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Irrationals, Transcendentals, & Infinity

Some questions to explore:

  • It has been suggested earlier that the set of prime numbers is infinite.  Primes are a subset of integers.
  • Integers are a subset of Rationals, which are a subset of Reals, and then on to Complex and beyond.
  • If the integers are infinite, and if the real numbers are infinite, are there the same number of both?
  • What about rational numbers?  What about irrational numbers? Are those two sets equal in size?
  • What is an algebraic number?  What is a transcendental number?  How large are those sets?
  • Are all of these infinities equal?  How many infinities are there?  Who thought all of this up?  Are there any other approaches?

A few hints to start with, from Wikipedia:

In mathematics, a transcendental number is a real or complex number that is not algebraic—that is, it is not a root of a nonzero polynomial equation with integer (or, equivalently, rational) coefficients. The best-known transcendental numbers are π and e. Though only a few classes of transcendental numbers are known (in part because it can be extremely difficult to show that a given number is transcendental), transcendental numbers are not rare. Indeed, almost all real and complex numbers are transcendental, since the algebraic numbers are countable while the sets of real and complex numbers are both uncountable. All real transcendental numbers are irrational, since all rational numbers are algebraic. The converse is not true: not all irrational numbers are transcendental; e.g., the square root of 2 is irrational but not a transcendental number, since it is a solution of the polynomial equation x2 − 2 = 0. Another irrational number that is not transcendental is the golden ratio, φ or ϕ, since it is a solution of the polynomial equation x2x − 1 = 0.

cantor dust


Fibonacci sequence the Golden Ratio

The Fibonacci sequence is the sequence defined by:

F0=0, F1=1, Fn=Fn1+Fn2,  for  n=2,3,4,....F_0 = 0,  F_1 = 1 ,  F_n = F_{n−1 }+ F_{n−2} \textrm{,  for }  n = 2, 3, 4, . . . .


First 15 sequence values:



The number

φ=1+52=1.61803... φ = \frac{1 + \sqrt{5}} {2} = 1.61803 . . .


is called the Golden Ratio. It satisfies the property

φ=1+1φ φ = 1 + \frac{1}{φ}


Can you show that the sequence of ratio of successive Fibonacci numbers

Fn+1Fn \frac{F_{n+1}}{F_n}


converges to the Golden Ratio?

limnFn+1Fn=φ \lim_{n\to\infty} \frac{F_{n+1}}{F_n} = φ


Observe that

F14/F13=377/2331.618025 F_{14} /F_{13} = 377/233 ≈ 1.618025


is already pretty close to φ.

What conditions must be met for this ratio to converge?

Fibonacci Spiral

Extension: Missing Squares

An elegant variation of this idea makes use of two rectangles of such proportion that they fit side by side to make a perfect 8 by 8 checkerboard. When the pieces are rearranged to make the larger rectangle, there is an apparent gain in area of one square unit.  Where did the extra square come from, and what does this have to do with Fibonacci?

fibonacci squares

Pythagorean Triples

Pythagorean triples are sets of three integers that are solutions to the associated identity for right triangles.

The question is this:

Can you find a formula or algorithm for generating Pythagorean Triples?

There may be more than one formula, and each might not be exhaustive. All we are after is a set of rules such that, given one number, can you find the other two that belong to the triplet?

pythagorean theorem

Games & Strategy

Game Theory, as described on Wikipedia, is:

the study of mathematical models of conflict and cooperation between intelligent rational decision-makers.

A classic example is the Prisoner’s Dilemma:

Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of communicating with the other. The prosecutors lack sufficient evidence to convict the pair on the principal charge, but they have enough to convict both on a lesser charge. Simultaneously, the prosecutors offer each prisoner a bargain. Each prisoner is given the opportunity either to betray the other by testifying that the other committed the crime, or to cooperate with the other by remaining silent. The offer is:

  • If A and B each betray the other, each of them serves two years in prison
  • If A betrays B but B remains silent, A will be set free and B will serve three years in prison (and vice versa)
  • If A and B both remain silent, both of them will only serve one year in prison (on the lesser charge).

Prisoner's Dilemma

Probability of Coprimality

This one is rather easy to pose, and can be difficult to solve, even with a proof sitting right in front of you. The result is a bit surprising as well, and gives some insight into the structure of the integers and prime numbers.

Given two random integers, what is the probability that they have no common factors besides 1?

or, how it is usually posed:

What is the probability of two random integers being coprime?

Where “coprime” means “the greatest common divisor is one”.

coprime number distribution

Blue Dot

There is an island with 100 women. 50 of the women have red dots on their foreheads, and the other 50 women have blue dots on their foreheads.

If a woman ever learns the color of the dot on her forehead, she must permanently leave the island in the middle of that night.

One day, an oracle appears and says “at least one woman has a blue dot on her forehead.” The woman all know that the oracle speaks the truth.

All the woman are perfect logicians (and know that the others are pefect logicians too). What happens next?


