect is common as well among bodies that are spinning or being throttled. A ‘wacky’ character has the option of self- replication only at manic high speeds and may ricochet off walls to achieve the velocity required.

Cartoon Law VII

Certain bodies can pass through solid walls painted to resemble tunnel entrances; others cannot.

This trompe l’oeil inconsistency has baffled generations, but at least it is known that whoever paints an entrance on a wall’s surface to trick an opponent will be unable to pursue him into this theoretical space. The painter is flattened against the wall when he attempts to follow into the painting. This is ultimately a problem of art, not of science.

Cartoon Law VIII

Any violent rearrangement of feline matter is impermanent.

Cartoon cats possess even more deaths than the traditional nine lives might comfortably afford. They can be decimated, spliced, splayed, accordion-pleated, spindled, or disassembled, but they cannot be destroyed. After a few moments of blinking self pity, they reinflate, elongate, snap back, or solidify.

Corollary: A cat will assume the shape of its container.

Cartoon Law IX

Everything falls faster than an anvil.

Cartoon Law X

For every vengea nce there is an equal and opposite revengeance.

This is the one law of animated cartoon motion that also applies to the physical world at large. For that reason, we need the relief of watching it happen to a duck instead.

Cartoon Law Amendment A

A sharp object will always propel a character upward.

When poked (usually in the buttocks) with a sharp object (usually a pin), a character will defy gravity by shooting straight up, with great velocity.

Cartoon Law Amendment B

The laws of object permanence are nullified for “cool” characters.

Characters who are intended to be “cool” can make previously nonexistent objects appear from behind their backs at will. For instance, the Road Runner can materialize signs to express himself without speaking.

Cartoon Law Amendment C

Explosive weapons cannot cause fatal injuries.

They merely turn characters temporarily black and smoky.

Cartoon Law Amendment D

Gravity is transmitted by slow-moving waves of large wavelengths.

Their operation can be wittnessed by observing the behavior of a canine suspended over a large vertical drop. Its feet will begin to fall first, causing its legs to stretch. As the wave reaches its torso, that part will begin to fall, causing the neck to stretch. As the head begins to fall, tension is released and the canine will resume its regular proportions until such time as it strikes the ground.

Cartoon Law Amendment E

Dynamite is spontaneously generated in “C-spaces” (spaces in which cartoon laws hold).

The process is analogous to steady-state theories of the universe which postulated that the tensions involved in maintaining a space would cause the creation of hydrogen from nothing. Dynamite quanta are quite large (stick sized) and unstable (lit). Such quanta are attracted to psychic forces generated by feelings of distress in “cool” characters (see Amendment B, which may be a special case of this law), who are able to use said quanta to their advantage. One may imagine C-spaces where all matter and energy result from primal masses of dynamite exploding. A big bang indeed.

© 1997 William Geoffrey Shotts. Last update: Thursday, December 4, 1997


Your task is to build a mousetrap powered car, built from wood, paper, plastic, metal, erector sets, pens, rulers, old toys, Legos, or other materials.

We need a fair comparison between race cars. Therefore it must be powered by only 1 mousetrap. You may not modify the mousetrap, such as by over-winding the metal coil, because that would unfairly increase its potential energy storage.

A rat trap, or trap for any other animal, is not safe or acceptable.

2 people may collaborate to make 1 car.

If you do not have your car on the day that it is due, you lose 5 points per day. After 3 late days the project will not be accepted at all.

I suggest working in groups, making your own local mousetrap racer “factory”. This approach is easier and more fun.

Clearly print your names somewhere on the car!
Tue Sept. 15 – Introduce the project.

Thur Sept. 24 – Bring in your mousetrap racer, even if it is not yet completed. Compare your car with the cars made by others. Test it out, and see what modifications need to be made.

Tue Sept. 29 – Mousetrap racers due today. We will have competitions:

(A) Fastest: Which car goes to the finish line in the shortest amount of time?

