Abstraction means isolation. Aspects of objects are often abstracted and studied separately from their origins. In philosophy, for example, beauty and virtue are abstracted and studied. In mathematics, quantity, shape and change are abstracted and studied. In computer science, interfaces to objects are an example.

The Clay Mathematics Institute announced seven individual million dollar prizes in 2000 for solutions to any of seven important unsolved mathematics problems. These problems are known as the Millennium Problems.

So far only the Poincare Conjecture has been solved. The winner was a Russian mathematician named Grigori Perelman. He turned down the prize money as well as the Fields Medal for this work!

Pythagoras believed that consonant intervals, or pleasant sounding frequency ratios, are fractions of small numbers such as 3 / 2, 4 / 3, 5 / 3, 5 / 4 and 6 / 5. The just scales are scales that contain these ratios. Unfortunately, they also contain many dissonant or unpleasant sounding intervals. The equal tempered scales have equal intervals between adjacent notes of octaves. The 12 note equal tempered scale is the only equal tempered scale that contains good (less than 1% error) approximations for all the aforementioned small number ratios, and, also contains more consonant intervals than dissonant intervals!

Pressing a piano key causes a hammer to hit a taut string. Struck taut strings vibrate, and, vibrating strings produce sound waves. The first piano key to the left of a properly tuned piano creates a sound wave with a frequency of 27.5 Hz. Moving one key to the right always increases the corresponding sound wave frequency by a factor of 21/12. Notice that moving by 12 keys doubles the frequency. Also notice that because a piano has 88 keys that the highest sound wave frequency is 4186.01 Hz.

The Mars Climate Orbiter was a $327,000,000 space probe launched by NASA on December 11, 1998 to study Martian weather in addition to other objectives.

Software aboard the space probe was designed to use the metric system. Due to human error, it was fed measurements from a different unit system. This led to it being in the wrong place and disintegrating in the Martian atmosphere on September 23, 1999.

The sine, cosine and tangent functions correspond to the coordinates along a unit circle. The hyperbolic versions of these functions correspond to the coordinates along a unit hyperbola x2 - y2 = 1:

The hyperbolic sine is denoted by sinh. The hyperbolic cosine is denoted by cosh. The hyperbolic tangent is denoted by tanh where tanh φ = sinh φ / cosh φ for all φ.

The hyperbolic angle associated with the point (cosh φ, sinh φ) is the region enclosed by the x axis, the unit hyperbola and the line segment through the origin and (cosh φ, sinh φ).

The size of an angle is equal to the area enclosed by twice that angle and a unit circle centered at the vertex. Likewise, the size of the hyperbolic angle φ in the figure above is equal to twice the area enclosed by φ. This is equal to the area of the blue region.

Traditional teachers lecture during class and assign homeworks for after class. In a "flipped classroom", students watch lecture videos before class and do their homeworks during class!

Lecture videos allow students to pause, rewind and replay material as often as desired. They also free up class time for other activities such as personalized tutoring.

Here is a simple procedure to calculate the mass of the Earth:

• Calculate the radius of the Earth as shown here.
• Calculate the time it takes any rock to fall from a known height.
• Calculate M, the mass of the Earth, using M = 2R2h / (Gt2). R is the radius of the Earth, h is the height, t is the elapsed time and G is the gravitational constant 6.673 x 10-11 m3 kg-1 s-2.

The gravitational force on a rock of mass m, on or near the surface of the Earth, is given by Newton's Law Of Universal Gravitation. This force is equal to GMm / R2. Newton's Second Law states that this force is also equal to ma where a is the acceleration of the rock. The kinematic equations for a constant acceleration, which can be found here, can be used to derive a = 2h / t2 for the rock. Therefore, the gravitational force on the rock is also equal to 2mh / t2. Setting GMm / R2 equal to 2mh / t2 and solving for M gives M = 2R2h / (Gt2).

Here is a simple procedure to calculate the radius of the Earth:

• Find someone over 500 km due north or due south of your position that can measure the height and length of the shadow of a vertical stick at noon on some agreed upon day for you.
• On the same day, measure the height and length of the shadow of a vertical stick at your location.
• Calculate R, the radius of the Earth, using R = s / (σ - φ) where:
  • s is the distance between both vertical sticks.
  • σ = Arctan(b / a)
  • φ = Arctan(d / c)
  • a is the height of their vertical stick.
  • b is the length of the shadow of their vertical stick,
  • c is the height of your vertical stick.
  • d is the length of the shadow of your vertical stick.

