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# Category Archives: Physics

## How to teach AP physics

It’s easy to teach physics in a wordy and complicated way – but taking a concept and breaking it down into simple steps, and presenting ideas in a way that are easily comprehensible to the eager student, is more challenging.

Yet that is what Nobel prize winning physicist Richard Feynman excelled at. The same skills that made one a good teacher also caused one to more fully understand the topic him/herself. This was Feynman’s basic method of learning.

1) Develop an array of hands-on labs that allow one to study basic phenomenon.

You can also use many wonderful online simulations, such as PhET or Physics Aviary.

2) Each day go over several problems in class. They need to see a master teacher take what appears to be a complex word problem, and turn it into equations.

3.) Insure that students take good notes. One way of doing this is having the occasional surprise graded notebook check (say, twice per month.)

4) Each week assign homework. Each day randomly call a few students to put one of their solutions on the board. Recall that the goal is not to get the correct numerical answer. (That sometime can come by luck or cheating.) Focus on the derivation. Does the student understand which basic principles are involved?

5) Keep track of strengths and weaknesses: Is there a weakness in algebra, trigonometry, or geometry? When you see a pattern emerge, assign problem sets that require mastering the weak area – not to punish them, but to build skills. Start with a few very easy problems, and slowly build in complexity. Let them work in groups if you like.

6) Don’t drown yourself in paperwork: Don’t grade every problem, from every student, every day. You could easily work 24 hours a day and still have more work to do. Only collect & grade some percent of the homework.

7) Focus on simple drawings – or for classes that uses programming to simulate physics phenomenon – simple animations. Are the students capable of sketching free-body diagrams that strip away extraneous info? Can they diagram out all the forces on an object?

8) Give frequent assessments that are easy to grade.

9) Get books such as TIPERS for Physics, or Ranking Task Exercises in Physics. They are diagnostic tools to check for misconceptions.. Call publishers for free sample textbooks and resources. For a textbook I happen to like Giancoli Physics; their teacher solution manual is very well thought out.

## Ferris wheel physics

A Ferris wheel is a large structure consisting of a rotating upright wheel, with multiple passenger cars. The cars are attached to the rim in such a way that as the wheel turns, they are kept upright by gravity.

The original Ferris Wheel was designed and constructed by George Washington Gale Ferris Jr. as a landmark for the 1893 World’s Columbian Exposition in Chicago. The generic term Ferris wheel is now used for all such structures, which have become the most common type of amusement ride at state fairs in the United States.

## Forces in the wheel itself

The wheel keeps its circular shape by the tension of the spokes, pulling upward against the lower half of the framework and downward against the huge axle.

Also see

## Classical relativity

This animation shows simultaneous views of a ball tossed up and then caught by a ferris wheel rider from one inertial and two non-inertial points of view. Although Newton’s predictions are easier to track from the inertial point of view, it turns out that they still work locally in accelerated frames and curved spacetime if we consider “geometric accelerations and forces” that act on every ounce of an object’s being and can be made to disappear by a suitable vantage point change.

Created by P. Fraundorf, licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.

## Net work done on the wheel = 0?

https://www.physicsforums.com/threads/ferris-wheel-work-done-by-net-force.715905/

## External resources

https://www.real-world-physics-problems.com/ferris-wheel-physics.html

https://physics.stackexchange.com/questions/205918/centripetal-force-on-a-ferris-wheel

How products are made: http://www.madehow.com/Volume-6/Ferris-Wheel.html

AP Physics problems: Ferris wheels and rotational motion

Build A Big Wheel, by Try Engineering, Lesson plan

AP Physics problem solving

http://faculty.washington.edu/boynton/114AWinter08/LectureNotes/Le8.pdf

## Learning Standards

TBA

## How records work

How record work (private for now)

https://kaiserscience.wordpress.com/physics/waves/how-records-work/

## Miniaturization

### Many sci-fi stories depend upon a technology called miniaturization. Isaac Asimov’s classic Fantastic Voyage; his more scientifically rigourous sequel, Fantastic Voyage II; DC Comics featuring The Atom, Marvel Comics featuring Antman and The Wasp.

