Tag Archives: Math

Educational Videos (Cauchy-Schwartz Inequality, Binomial Theorem, Hermitian Matrices)

I feel that there is not a large enough presence of proof-based mathematics on YouTube. I would like to try to change that.

Here are a couple new ones I made (Hermitian and Unitary matrix video [not proof based], and Cauchy-Schwartz Inequality) and my most beloved video (Binomial Theorem Proof).

Cauchy-Schwartz Inequality Proof using Inner Product and Complex Analysis

Continue reading Educational Videos (Cauchy-Schwartz Inequality, Binomial Theorem, Hermitian Matrices)

Elementary Proofs in Geometry, a Math Writing Assignment

I am taking a math writing course and our first assignment was to create an essay from lecture notes. This is what I came up with.

Also, a nice read on writing mathematics is found at this pdf: http://homepages.math.uic.edu/~kauffman/SuGuidelines.pdf

 

Elementary Proofs in Geometry Using Features of Pi

Pi is an extraordinarily beautiful number in mathematics. One of pi’s most amusing features is that the measurement of a straight angle is pi radians. Similarly, the sum of the interior angles of a triangle is also equal to pi radians. We seek to prove these facts and use them to determine if parallel lines can ever meet.

First, we will prove that all vertical angles are congruent. After proving this theorem, we will show that parallel lines never meet. To prove this theorem, we will give a definition of what it means for two lines to be parallel and then use the theorem that the sum of the interior angles of a triangle is equal to a straight angle to prove it.

Theorem 1: Vertical angles are congruent.

To prove that vertical angles are congruent we will utilize two axioms.

Continue reading Elementary Proofs in Geometry, a Math Writing Assignment

Paul Erdös, The Man Who Loves Numbers

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I have recently learned about the life of Paul Erdös.

Paul Erdös was a highly eccentric mathematician whose existence could be defined by the pursuit of math.

Here are some excerpts from a review of the book: The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth :

Click here for full article.

“Erdös first did mathematics at the age of three, but for the last twenty-five years of his life, since the death of his mother, he put in nineteen-hour days, keeping himself fortified with 10 to 20 milligrams of Benzedrine or Ritalin, strong espresso, and caffeine tablets. “A mathematician,” Erdös was fond of saying, “is a machine for turning coffee into theorems.” When friends urged him to slow down, he always had the same response: “There’ll be plenty of time to rest in the grave.”

“To communicate with Erdös you had to learn his language. “When we met,” said Martin Gardner, the mathematical essayist, “his first question was `When did you arrive?’ I looked at my watch, but Graham whispered to me that it was Erdös’s way of asking, `When were you born?'” Erdös often asked the same question another way: “When did the misfortune of birth overtake you?” His language had a special vocabulary–not just “the SF” and “epsilon” but also “bosses” (women), “slaves” (men), “captured” (married), “liberated” (divorced), “recaptured” (remarried), “noise” (music), “poison” (alcohol), “preaching” (giving a mathematics lecture), “Sam” (the United States), and “Joe” (the Soviet Union). When he said someone had “died,” Erdös meant that the person had stopped doing mathematics. When he said someone had “left,” the person had died.”


 

Some YouTube videos that catch a glimpse of the man’s eccentricity:
Note: SF or “Supreme Fascist” refers to “God” or “the higher power”.

Paul Erdös is introducing himself:

Paul Erdös – SF means Supreme Facist

Paul Erdös – The Purpose of Life

Proof that 2=1… or not

Federico Pistono posted about the Wall Street Journal’s opinion piece on science proving the existence of God… or not, so I am going to post (in similar fashion) 2=1… or not.

Many involved with mathematics have already heard of this fallacious proof, however it is an eloquent one and I wanted to share it:

Objective: To prove that all numbers equal each other, namely, 2=1.

Proof

1. Let a and b be equal non-zero numbers.
a=b

2. Multiply both sides by a
a^2=ab

3. Add a^2-2ab to both sides
a^2+a^2-2ab=ab+a^2-2ab

4. Factor:
2(a^2-ab)=a^2-ab

5. Divide both sides by a^2-ab
2=1

Ah, brilliant, we have just proved step-by-step that 2 is equal to 1. Using these same steps we can prove that any number is equal to each other, right?