(B) Furthest distance: Which car goes the furthest?

Much information on mouse trap racers is available online. However, you may not use a kit to build your racer.

Websites with information on how to make mousetrap cars:



Gallery of great mousetrap racers

What is a mousetrap powered car? How does it work?

It is a vehicle powered by a mousetrap spring. We tie one end of a string to the tip of a mousetrap’s snapper arm, and the other end of the string has a loop that is designed to “catch” a hook that is glued to a drive axle. Once the loop is placed over the axle hook, the string is wound around the drive axle by turning the wheels in the opposite direction to the vehicle intended motion.

As the string is wound around the axle, the lever arm is pulled closer to the drive axle causing the mousetrap’s spring to “wind-up” and store energy. When the drive wheels are released, the string is pulled off the drive axle by the mousetrap, causing the wheels to rotate.

How do you build a mouse trap powered racer?
There is no one “right way” to build a mousetrap powered vehicle. The first step to making a good mouse trap powered car is simple: put something together and find out how it works.

Once you have something working you can begin to isolate the variables that are affecting the performance and learn to adjust to improve your results. You build, you test and experiment, you change, and you do it all over again.

What’s the difference between a FAST Racer and a LONG distance traveler?

When you build a mouse-trap car for distance, you want a small energy consumption per second or a small power usage. Smaller power outputs will produce less wasted energy and have greater efficiency. When you build a vehicle for speed, you want to use your energy quickly or at a high power output. You can change the power ratio of your vehicle by changing one or all of the following:

* where the string attaches to the mouse-trap’s lever arm
* the drive wheel diameter
* the drive axle diameter.

The amount of energy released by using a short lever arm or a long lever arm is the same, but the length of the lever arm will determine the rate at which the energy is released and this is called the power output. Long lever arms decrease the pulling force and power output but increase the pulling distance. Short lever arms increase the pulling force and the power output by decrease the pulling distance but increasing the speed.

If you are building a mouse-trap car for speed, you will want to maximize the power output to a point just before the wheels begin to spin-out on the floor. Maximum power output means more energy is being transferred into energy of motion in a shorter amount of time. Greater acceleration can be achieved by having a short length lever arm and/or by having a small axle to wheel ratio.

If you are building a distance vehicle, you want to minimize the power output or transfer stored energy into energy of motion at a slow rate. This usually means having a long lever arm and a large axle-to-wheel ratio.

If you make the lever arm too long, you may not have enough torque through the entire pulling distance to keep the vehicle moving, in which case you will have to attach the string to a lower point or change the axle-to wheel ratio.


Most parts are scavenged from toys, or recycled materials. You may also consider stores such as Michael’s Art Supply, Home Depot, or A. C. Moore. Mousetraps are available in 2 packs, for less than $2, from all large supermarkets.

Anomalous sounds

Here’s an actual news story: “Loud booms heard across Southern New Hampshire: Source of the noise still unclear.”

Nashua police say they don’t know what caused several loud “booms” Saturday afternoon that were heard across Southern New Hampshire. Many reports came from Nashua and surrounding towns, but the sounds were reported as far north as Manchester and as far south as Westford, Massachusetts. Some who heard it in Nashua said they felt their houses shake. Police and fire departments said they have not been alerted to any incidents related to the noise in the area. The cause is still unclear.
– WMUR 9 News. (An ABC affiliated TV station) 2/10/18

How is it possible that such loud, possibly building shaking sounds could be heard in some parts of this town – yet in other parts of the city other residents reported no sound? Also, in a town next door no reports have yet surfaced of anyone hearing them – yet in a town after that, some residents also reported these booming sound.