Related Problem:

Three Masters of Logic wanted to find out who was the wisest amongst them. So they turned to their Grand Master, asking to resolve their dispute. “Easy,” the old sage said. “I will blindfold you and paint either red, or blue dot on each man’s forehead. When I take your blindfolds off, if you see at least one red dot, raise your hand. The one, who guesses the color of the dot on his forehead first, wins.” And so it was said, and so it was done. The Grand Master blindfolded the three contestants and painted red dots on every one. When he took their blindfolds off, all three men raised their hands as the rules required, and sat in silence pondering. Finally, one of them said: “I have a red dot on my forehead.”

How did he guess?

Ten Envelopes

A man told his son that he would give him $1000 if he could accomplish the following task. The father gave his son ten envelopes and a thousand dollars, all in one dollar bills. He told his son, “Place the money in the envelopes in such a manner that no matter what number of dollars I ask for, you can give me one or more of the envelopes, containing the exact amount I asked for without having to open any of the envelopes. If you can do this, you will keep the $1000.” When the father asked for a sum of money, the son was able to give him envelopes containing the exact amount of money asked for. How did the son distribute the money among the ten envelopes?

envelope full of one dollar bills
Envelope Dollar E Mail Money Bill Gift Christmas

Ten Digit Number

Is there a 10-digit number where the first digit is equal to how many 0’s are in the number, the second digit is equal to how many 1’s are in the number, the third digit is equal to how many 2’s are in the number, all the way up to the last digit, which is equal to how many 9’s are in the number?

ten digit number grid

Common Fractions & Decimal Equivalents

Step 1:  Use long division (no calculators please!) to convert the fractions (in this case, specifically reciprocal integers) to decimal representations.

Step 2:  Look at the results.  What do you notice?  Write down all the questions that come up.

Step 3:  As a group, we will make a list of all the questions.

Step 4: Think and write up your answers to the questions.

Step 5:  Discuss!


Note:  Work with a partner to divide up the work in Stages 1, 2, and 4.


1/2 = __________                         1/11 = __________

1/3 = __________                         1/12 = __________

1/4 = __________                         1/13 = __________

1/5 = __________                         1/14 = __________

1/6 = __________                         1/15 = __________

1/7 = __________                         1/16 = __________

1/8 = __________                         1/17 = __________

1/9 = __________                         1/18 = __________

1/10 = __________                        1/19 = __________

     1/20 = __________


Reciprocal Integers

Pirates & Coins

The Game:

There are 5 rational pirates (in strict order of seniority A, B, C, D and E) who found 100 gold coins. They must decide how to distribute them.

The pirate world’s rules of distribution say that the most senior pirate first proposes a plan of distribution. The pirates, including the proposer, then vote on whether to accept this distribution. If the majority accepts the plan, the coins are dispersed and the game ends. In case of a tie vote, the proposer has the casting vote. If the majority rejects the plan, the proposer is thrown overboard from the pirate ship and dies, and the next most senior pirate makes a new proposal to begin the system again. The process repeats until a plan is accepted or if there is one pirate left.

Pirates base their decisions on four factors:

  1. First of all, each pirate wants to survive.
  2. Second, given survival, each pirate wants to maximize the number of gold coins he receives.
  3. Third, each pirate would prefer to throw another overboard, if all other results would otherwise be equal.
  4. And finally, the pirates do not trust each other, and will neither make nor honor any promises between pirates apart from a proposed distribution plan that gives a whole number of gold coins to each pirate.

Five Pirates, One Hundred Coins


Extension:  Keep everything above the same, except change it to 500 pirates rather than 5.   Can the same strategy be applied?


Prisoner Problem

During an ancient war three prisoners were brought into a room. In the room was a large box containing three white hats and two black hats. Each man was blindfolded, and one of the hats was placed on his head.

The men were lined up, one behind the other, facing the wall. The blindfold of the man farthest from the wall (man C) was removed, and he was permitted to look at the hats of the two men in front of him. If he knew (not guessed) the color of the hat on his head, he would be freed. However, he was unable to tell. The blindfold was then taken from the head of the next man (man B), who could see only the hat of the one man in front of him. This man had the same chance for freedom, but he, too, was unable to tell the color of his hat. The remaining man (man A) then told the guards the color of the hat he was wearing and was released. What color hat was he wearing, and how did he know?

three black hats and two white hats

Ask Marylin About Doors

Adapted from Parade magazine, April 15th, 1990:

There are 1,000 tenants and 1,000 apartments. The first tenant opens every door. The second closes every other door. The third tenant goes to every third door, closing it if it is open and opening it if it is closed. The fourth tenant goes to every third door, closing it if it is open and opening it if it is closed. This continues with every tenant until the 1,000th tenant opens the 1,000th door. Which doors are open?

1000 apartment doors


Bonus question: Can you be sure ahead of time that the 1,000th tenant closes the 1,000th door rather than opens it? How can you be sure?

Another problem from Parade (same column, same year) about doors:

Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?

three doors

List of Possible Topics

  • Euler's Identity
  • Transcendental Numbers
  • Hat/Dot logic puzzles
  • Mathematics of Paradoxes
  • Squaring the Circle
  • Geometric Constructions
  • Machine Learning
  • Gabriel’s Horn
  • Banach-Tarski Paradox
  • Shor’s Algorithm
  • Additional Topics
    • Types of Numbers:  Frobenius, McNugget, Versatile, etc.
    • Cassini’s Identity

…stay tuned for more, with links and details.