Well, obviously, there is something deeply wrong about this proof. Try to consider what might be wrong for a moment, it is a fun challenge. When you’ve done that, continue reading.

Lame_Math_Joke_by_ichiko_wind_griffin

The problem in this fallacious proof lies in step 5. In step 1 we stated that a=b, so dividing by a^2-ab would mean we would be dividing by 0. Strange things happen when you divide by 0, and this result is one of them. Therefore, this proof is invalid and 2 does not equal 1. Mathematics is still fundamentally sound and all can remain at peace.

Math is Beautiful

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Mathematics has always been beautiful to me. It has a certain purity in it when explaining the universe that allows no misunderstandings or objections, only truth. In my studies of mathematical series, I’ve finally came across Euler’s Formula and Euler’s Identity. The results of these concepts are truly profound and ought to be understood by every human being on this planet. For some reason, they are hidden away behind complex mathematics, but the logic behind it is rather simple. I wonder how education would change if concepts like these were presented to younger minds from the get-go. Would more people be interested in mathematics and thus, science? I digress…

Euler’s Formula
Euler’s formula can be derived:
1. Using the Maclaurin Series of cos(x) and sin(x).

cos(x) = 1 - x^2/(2!)+x^4/(4!)-x^6/(6!)+x^8/(8!)-x^10/(10!)...

sin(x) = x - x^3/(3!)+x^5/(5!)-x^7/(7!)+x^9/(9!)-x^11/(11!)...

2. Using the Maclaurin Series of e^x

e^x approx 1 + x^2/(2!)+x^3/(3!)+x^4/(4!)+x^5/(5!)+x^6/(6!)...

3. Using the imaginary unit for Maclaurin Series of e^ix

e^ix approx 1 + (ix)^2/(2!)+(ix)^3/(3!)+(ix)^4/(4!)+(ix)^5/(5!)+(ix)^6/(6!)...

By applying the rules of imaginary unit, i, the Maclaurin Series of e^ix can be rearranged to form…

e^ix approx (1 - x^2/(2!)+x^4/(4!)-x^6/(6!)+x^8/(8!)...) + i(x-x^3/(3!)+x^5/(5!)-x^7/(7!)+x^9/(9!)...)

4. Therefore, e^ix = cos(x)+isin(x)

This has plenty of further uses in math, but taking a ‘poetic’ dive into math, let us plug in x=pi

5. e^ix with x=pi

e^ix=cos(pi)+isin(pi)

e^ix=-1

e^ix+1=0

This is Euler’s Identity.

What is profound about this? Well, you are relating Euler’s number (e), an important mathematical constant in finance (namely compound interest), with pi which relates a circle’s circumference to its diameter, with imaginary unit i, which is commonly used in engineering disciplines to find roots of polynomials. You relate these three unique constants of mathematics with the two most simple numbers of one and zero. You are connecting vast areas of life together in one beautiful and simple mathematic equation.

If this doesn’t make you feel emotional in any way, then I don’t know what will. This tells us that the universe, in some way, is connected.

The Mysteries of Benford’s Law

Benford’s Law, in the most elementary form of understanding, states that the number “1” transpires as the leading digit 30% of the time compared to higher digits such as 9 which occurs 5% of the time. This occurs for all kinds of data sets ranging from electricity bills, street addresses, stock prices, to even physical and mathematical constants. Yes, that’s right, the physical and mathematical constants of the universe follow this mysterious law.

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This graph showcases the percentage to be expected based on the results of Benford’s Law. The number 1 representing 30% frequency rate as the leading digit to the number 9 representing a mere 4.6%. Even more strange, the percentage decrease in order from 1 to 9.

 

The law has been used in court cases to detect fraud based on the ‘plausible’ assumption that people who make up numbers evenly distribute them. It has been used to detect election/voting frauds, fraudulent macroeconomic data, and even scientific fraud.

Continue reading The Mysteries of Benford’s Law