Finding someone due north or due south of your position and s will likely require a map. σ = Arctan(b / a) because tan σ = b / a. φ = Arctan(d / c) because tan φ = d / c. R = s / (σ - φ) because σ - φ = s / R in radians. The two appearances of σ and φ in the diagram are due to the congruency of alternate interior angles formed from parallel light rays from the sun and the transversals collinear with the vertical sticks.

Eratosthenes performed a similar calculation about 2200 years ago!

Newton's Third Law implies that if any object applies a force on a second object, then the second object will always simultaneously apply an equal, opposite and collinear force of the same type on the first object.

Imagine a stationary box on the ground experiencing a downward gravitational force and an equal and opposite upward normal force. These two forces cannot be a Newton's Third Law pair because each force of a Newton's Third Law pair must operate on different objects.

Imagine two stationary charged balls where the gravitational and electrostatic forces are of equal strength. Each charged ball exerts a gravitational force on the other ball that is equal, opposite and collinear to the electrostatic force it experiences from the other ball. Nevertheless, a gravitational force and an electrostatic force cannot be a Newton's Third Law pair because both forces are not of the same type.

"The highest form of pure thought is in mathematics."

Inelastic collisions are collisions in which the total kinetic energy is not conserved. Perfectly inelastic collisions are inelastic collisions in which the objects stick together.

An engineer, scientist and mathematician were riding on a train through Kansas when they spotted a cow similar to the following:

The engineer said, "Look! All the cows in Kansas are black!".

The scientist then corrected the engineer, "No! Some cows are black."

The mathematician then corrected the scientist, "In Kansas, there is at least one cow, at least one side of which appears to be black."

Write ups of mathematical and scientific work communicate results as well as describe how to repeat research. It is a resource the entire world can benefit from for all time.

The standard format for math and science papers consists of the following:

• title - description of the main points
• abstract - summary of the purpose, results and conclusions
• introduction - description of the purpose and context
• materials and methods - description of the equipment and procedures
• results - presentation of the results
• discussion - discussion of the results
• references - list of the cited articles and books

Titles are typically ten words or less. Abstracts are typically 200 words or less. A common purpose is to test a hypothesis. Context includes existing related knowledge. A guideline for the level of detail to include in the materials and methods section is that someone of similar training and ability should be able to repeat the work. Results sections often include graphs and tables. Discussion sections should include interpretations of results. Discussion sections can also contain suggestions for improvements, suggestions for further work, and, speculations based on the results. Reference sections can be written in many styles. Different journals will require different formats for references.

All pieces of information are answers to questions. And, all pieces of information have sizes. A bigger size implies an answer to a more complex question.

All questions are equivalent to one or more yes/no questions. A more complex question implies more yes/no questions.

The size of information needed to answer a single yes/no question is referred to as a bit. Different information sizes correspond to different numbers of bits.

The Greek alphabet provides even more variable name choices:

Microprocessors are processors on a single integrated circuit. Microcomputers are computers that contain microprocessors. Microcontrollers are computers on a single integrated circuit!

Emmy Noether (1882 - 1935) made fundamental contributions to mathematics and physics. Albert Einstein described her as "the most significant creative mathematical genius thus far produced since the higher education of women began".

Noether's Theorem provides a way to determine the conservation laws of systems from the symmetries of Lagrangians.

One hydrogen atom has a diameter of about a tenth of a nanometer. Therefore, approximately a billion trillion would fit in a cubic millimeter!

Kinematic equations describe positions, velocities and accelerations without forces. The following are the kinematic equations assuming a constant acceleration and one dimension:

• Δx = voΔt + a(Δt)2/2
• Δx=(vo + vf)Δt/2
• Δv = aΔt
• Δv2 = 2aΔx

More dimensions or more complicated accelerations would require different kinematic equations.

There are many reasons some high quality educational materials are free. The author may be hoping the free materials will increase the sales of other formats, other works and/or related services. Also, the materials may have entered the public domain.

Open source materials have additional freedoms such as the ability to create and distribute derivative works for free.

The internet allows many to quickly and easily enjoy the benefits of free and open source digital materials. Imagine the impact high quality free and open source digital educational materials can have on homeschooling and education in the Third World as just two examples.

Mathematics is the science of patterns. Different branches of mathematics study the patterns in different areas.

