Follow Your Passion in Math

Grow your passion by motivating practice like the 11-year-old Math Marvel

“Chess is a mathematical poem” – Aarushi Maheshwari 

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Mathematical Beauty

e^iπ + 1 = 0

Euler’s identity is often cited as an example of deep mathematical beauty.

Constance Reid: Euler’s identity is “the most famous formula in all mathematics”.

Euler’s Identity is a special case of his formula from complex analysis suggested by Swiss mathematician Euler in  1748.

The identity links five fundamental mathematical constants:

• The number 0
• The number 1
• The number π (π = 3.141 . . .)
• The number e (e = 2.718 . . .)
• The number i, the imaginary unit of the complex numbers.

Euler’s Identity “is absolutely paradoxical; we cannot understand it, and we don’t know what it means, but we have proved it, and therefore we know it must be the truth”. – Benjamin Peirce.

The Largest Known Prime Number

The largest known prime number was discovered in 2016 by a computer laboratory of the University of Central Missouri .

2^57,885,161 – 1

is one of the Mersenne primes that can be written in form 2ⁿ – 1, where n is an integer.  They are named after 17-century French mathematician Marin Mersenne.

The number written in the decimal form contains 17 million digits.

Why Is X The Unknown?

Terry Moore explains that the inventors of algebra were Arabic scientists. Their definition:

X = something, some undefined unknown thing = the unknown thing.

When the books were translated from Arabic to Spanish the Arabic unknown was “SH” that did not exist inSpanish. They replaced it by “X“.  It migrated to other European languages 600 years ago.

X is “X” because Spanish language does not have “SH“.

Selfie Math

Jim is going on a tour around the world. He has 5 tops, 4 bottoms, and 3 pairs of footwear. He wants to post a selfie everyday to show different places and his different outfits to his girl-friend Mary, who stays at home. He does not want to wear the same outfit twice. ## For how many days does he have enough clothing?

Leslie Green gives the answer : 96 days.

Can you explain why it is 96? What is logic behind?

 

 

Guesstimate Methods

Guesstimate is an estimate based on a mixture of guesswork and calculation.

“Everything should be made as simple as possible, but not simpler.” – Albert Einstein

“The purpose of computing is insight, not numbers.” – Richard Hamming

Guesstimate Methods and Examples from Igor Kokcharov, Ph.D, PMP

Famous Math and Logic Paradoxes

Math and Logic are full of paradoxes.

1. Achilles and the Tortoise The Paradox of Achilles and the Tortoise was described by the Greek philosopher Zeno of Elea in the 5th century BC. The great hero Achilles challenges a tortoise to a footrace. He agrees to give the tortoise a head start of 100m. When the race begins, Achilles starts running, so that by the time he has reached the 100m mark, the tortoise has only walked 10m. But by the time Achilles has reached the 110m mark, the tortoise has walked another 1m. By the time he has reached the 111m mark, the tortoise has walked another 0.1m, then 0.01m, then 0.001m, and so on. The tortoise always moves forwards while Achilles always plays catch up.  Why is Achilles always behind the tortoise?

2. Bermuda Triangle Paradox. Why the sum of the interior angles of the Bermuda triangle is not 180 degrees?

3. Simpson Paradox. The average score for dance of boys and girls in class A are 16 and 21, respectively.  The average score of boys and girls in class B are 15 and 20, respectively.  Twenty percent of class A students are girls. Forty percent of class B students are girls.  Which class has a higher average score?

4.  Braess paradox. The diagram shows a road network. All cars drive in one direction from A to F. The numbers represent the maximum flow rate in vehicles per hour. Engineers want to construct a new road with a flow rate of 100 vehicles per hour. Drivers randomly choose the road at  crossroads. What new road decreases the capacity of the network (the number of vehicles at point F)?

5. Barber Paradox  In a city, the barber is the ‘one who shaves all those, and those only, who do not shave themselves.’
Who shaves the barber?

6. The Two Envelope Paradox. One envelope has twice as much money as the second one. Gerry does not know which envelope contains the larger  amount. He takes one of the envelopes, counts the money, and is offered the chance to switch the envelope. He thinks “If the amount of money in the chosen envelope is X dollars, then the other envelope contains either 2X of 0.5X dollars, with equal probability of 0.5. The expected value of switching is  0.5 (2X) + 0.5 (0.5X) = 1.25X. This is greater than the value in the initially chosen envelope.  It is better to switch.”  What is your advice?

7. Potato Paradox. I have 100kg of potatoes, which are 99 percent water. I dry them until they are 98 percent water.  How much do they weigh now?

8.  Leonard Euler’s Paradox.   Why the average of all  of the numbers is not a zero?

1, -1, 2, -2, 3, -3, . . .

9. Friendship Paradox.  Your friends have more friends than you. Why?

10. Uninteresting Number Paradox. How many uninteresting numbers exist?

11. Gabriel’ Horn Paradox. The shape obtained from rotating the equation about x-axis resembles a trumpet. If we need an infinite volume of paint to paint the infinite horn, how much paint does the horn can contain inside itself?

12. Pop Quiz Paradox. A teacher announces that there will be a quiz one day during the next week. The teacher gives the definition that they would not when they come in to the class that the quiz was going to be given that day. The brightest student says that the quiz cannot be on Friday because they will know the day. With the same technique, she eliminates Thursday, Wednesday, Tuesday, and Monday. “You cannot give us a pop quiz next week” she says. When does the teacher give the pop quiz?  I know the paradox from Charles Carter Wald. Probably, Martin Gardner described it for the first time in The Colossal Book of Mathematics.

 

Famous Math and Logic Paradoxes from Igor Kokcharov, Ph.D, PMP

Answers

1. Achilles and the Tortoise

2. Bermuda Triangle Paradox

3. Simpson Paradox

4.  Braess paradox

5. Barber Paradox

6. The Two Envelope Paradox

7. Potato Paradox

8.  Leonard Euler’s Paradox

9. Friendship Paradox

10. Uninteresting Number Paradox

11. Gabriel’s Horn Paradox

12. Pop Quiz Paradox