### Is miniaturization real? Could it be real? What would be the results if it was real?

## Miniaturization in fiction

### 1940’s movie – Dr. Cyclops. People are reduced to less than a foot in size by the titular mad scientist, and are subjugated to his whims.

### 1957 movie – The Incredible Shrinking Man inspired a boom in science fiction films that made use of size-alteration.

### 1961 the Silver Age comic book character, the Atom, Dr. Ray Palmer, created by DC Comics.

### 1960’s Ant-Man is a Marvel Comics superhero.

### 1966 movie – Fantastic Voyage

### 1987 movie Innerspace

### 1989 movie, Honey, I Shrunk the Kids.

## What would happen if we just compressed someone to a small size?

### Neil Degrasse Tyson probably shouldn’t write any more comics 😉

### Details: TBA

## Physics: How would one try to do this?

### There are no practical ways to actually do this. However, science fiction stories speculate on how this could be done. Interestingly, sustained thought and speculation on science fiction technologies has allowed scientists to develop real-world technologies.

### A. Compression / increasing density

“Why are you so certain miniaturization is impossible?”

“If you reduce a man to the dimensions of a fly, then all the mass of a man would be crowded into the volume of a fly. You’d end up with a density of something like -” he paused to think – “a hundred and fifty thousand times that of platinum. ”

### B. Removing atoms

From Fantastic Voyage II:

“But what if the mass were reduced in proportion?” – “Then you end up with one atom in the miniaturized man for every three million in the original. The miniaturized man would not only have the size of a fly but the brainpower of a fly as well. ”

### C. Changing Planck’s constant

### This is a major science-plot point in Isaac Asmimov’s book, Fantastic Voyage II (1988)

From Fantastic Voyage II

“And if the atoms are reduced, too?”

“If it is miniaturized atoms you are speaking of, then Planck’s constant, which is an absolutely fundamental quantity in our Universe, forbids it. Miniaturized atoms would be too small to fit into the graininess of the Universe. ”

“And if I told you that Planck’s constant was reduced as well, so that a miniaturized man would be encased in a field in which the graininess of the Universe was incredibly finer than it is under normal conditions?”

“Then I wouldn’t believe you. ”

“Without examining the matter? You would refuse to believe it as a result of preconceived convictions, as your colleagues refuse to believe you?”

And at this, Morrison was, for a moment, silent….

…Well over half an hour had passed before Morrison felt convinced that the objects he could see outside the ship were shrinking and were receding perceptibly toward their normal size.

Morrison said, “I am thinking of a paradox.”

“What’s that?” said Kalinin, yawning. She had obviously taken her own advice about the advisability of relaxing.

“The objects outside the ship seemed to grow larger as we shrink. Ought not the wavelengths of light outside the ship also grow larger, becoming longer in wavelength, as we shrink? Should we not see everything outside turn reddish, since there can scarcely be enough ultraviolet outside to expand and replace the shorter-wave visible light?”

Kalinin said, “If you could see the light waves outside, that would indeed be how they would appear to you. But you don’t. You see the light waves only after they’ve entered the ship and impinged upon your retina. And as they enter the ship, they come under the influence of the miniaturization field and automatically shrink in wavelength, so that you see those wavelengths inside the ship exactly as you would see them outside.”

“If they shrink in wavelength, they must gain energy.”

“Yes, if Planck’s constant were the same size inside the miniaturization field as it is outside. But Planck’s constant decreases inside the miniaturization field — that is the essence of miniaturization. The wavelengths, in shrinking, maintain their relationship to the shrunken Planck’s constant and do not gain energy. An analogous case is that of the atoms. They also shrink and yet the interrelationships among atoms and among the subatomic particles that make them up remain the same to us inside the ship as they would seem to us outside the ship.”

“But gravity changes. It becomes weaker in here.”

“The strong interaction and the electroweak interaction come under the umbrella of the quantum theory. They depend on Planck’s constant. As for gravitation?” Kalinin shrugged. “Despite two centuries of effort, gravitation has never been quantized. Frankly, I think the gravitational change with miniaturization is evidence enough that gravitation cannot be quanitzed, that it is fundamentally nonquantum in nature.”