The answer? It’s complicated, but basically:

(a) there are a wide variety of ways that sounds are produced – including some bizarre ways that most people have never heard of

(b) Sound waves don’t always move in a straight path like many people imagine; changing temperature/density of the air can cause sound waves to bend and diffract, so:

(b1) sound can sometimes travel much further distances than one would expect

(b2) sound can come from a location very different from what “seems obvious” just by listening

(b3) local wind can mask sound, so the same loud sound might be heard in one neighborhood, yet be undetectable by people just a mile away.

Basic physics idea:

Right off the bat, let’s realize that sound doesn’t move in a straight line: It spreads out radially from it’s source, and then – because of a phenomenon known as diffraction – it can even bend around obstacles.

Diffraction of sound Hyperphysics

Source: Hyperphysics, Diffraction of sound, http://hyperphysics.phy-astr.gsu.edu/


“If the air above the earth is warmer than that at the surface, sound will be bent back downward toward the surface by refraction.” – Hyperphysics

sound refraction Hyperphysics 2


Normally, only sound initially directed toward the listener can be heard, but refraction can bend sound downward – effectively amplifying the sound. This can occur over cool lakes.

sound refraction Hyperphysics 3

Sounds also can bounce off of objects, and come to our ears from a direction totally different than the original source.


ABD Engineering writes:

…wind alters sound propagation by the mechanism of refraction; that is, wind bends sound waves. Wind nearer to the ground moves more slowly than wind at higher altitudes, due to surface characteristics such as hills, trees, and man-made structures that interfere with the wind.

This wind gradient, with faster wind at higher elevation and slower wind at lower elevation causes sound waves to bend downward when they are traveling to a location downwind of the source and to bend upward when traveling toward a location upwind of the source.

Waves bending downward means that a listener standing downwind of the source will hear louder noise levels than the listener standing upwind of the source.

Temperature gradients in the atmosphere. On a typical sunny afternoon, air is warmest near the ground and temperature decreases at higher altitudes. This temperature gradient causes sound waves to refract upward, away from the ground and results in lower noise levels being heard at the listener’s position.

In the evening, this temperature gradient will reverse, resulting in cooler temperatures near the ground. This condition, often referred to is a temperature inversion will cause sound to bend downward toward the ground and results in louder noise levels at the listener position.


How Weather Affects an Outdoor Noise Study by ABD Engineering and Design

Cheung Kai-chung, from Physics World (Hong Kong), (Translation by Janny Leung) offers this explanation

Sound wave will be refracted to the ground when traveling with the wind.

Sound waves refracted 1

Sound wave will be refracted upwards when traveling against the wind.

sound waves refracted 2

Source:   Why can a distant sound be heard easier when it travels with the wind? Why does it become weaker if it travels against the wind?

Can wind mask even loud sounds?

A discussion to consider, from Physics forums, includes this phenomenon: “Yes. I have a freeway about 10 blocks South of my house. I can hear the traffic very clearly with no wind, or a South wind. If there is even a slight North wind, the traffic noise becomes almost inaudible. If there is a brisk North wind (over 15 MPH), the sound is completely gone.”



Sound refraction due to cold air:

Also this “…if the air close to the ground is colder than the air above it then sound waves traveling upwards will be bent downwards. This is called Refraction. These refracted sound waves can act to amplify the sound to someone standing far away.”


Sound seems amplified when traveling over water.

In School-for-Champions we read

“If you are sitting in a boat, a sound coming from the shore will seem louder than the same sound heard by a person on land. Sound seems to be amplified when it travels over water. The reason is that the water cools the air above its surface, which then slows down the sound waves near the surface. This causes refraction or bending of the sound wave, such that more sound reaches the boat passenger. Sound waves skimming the surface of the water can add to the amplification effect, if the water is calm.”


See their full lesson here School-for-champions.com: Sound_amplified_over_water

Can snow on the ground affect sound?