Diophantus was a Greek mathematician who studied arithmetic. He wrote Arithmetica which greatly influenced the history of arithmetic much like Euclid's Elements greatly influenced the history of geometry.

The Mandelbrot Set is the set of complex numbers z for which the terms of the sequence f(0), f(f(0)), f(f(f(0))), ... are bounded where f(u) = u2 + z.

The plot of the Mandelbrot Set is a fractal :

Computer memories only store naturals. Rationals can be encoded in naturals but not irrationals. Therefore, in general computer arithmetic is inexact. The IEEE 754 standard specifies how computers should implement arithmetic.

Electronic digital computers can be divided into generations based on the following technologies:

• vacuum tubes
• transistors
• integrated circuits
• microprocessors

Symmetry is a property of mathematical objects. Symmetry implies some property remains unchanged after some operation. Mirror symmetry is just one example. Other examples include commutativity, associativity, linearity and periodicity.

Hyperreal numbers include the real numbers as well as numbers that model infinite and infinitesimal quantities.

Fractals are geometric objects with self similar shapes. Self similarity implies any size part is approximately similar to the whole. Examples of fractals in nature include trees, lightning and coastlines.

Success is 99% failure.

Negatives can be replaced by naturals in modular arithmetic:

• -1 = N - 1 (mod N)
• -2 = N - 2 (mod N)
• -3 = N - 3 (mod N)
• ...etc.

This simplifies implementations of negatives, and therefore subtraction, in computers.

Almost identities are "almost true". Here is an example.

Platonists believe mathematical objects actually exist somehow somewhere beyond our universe. They believe mathematics involves making discoveries about these otherworldly objects. Famous Platonists include Plato and Kurt Gödel.

Much, but not not all, of mathematics can be founded on the system known as ZFC. It includes Zermelo–Fraenkel set theory and the axiom of choice.

"I am not discouraged, because every wrong attempt discarded is another step forward."

"I have not failed. I've just found 10,000 ways that won't work."

"Many of life's failures are men who did not realize how close they were to success when they gave up."

"Number is the essence of all things."

Differential equations contain derivatives. Solutions to differential equations are functions rather than numbers. Often only approximate solutions can be found. The following formula is helpful for finding approximate function values: y(b) ≅ y(a) + y'(a) (b - a) for |a - b| << 1. It is also helpful to notice y''(t) = f(t, y(t), y'(t)) is equivalent to the pair: y'(t) = u(t) and u'(t) = f(t, y(t), u(t)). Here are some examples of differential equations and plots of their solutions:

• • x'(t) = -y(t) - x(t)2
  • y'(t) = 2x(t) - y(t)3
  • x(0) = 1
  • y(0) = 1

• • x''(t) = -x(t) / r(t)2 - x2(t) / r2(t)2
  • y''(t) = -y(t) / r(t)2 - y(t) / r2(t)2
  • x(0) = 0
  • y(0) = 4
  • x'(0) = -1
  • y'(0) = 0
  • (r(t)2 = x(t)2 + y(t)2)
  • (x2(t) = x(t) - 2)
  • (r2(t)2 = x2(t)2 + y(t)2)

Identities are equations that are true for all variable values. An example is sin2x + cos2x = 1.

Equality is a property of numbers. Congruency is a property of geometric figures. They are not the same thing.

Weierstrass defined a function that is continuous everywhere and differentiable nowhere. Here it is.

M Theory is a theory of the universe based on objects of various dimensions called "branes". In M Theory, the universe has ten spatial dimensions and one time dimension.

Incommensurable line segments have length ratios that are not rational. No square tiles, no matter how small, can completely tile a rectangular room with incommensurable sides! The discovery of incommensurables was a crisis for the Pythagoreans who believed commensurables were sufficient to describe the world. Legend has it Hippasus was thrown overboard by the Pythagoreans for proving the sides and diagonals of unit squares are incommensurable.

Random Access Machines (RAMs) are computable functions. RAMs consist of registers that contain natural numbers representing instructions and data:

Here are the possible instructions:

• Increment the value in some register.
• Decrement the value in some register.
• Jump to some register if the value in some register is zero.
• Stop.

RAMs execute instructions until a stop instruction is reached. Subsequent instructions operate on subsequent registers unless a jump occurs. Jumps specify alternative registers to operate on next.