“I can’t believe that,” said Morrison. “Two centuries of failure can merely mean we haven’t managed to get deep enough into the problem yet. Superstring theory nearly gave us out unified field at last.” (It relieved him to discuss the matter. Surely he couldn’t do so if his brain were heating in the least.)

“Nearly doesn’t count,” said Kalinin. “Still, Shapirov aagreed with you, I think. It was his notion that once we tied Planck’s constant to the speed of light, we would not only have the practical effect of miniaturizing and deminiaturizing in an essentially energy-free manner, but that we would have the theoretical effect of being able to work out the connection between quantum theory and relativity and finally have a good unified field theory. And probably a simpler one than we could have imagined possible, he sould say.”

“Maybe,” said Morrison. He didn’t know enough to comment beyond that.

### D. Nanotechnology as a practical replacement for miniaturization of large objects

### “…The ideas and concepts behind nanoscience and nanotechnology started with a talk entitled “There’s Plenty of Room at the Bottom” by physicist Richard Feynman at an American Physical Society meeting at the California Institute of Technology (CalTech) on December 29, 1959, long before the term nanotechnology was used. In his talk, Feynman described a process in which scientists would be able to manipulate and control individual atoms and molecules. Over a decade later, in his explorations of ultraprecision machining, Professor Norio Taniguchi coined the term nanotechnology. It wasn’t until 1981, with the development of the scanning tunneling microscope that could “see” individual atoms, that modern nanotechnology began.”

Nano.gov What is nanotechnology?

## References

Miniaturization: Technovelgy article

Excerpt from Fantastic Voyage: II A Novel By Isaac Asimov, 1988

## Learning Standards

**Next Generation Science Standards: Science & Engineering Practices**

● Ask questions that arise from careful observation of phenomena, or unexpected results, to clarify and/or seek additional information.

● Ask questions that arise from examining models or a theory, to clarify and/or seek additional information and relationships.

● Ask questions to determine relationships, including quantitative relationships, between independent and dependent variables.

● Ask questions to clarify and refine a model, an explanation, or an engineering problem.

● Evaluate a question to determine if it is testable and relevant.

● Ask questions that can be investigated within the scope of the school laboratory, research facilities, or field (e.g., outdoor environment) with available resources and, when appropriate, frame a hypothesis based on a model or theory.

● Ask and/or evaluate questions that challenge the premise(s) of an argument, the interpretation of a data set, or the suitability of the design

**MA 2016 Science and technology**

Appendix I Science and Engineering Practices Progression Matrix

Science and engineering practices include the skills necessary to engage in scientific inquiry and engineering design. It is necessary to teach these so students develop an understanding and facility with the practices in appropriate contexts. The Framework for K-12 Science Education (NRC, 2012) identifies eight essential science and engineering practices:

1. Asking questions (for science) and defining problems (for engineering).

2. Developing and using models.

3. Planning and carrying out investigations.

4. Analyzing and interpreting data.

5. Using mathematics and computational thinking.

6. Constructing explanations (for science) and designing solutions (for engineering).

7. Engaging in argument from evidence.

8. Obtaining, evaluating, and communicating information.

Scientific inquiry and engineering design are dynamic and complex processes. Each requires engaging in a range of science and engineering practices to analyze and understand the natural and designed world. They are not defined by a linear, step-by-step approach. While students may learn and engage in distinct practices through their education, they should have periodic opportunities at each grade level to experience the holistic and dynamic processes represented below and described in the subsequent two pages… http://www.doe.mass.edu/frameworks/scitech/2016-04.pdf

## Facts and Fiction of the Schumann Resonance

Excerpted from Facts and Fiction of the Schumann Resonance, by Brian Dunning, Skeptoid Podcast #352

It’s increasingly hard to find a web page dedicated to the sales of alternative medicine products or New Age spirituality that does not cite the Schumann resonances as proof that some product or service is rooted in science. … Today we’re going to see what the Schumann resonances actually are, how they formed and what they do, and see if we can determine whether they are, in fact, related to human health.