“When the ground has a thick layer of fresh, fluffy snow, sound waves are readily absorbed at the surface of the snow. However, the snow surface can become smooth and hard as it ages or if there have been strong winds. Then the snow surface will actually help reflect sound waves. Sounds seem clearer and travel farther under these circumstances.” – Colorado State Climatologist Nolan Doesken

Related topic: The Hum is a phenomenon, or collection of phenomena, involving widespread reports of a persistent and invasive low-frequency humming,rumbling, or droning noise not audible to all people. Hums have been widely reported by national media in the UK and the United States. The Hum is sometimes prefixed with the name of a locality where the problem has been particularly publicized: e.g., the “Bristol Hum” or the “Taos Hum”. It is unclear whether it is a single phenomenon; different causes have been attributed. ”

Human reactions to infrasound – https://en.wikipedia.org/wiki/Infrasound#Human_reactions

Skyquakes or mystery booms are unexplained reports of a phenomenon that sounds like a cannon or a sonic boom coming from the sky. They have been heard in several locations around the world. – https://en.wikipedia.org/wiki/Skyquake

And: The microwave auditory effect, also known as the microwave hearing effect or the Frey effect, consists of audible clicks (or, with speech modulation, spoken words[citation needed]) induced by pulsed/modulated microwave frequencies. The clicks are generated directly inside the human head without the need of any receiving electronic device. The effect was first reported by persons working in the vicinity of radar transponders during World War II. (Wikipedia)

Find  The Guns of Barisal and Anomalous Sound Propagation


Learning Standards


In this topic we are engaging in skeptical analysis of what some term ‘unexplained phenomenon’.

The Massachusetts STEM Curriculum Framework addresses “Understandings about the Nature of Science”

Scientific inquiry is characterized by a common set of values that include: logical thinking, precision, open-mindedness, objectivity, skepticism, replicability of results, and honest and ethical reporting of findings.

Science disciplines share common rules of evidence used to evaluate explanations about
natural systems. Science includes the process of coordinating patterns of evidence with current theory.

Most scientific knowledge is quite durable but is, in principle, subject to change based on new evidence and/or reinterpretation of existing evidence.

The “College Board Standards for College Success: Science” addresses these same skeptical inquiry methods in Standard SP.1: Scientific Questions and Predictions. Asking scientific questions that can be tested empirically and structuring these questions in the form of testable predictions.

Students recognize, formulate, justify and revise scientific questions that can be addressed by science in order to construct explanations.

Students make and justify predictions concerning natural phenomena. Predictions and justifications are based on observations of the world, on knowledge of the discipline and on empirical evidence.

Students determine which data from a specific investigation can be used as evidence to address a scientific question or to support a prediction or an explanation, and distinguish credible data from noncredible data in terms of quality.

Students construct explanations that are based on observations and measurements of the world, on empirical evidence and on reasoning grounded in the theories, principles and concepts of the discipline.

The “Benchmarks for Science Literacy” (AAAS) addresses these same skeptical inquiry methods:

In science, a new theory rarely gains widespread acceptance until its advocates can show that it is borne out by the evidence, is logically consistent with other principles that are not in question, explains more than its rival theories, and has the potential to lead to new knowledge. 12A/H3** (SFAA)

Scientists value evidence that can be verified, hypotheses that can be tested, and theories that can be used to make predictions. 12A/H4** (SFAA)

Curiosity motivates scientists to ask questions about the world around them and seek answers to those questions. Being open to new ideas motivates scientists to consider ideas that they had not previously considered. Skepticism motivates scientists to question and test their own ideas and those that others propose. 12A/H5*

SAT subject test in Physics: Waves and optics

• General wave properties, such as wave speed, frequency, wavelength, superposition, standing wave diffraction, and Doppler effect



China’s Floating City Mirage

China’s Floating City – Was this a real mirage, a misinterpretation of a reflection, or a hoax?

from “Floating Cities are Generally not Fata Morgana Mirage.” Discussion by Mick West, Oct 20, 2015, on Metabunk.org.