Here is a "proof" that 20 + 21 + 22 + ... = -1:

• S = 20 + 21 + 22 + ...
• 2S = 21 + 22 + 23 + ...
• 2S - S = S
• 2S - S = -20
• S = -20
• S = -1
• 20 + 21 + 22 + ... = -1
• QED

Two masses separated by empty space exert gravitational forces on each other. Two charges separated by empty space exert electromagnetic forces on each other. These forces exist although nothing travels through the empty space separating the objects.

How can these forces traverse vacuums? Einstein showed how contortions in space itself communicate gravitational forces between masses. Kaluza and Klein tried to also describe electromagnetism by contortions in space. Although Einstein's theory has been experimentally confirmed, Kaluza-Klein theory has not. This may change in a few years.

Computable functions can be computed by machines or by humans following instructions. Functions that are computable include random access machines, Turing machines, general recursive functions and lambda calculus functions. Not all functions are computable!

Machines, and therefore computable functions, must obey the laws of physics. Communicating processes in terms of computable functions removes ambiguity.

Math is more than doing hand calculations. Is biology more than using a microscope? Is astronomy more than using a telescope? Is writing more than using a word processor?

The set of naturals is denoted by ℕ. The set of integers is denoted by ℤ. The set of rationals is denoted by ℚ. The set of reals is denoted by ℝ.

These symbols are in a style known as blackboard bold. This style of lettering originated from attempts to denote bold letters on chalkboards.

√2 cannot be expressed as a fraction. However, there is a way to denote √2 with two integers using a polynomial equation: x2 - 2 = 0. Irrationals such as √2 that are solutions to polynomial equations containing only integers are referred to as algebraic numbers.

Are there irrationals that are not algebraic? Yes! Two examples are π and e. These numbers are called transcendental numbers.

Imagine you have a choice of gifts: $1000.00 or a some pennies on a chessboard. The pennies are arranged on the squares of the chessboard in a pattern. The first square has 1 penny. The second square has 2 pennies. The third square has 4 pennies, etc. Which would you choose?

The table below shows the number of pennies on each square:

That implies a total sum of 1.84 x 1019 pennies or 184,000 trillion dollars!

The Hindu-Arabic numeral system is the familiar system composed of the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. This numeral system was developed in India around 500 A.D. It is often referred to as the Arabic numeral system because it was introduced to Europe by North African Arabs in the tenth century.

The publication of Fibonacci's Liber Abaci is credited with widely promoting the use of Hindu-Arabic numerals in the West. For centuries previously, Europeans mainly used the relatively clumsy Roman numeral system.

Arithmetic is the study of quantity. Elementary algebra is largely techniques to calculate quantities. Advanced algebra goes beyond calculating quantities. Therefore, algebra is more than arithmetic even though initially it may not appear that way.

The following numbers are prime:

• 31
• 331
• 3331
• 33331
• 333331
• 3333331
• 33333331

But, 333333331 is not prime.

Imagine an equilateral triangle with sides of one unit length:

The length of the path across the base of the triangle is one unit length. The length of the path over the triangle is two unit lengths. The area of the triangle is √3/ 4.

Imagine two equilateral triangles with sides of one half unit length:

The length of the path across the bases of the triangles is one unit length. The length of the path over the triangles is two unit lengths. The total area of the triangles is √3 / 8.

Imagine four equilateral triangles with sides of one quarter unit length:

The length of the path across the bases of the triangles is one unit length. The length of the path over the triangles is two unit lengths. The total area of the triangles is √3/ 16.

Continue this process creating more and more equilateral triangles. The length of the path across the bases of the triangles will always be one unit length. The length of the path over the triangles will always be two unit lengths. The total area of 2n triangles will always be √3 / 2n + 2.

As the number of equilateral triangles increases, the total area of the triangles gets smaller and smaller. And, the path across the bases of the triangles resembles more and more the path over the triangles.

How can this be when one of that paths is always twice as long as the other?

Elementary functions are single variable functions built from ex, ln(x), nth roots and constants by a finite number of compositions, sums, differences, products and quotients. The nth roots include x, x1/2, x1/3, x1/4, ..., etc. The composition of f(x) and g(x) is f(g(x)).

Solutions to equations are not always elementary. For example, the solution to y = x ln(y) for x ≠ 0 is y = -x W(-1 / x). This involves the Lambert W function which does not have an elementary representation. Equations with no elementary solutions often lead to new functions defined to be the solutions to those equations.