In physics, Schumann resonances are the name given to the resonant frequency of the Earth’s atmosphere, between the surface and the densest part of the ionosphere.

They’re named for the German physicist Winfried Otto Schumann (1888-1974) who worked briefly in the United States after WWII, and predicted that the Earth’s atmosphere would resonate certain electromagnetic frequencies.

[What is a resonant frequency? Here is a common example. When you blow on a glass bottle at a certain frequency, you can get the bottle to vibrate at the same frequency]

This bottle has a resonant frequency of about 196 Hz.

That’s the frequency of sound waves that most efficiently bounce back and forth between the sides of the bottle, at the speed of sound, propagating via the air molecules.

Electromagnetic radiation – like light, and radio waves – is similar, except the waves travel at the speed of light, and do not require a medium like air molecules.

The speed of light is a lot faster than the speed of sound, but the electromagnetic waves have a lot further to go between the ground and the ionosphere than do the sound waves between the sides of the bottle.

This atmospheric electromagnetic resonant frequency is 7.83 Hz, which is near the bottom of the ELF frequency range, or Extremely Low Frequency.

The atmosphere has its own radio equivalent of someone blowing across the top of the bottle: lightning.

Lightning is constantly flashing all around the world, many times per second; and each bolt is a radio source. This means our atmosphere is continuously resonating with a radio frequency of 7.83 Hz, along with progressively weaker harmonics at 14.3, 20.8, 27.3 and 33.8 Hz.

These are the Schumann resonances. It’s nothing to do with the Earth itself, or with life, or with any spiritual phenomenon; it’s merely an artifact of the physical dimensions of the space between the surface of the Earth and the ionosphere.

Every planet and moon that has an ionosphere has its own set of Schumann resonances defined by the planet’s size.

Biggest point: this resonated radio from lightning is a vanishingly small component of the electromagnetic spectrum to which we’re all naturally exposed.

The overwhelming source is the sun, blasting the Earth with infrared, visible light, and ultraviolet radiation. All natural sources from outer space, and even radioactive decay of naturally occurring elements on Earth, produce wide-spectrum radio noise. Those resonating in the Schumann cavity are only a tiny, tiny part of the spectrum.

Nevertheless, because the Schumann resonance frequencies are defined by the dimensions of the Earth, many New Age proponents and alternative medicine advocates have come to regard 7.83 Hz as some sort of Mother Earth frequency, asserting the belief that it’s related to life on Earth.

The most pervasive of all the popular fictions surrounding the Schumann resonance is that it is correlated with the health of the human body.

There are a huge number of products and services sold to enhance health or mood, citing the Schumann resonance as the foundational science.

A notable example is the Power Balance bracelets. Tom O’Dowd, formerly the Australian distributor, said that the mylar hologram resonated at 7.83 Hz.

When the bracelet was placed within the body’s natural energy field, the resonance would [supposedly] “reset” your energy field to that frequency.

Well, there were a lot of problems with that claim.

First of all, 7.83 Hz has a wavelength of about 38,000 kilometers. This is about the circumference of the Earth, which is why its atmospheric cavity resonates at that frequency. 38,000 kilometers is WAY bigger than a bracelet! There’s no way that something that tiny could resonate such an enormous wavelength. O’Dowd’s sales pitch was implausible, by a factor of billions, to anyone who understood resonance.

This same fact also applies to the human body. Human beings are so small, relative to a radio wavelength of 38,000 kilometers, that there’s no way our anatomy could detect or interact with such a radio signal in any way.

Proponents of binaural beats cite the Schumann frequency as well. These are audio recordings which combine two slightly offset frequencies to produce a third phantom beat frequency that is perceived from the interference of the two.

Some claim to change your brain’s encephalogram, which they say is a beneficial thing to do. Brain waves range from near zero up to about 100 Hz during normal activity, with a typical reading near the lower end of the scale. This happens to overlap 7.83 — suggesting the aforementioned pseudoscientific connection between humans and the Schumann resonance — but with a critical difference. An audio recording is audio, not radio. It’s the physical oscillation of air molecules, not the propagation of electromagnetic waves. The two have virtually nothing to do with each other.