A video is being widely shared on social media (and the “weird news” sections of more traditional media) claiming to show the image of an impossibly large city rising above the fog in the city of Foshan (佛山), Guangdong province, China. Here is a composite image from the video.

Mirage hoax China city

Some have said this is an example of a fata morgana, a type of mirage where light is bent though the atmosphere in such a way to create the illusion of buildings on the horizon.

This is utterly impossible in this case, as fata morgana only creates a very thin strip of such an illusion very close to the horizon, and appears small and far away. It does not create images high in the sky.

Fata Morgana Mirage in Greenland by Jack Stephens

Besides, a fata morgana might create the illusion of buildings by stretching landscape features, or it might distort existing buildings. But what it cannot do it create a perfect image of existing nearby buildings, complete with windows.

China floating city illusion

It is important to note that no expert has actually looked at this video and said it was a fata morgana.

The second and more common type of “floating city” illusions is with buildings that are simply rising up out of clouds or low fog, and hence appear to be floating above them. This has led to “floating city” stories in the past, with this recent example, also from China.

China city in clouds

This is simply a photo of building across the river, but when cropped it appears like they are floating, which led to all kinds of wild stories of “ghost cities”.

This actually came from mistranslations of the original news reports, where local people (who knew exactly what they were looking at) were simply marveling at how pretty the scene looked, with the buildings appearing to float above clouds.

Could the Foshan video be of real buildings obscured by clouds? It does not appear so. Look at some real buildings in Foshan (and keep in mind it’s not entirely clear if Foshan is the actual setting of either the top or the bottom of the video.

Consider what it would take for these buildings to appear like they do in the video, with the road beneath them. The scale is simply impossible. The image has to be composited somehow, and the possibilities are:

  • Computer generated buildings spliced into the video of the road.

  • Two different videos spliced together

  • The video is shot though glass, and the buildings are behind the camera, or to the side (with the glass at around 45°, like a half open window/door)

It’s unfortunate that many people leap for the “fata morgana” or other mirage explanation when it’s quite clear that this is far too high in the sky to be anything like that.

Types floating city illusion hoax





An Introduction to Mirages, Andrew T. Young

Fata Morgana between the Continental Divide and the Missouri River

Learning Standards

2016 Massachusetts Science and Technology/Engineering Curriculum Framework

HS-PS4-3. Evaluate the claims, evidence, and reasoning behind the idea that electromagnetic radiation can be described by either a wave model or a particle model, and that for some situations involving resonance, interference, diffraction, refraction, or the photoelectric effect, one model is more useful than the other.

A Framework for K-12 Science Education: Practices, Crosscutting Concepts, and Core Ideas (2012)

Core Idea PS4: Waves and Their Applications in Technologies for Information Transfer
When a wave passes an object that is small compared with its wavelength, the wave is not much affected; for this reason, some things are too small to see with visible light, which is a wave phenomenon with a limited range of wavelengths corresponding to each color. When a wave meets the surface between two different materials or conditions (e.g., air to water), part of the wave is reflected at that surface and another part continues on, but at a different speed. The change of speed of the wave when passing from one medium to another can cause the wave to change direction or refract. These wave properties are used in many applications (e.g., lenses, seismic probing of Earth).

The wavelength and frequency of a wave are related to one another by the speed of travel of the wave, which depends on the type of wave and the medium through which it is passing. The reflection, refraction, and transmission of waves at an interface between two media can be modeled on the basis of these properties.

All electromagnetic radiation travels through a vacuum at the same speed, called the speed of light. Its speed in any given medium depends on its wavelength and the properties of that medium. At the surface between two media, like any wave, light can be reflected, refracted (its path bent), or absorbed. What occurs depends on properties of the surface and the wavelength of the light.

SAT Subject Area Test in Physics

Waves and optics:

  • Reflection and refraction, such as Snell’s law and changes in wavelength and speed
  • Ray optics, such as image formation using pinholes, mirrors, and lenses


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