The golden ratio φ is believed to be aesthetically pleasing. It has some interesting properties that can be seen here. A rectangle with sides in proportion to φ is called a golden rectangle:

If you remove a square from the interior of a golden rectangle, you get another golden rectangle!:

There are many purported sightings of φ in famous works of art and architecture. The intentionality of many of these sightings is disputed:

Many see φ in other places too:

A Mathematician's Lament by Paul Lockhart is a brilliant essay on what he thinks is wrong with mathematics education today. Here it is.

Arithmetic is the study of quantity. The word arithmetic is the Greek word for number.

Geometry is the study of space. The word geometry comes from "geo" and "metria", the Greek words for Earth and measurement.

Algebra is the study of operations and relations. The word algebra comes from "al-jabr", the Arabic word for reunion.

Calculus is the study of change. The word calculus is the Latin word for counting stone.

"Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry."

"Mathematics is the language with which God wrote the Universe."

QED is an acronym for "quod erat demonstrandum". This Latin phrase means "which was to be demonstrated". It is traditionally added to the end of proofs.

The product of two negatives is a positive. Rules such as this one are chosen. For example, it is possible to chose the rule that the product of two negatives gives a negative. It is even possible to chose the rule that the product of two negatives is always 1792!

What is the general criteria for selecting which rules to adopt? Beauty, simplicity and utility are some considerations. Here is an argument for the product of two negatives giving a positive:

• Assume a < 0 and b < 0.
• a (b + (-b)) = 0 is desirable.
• Therefore, ab = - (a(-b)) is desirable.
• -(a(-b)) > 0
• Therefore, ab > 0 is also desirable.

ln(ab) = ln(a) + ln(b) for a, b > 0. This implies a property for the area under the plot of 1 / t that is not obvious. Here is the proof.

The meaning of ba is clear when a and b are rational. It is even understandable, in terms of limits, when one of them is irrational. What about when a and b are both irrational? For all reals a and b, exp(a ln(b)) is a useful definition. ln(x) is the area under the plot of 1 / t from t = 1 to t = x for x > 0. exp(x) is the inverse of ln(x).

"[The] beautiful system of the sun, planets and comets could only proceed from the counsel and dominion of an intelligent and powerful Being."

"Atheism is so senseless and odious to mankind that it never had many professors."

"There is one God the Father ever-living, omnipresent, omniscient, almighty, the maker of heaven and earth, and one Mediator between God and Man the Man Christ Jesus."

"We account the Scriptures of God to be the most sublime philosophy. I find more sure marks of authenticity in the Bible than in any profane history whatever."

Does rearranging the pieces causes some area to disappear?

Here is a "proof" that 2x = x:

• x2 = x(x)
• x2 = x + x + x + ... + x
• (x2)' = (x + x + x + ... + x)'
• 2x = 1 + 1 + 1 + ... + 1
• 2x = x
• QED

Here is a proof that √2 cannot be written as a fraction and hence is irrational:

• Every rational has a reduced form.
• A reduced form cannot have two evens.
• Every square of an odd is an odd.
• Every square of an even is an even.
• If m / n is the reduced form of √2, then m2 = 2n2.
• That implies m2 is even.
• That implies m is even.
• That implies m = 2k for some integer k.
• That implies n2 = 2k2 for some integer k.
• That implies n2 is even.
• That implies n is even.
• m and n cannot both be even.
• m / n does not exist.
• √2 cannot be written as a fraction.
• QED

Rationals are numbers that can be written as fractions. There are an infinite number of rationals between any two rationals. All those rationals cannot describe all possible lengths. Some lengths, such as √2, are not rational. They are irrational. An irrational can be specified by giving a function that determines whether any rational is bigger or smaller.

From algebra, we know that we must do the same thing to both sides of an equals sign. Following that rule, here is the "proof" that if x = 1, then x = 0:

• x = 1
• x2 = x
• x2 - 1 = x - 1
• (x + 1)(x - 1) = (x - 1)
• x + 1 = 1
• x = 0
• QED

An intriguing teaching method is the "Moore Method" by Robert L. Moore. Moore was a distinguished mathematician who taught mathematics at the University of Texas for many decades. He believed the right way to learn mathematics was for everyone to develop it for themselves. Moore did not lecture. He would introduce a few definitions then assign various problems. His role was like a coach. Books and collaboration were banned. Many variations on the Moore Method exist today. Some classes allow one book. Some incorporate tiny lectures.