Irrationals like √2 cannot be be represented by fractions. However, there is a way to represent √2 with rationals using polynomials: √2 = "that number x such that x2 = 2".

Do there exist irrationals that transcend even rational polynomials? Yes! Two examples are π and e. These numbers are called transcendental numbers. Irrationals such as √2 that can be expressed with rational polynomials are referred to as algebraic numbers.

Imagine you have a choice of gifts: $1000.00 or a special chessboard of pennies. The first chessboard square has 1 penny. The second chessboard square has 2 pennies. The third chessboard square has 4 pennies. This pattern continues for all 64 squares. Which would you choose?

The table below shows the amount of pennies on each checkerboard square:

That implies a total sum of 1.84 x 1019 pennies or 184 quadrillion dollars!

Math practitioners today universally use the Hindu-Arabic numeral system. This is the familiar system composed of the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.

The Hindu-Arabic 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, Europe mainly used the relatively clumsy Roman numeral system.

Arithmetic is the study of quantity. Elementary algebra is largely just techniques to calculate quantities. So is algebra a part of arithmetic? Advanced algebra includes areas 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.

Here is a "proof" that 1 = 2. 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 equillateral triangles with sides of one half unit length:

The length of the path across the base 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 equillateral triangles with sides of one quarter unit length:

The length of the path across the base 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 base 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

...etc.

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

Therefore, 1 = 2. QED

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, ... . The composition of f(x) and g(x) is f(g(x)). Trigonometric functions are also included because they can be defined in terms of elementary functions.

Solutions to equations are not always elementary. For example, the solution to x = a ln(x) for a ≠ 0 is x = -a W(-1/a). 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. If this seems a little hokey, keep in mind that many elementary functions also came about because there was no simpler way to express the solutions to certain problems. For example, sin(x), cos(x) and tan(x) were invented because there was no simpler way to define the ratios of the lengths of right triangles.

The Golden Ratio φ has some interesting properties that can be seen here.

Architects, painters and musicians have tried to incorporate φ into their work for centuries. A rectangle with sides in proportion to φ is called a Golden Rectangle. It is believed to be aesthetically pleasing:

If you remove a square from the interior of a Golden Rectangle, you get another Golden Rectangle!

There are many famous purported examples of φ in famous works of art and architecture. The intentionality of many of these φ sightings is disputed by some. Many see surprising examples of φ in other places too:

Mathematician and educator Paul Lockhart has written a brilliant essay on what he thinks is wrong with mathematics education today. Here it is if you are interested.

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 relations. The word algebra comes from "al-jabr", the Arabic word for reunion. This is referring to the technique of "reuniting" terms in an equation on opposite sides of the equals sign. Algebra introduces variables to focus on relations rather than numbers.

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

"Mathematics is the language with which God wrote the Universe." -- Galileo Galilei

"Mathematics, rightly viewed, possesses not only truth, but supreme 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." -- Bertrand Russell

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 gives a positive. Where did this rule come from? Rules such as these 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 aesthetic 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 > 0 and b > 0. This property says something about the area under the plot of 1 / t that isn't obvious. Here is the proof.

The meaning of ba is clear when a and b are integers or rationals. It is even understandable, in terms of limits, when only one of them is irrational. What about when a and b are both irrational? ba is defined for all reals a and b as exp(a ln(b)). 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).

Where did exp(a ln(b)) come from? Suppose m(x) = rx and n(x) = logr(x) for some rational base r > 0. Then the following properties must be true for m(x), n(x) and all rationals:

• m(x) and n(x) are inverses.
• n(xy)) = y n(x)
• m(x + y) = m(x) m(y)
• m(1) = r
• m(0) = 1

These imply the following:

• xy = m(y n(x))
• m(-x) = 1 / m(x)
• n(x y) = n(x) + n(y)
• n(x / y) = n(x) - n(y)
• n(r) = 1
• n(1) = 0

Notice how xy = m(y n(x)) was implied by m(x) = rx, n(x) = logr(x) and a few properties. By analogy, imagine trying to define xy as f(y g(x)) for all reals where f(x) = sx and g(x) = logs(x) for some irrational base s > 0. Ideally, definitions for f(x) and g(x) would be found where the aforementioned properties would hold for all reals. f(x) = exp(x) and g(x) = ln(x) satisfy all of these conditions for s = e!

"[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."

Many years ago I proved a number of basic calculus theorems such as the Mean Value Theorem, the Extreme Value Theorem and the Fundamental Theorem of Calculus for continuous functions . They were intended for my eyes only so some parts may require an explanation: "wrt" stands for "with respect to", "s.t." stands for "such that" and "etc." is used to represent trivial details. Here are the proofs if you are interested.

Does rearranging the pieces causes some area to disappear?

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 2x = x where is x some real:

• 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.
• Therefore, m / n does not exist.
• QED

Rationals are numbers that can be written as a fraction. There are an infinite number of rationals between any two rationals.

All those rationals aren't sufficient to describe all possible lengths. Some lengths, such as √2, are not rational. They are irrational.

An irrational can be specified by giving a function, over rationals, that returns whether the given 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

Paradoxes and weird results make mathematics fun. Here are a few zingers you may enjoy:

• The Halting Theorem
• The Hilbert Hotel
• Gödel's Theorem
• Aperiodic Penrose Tilings

You're probably used to the lecture model of teaching where an expert talks at you for an hour or so.

There are alternative promising ways of teaching. An intriguing one is the "Moore Method" by Robert L. Moore.

Robert L. 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 inspired classes will allow one book. Some incorporate only tiny lectures.

A fascinating current area of research is the effectiveness of the Moore Method for online teaching incorporating free open source computer algebra systems such as Sage.

In college I once got this idea to redo all the homework for the entire quarter three times. This ended up being my first A+ in college! The final was easy!

By redoing all the homework multiple times, I memorized the steps to solve most types of problems. I could recall the steps quickly and easily. That was the reason for my success.

You can use this technique too. Do lots of problems. Do extra problems above what is assigned in class. Do them multiple times before tests.

Your brain needs lots of exposure to develop deep roots. Repetition. Repetition. Repetition. Repetition is the key.

I am a big fan of inspirational movies and intriguing documentaries. Here is my short list of some of the best:

• Stand And Deliver - This is the true story of an AP Calculus teacher at an inner city L.A. high school in the 1980s.
• Dead Poet's Society - This is a story about all that life can be. It inspires students about their education and life.
• Fermat's Last Theorem - This is a documentary about real mathematicians in the real world. It puts a human face to a subject that may seem dry and lifeless to some.
• Dangerous Knowledge - This is a documentary that explores the lives and works of some amazing mathematicians and scientists.

Word problems can cause a lot of grief. Here is my recipe I've developed after years of experience:

1. Make a picture of what is going on.
2. Write down everything you are given.
3. Write down what you are looking for.
4. Only then try to solve the problem.
5. Don't plug in numbers until the very end.