[Other salespeople claim] that our bodies’ energy fields need to interact with the Schumann resonance, but can’t because of all the interference from modern society [and so they try to sell devices that supposedly connect our body to the Schumann resonance.]

It’s all complete and utter nonsense. Human bodies do not have an energy field: in fact there’s not even any such thing as an energy field. Fields are constructs in which some direction or intensity is measured at every point: gravity, wind, magnetism, some expression of energy. Energy is just a measurement; *it doesn’t exist on its own as a cloud or a field or some other entity. The notion that frequencies can interact with the body’s energy field is, as the saying goes, so wrong it’s not even wrong*.

Another really common New Age misconception about the Schumann resonance is that it is the resonant frequency of the Earth. But there’s no reason to expect the Earth’s electromagnetic resonant frequency to bear any similarity to the Schumann resonance.

Furthermore, the Earth probably doesn’t even have a resonant electromagnetic frequency. Each of the Earth’s many layers is a very poor conductor of radio; combined all together, the Earth easily absorbs just about every frequency it’s exposed to. If you’ve ever noticed that your car radio cuts out when you drive through a tunnel, you’ve seen an example of this.

Now the Earth does, of course, conduct low-frequency waves of other types. Earthquakes are the prime example of this. The Earth’s various layers propagate seismic waves differently, but all quite well. Seismic waves are shockwaves, a physical oscillation of the medium. Like audio waves, *these are unrelated to electromagnetic radio waves*.

Each and every major structure within the Earth — such as a mass of rock within a continent, a particular layer of magma, etc. — does have its own resonant frequency for seismic shockwaves, but there is (definitively) no resonant electromagnetic frequency for the Earth as a whole.

So our major point today is that you should be very skeptical of any product that uses the Schumann resonance as part of a sales pitch.

The Earth does not have any particular frequency. Life on Earth is neither dependent upon, nor enhanced by, any specific frequency.

Source: skeptoid.com/episodes/4352

## Resonance

Resonance: When a vibrating system drives another system to oscillate with greater amplitude at specific frequencies.

from Physicsclassroom.com:

Musical instruments are set into vibrational motion at their natural frequency when a person hits, strikes, strums, plucks or somehow disturbs the object.

Each natural frequency of the object is associated with one of the many standing wave patterns by which that object could vibrate. The natural frequencies of a musical instrument are sometimes referred to as the harmonics of the instrument.

An instrument can be forced into vibrating at one of its harmonics (with one of its standing wave patterns) if another interconnected object pushes it with one of those frequencies. This is known as resonance – when one object vibrating at the same natural frequency of a second object forces that second object into vibrational motion.

Physics Classroom – Sounds – Lesson 5 – Resonance

(work in progress)

RedGrittyBrick, a physicist writing on skeptics.stackexchange.com, notes that a bridge can be susceptible to mechanical resonance:

Mechanical structures usually have one or more frequencies at which some part of the structure oscillates. A tuning fork has a well-defined natural frequency of oscillation. More complex structures may have a dominant natural frequency of oscillation. If some mechanical inputs (such as the pressure of feet walking in unison) have a frequency that is close to a natural frequency of the structure, these inputs will tend to initiate and, over a short time, increase the oscillating movements of the structure. Like pushing a child’s swing at the right time.

One example is London’s Millennium Bridge which was closed shortly after opening because low-frequency vibrations in the bridge were causing large groups of pedestrians to simultaneously shift their weight and reinforcing the oscillation. Dampers were fitted.

Related topics

Nikola Tesla and wireless power transmission

### Related topics

Facts and Fiction of the Schumann Resonance: On this website

### External links

Facts and Fiction of the Schumann Resonance : On Skeptoid

Learning Standards

**2016 Massachusetts Science and Technology/Engineering Curriculum Framework**

HS-PS4-5. Communicate technical information about how some technological devices use the principles of wave behavior and wave interactions with matter to transmit and capture information and energy. Examples of principles of wave behavior include resonance, photoelectric effect, and constructive and destructive interference.

## Uses of imaginary numbers

## 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