## Friday, August 30, 2002

### Experience Mathematics # 11 -- Counting

How can you tell if there are enough chairs in your classroom so that every student has a chair? One way is to count the chairs and the number of students in your class. A simpler way is to ask each student to sit down. If all the students are able to sit down, and no chair is left over, the number of students and chairs is equal.

This fundamental idea is at the heart of mathematics, because it deals with counting. Just like there is a weight that represents a kilogram, there is a unit in mathematics that represents a number $n$. Let $I(n)$ be the set containing the first $n$ natural numbers. The set $I(n)$ is the “unit” that represents the number $n$. For example, the set $A$ with elements $a, b, c, \dots, z$, has $26$ elements because this set can be put in one to one correspondence with $I(26)$.

But what about infinite sets? The set $N$ of natural numbers is a unit for infinite sets. We say a set $S$ is countable if it can be put in one to one correspondence with $N$. It is remarkable that the following subsets of $N$ are countable. Can you find the one to one correspondence between $N$ and these sets?

1. The set of all even numbers.

2. The set with elements $1, 4, 7, 10, \dots$

3. The set of all rational numbers.

Can you show that the set of prime numbers is countable?

When it comes to infinite sets, a part can be equal to the whole.

## Wednesday, August 21, 2002

### Experience Mathematics #10 -- Sets

A fundamental object in mathematics is a set. You can think of a set as a collection of objects. We are familiar with the set $N$ of natural numbers. The empty set is a set with no elements.

We say that a set $B$ is a subset of a set $A$ if each element of $B$ is also in $A$. Two sets are equal if they are subsets of each other. The empty set is a subset of every set. For example the set of even numbers $\left\{ 2, 4, 6, \dots\right\}$ is a subset of $N$.

Q1. How many subsets does the empty set have? (Hint: There is atleast one subset.)

The members of the set are called its elements.

Let $I(n)$ denote the set containing the first $n$ natural numbers. The set $I(n)$ has n elements.

Q2. List all the subsets of the set $I(1)$. How many subsets does $I(1)$ have?

Q3. List all the subsets of the set $I(2)$. How many subsets does $I(2)$ have?

Q4. List all the subsets of the set $I(3)$. How many subsets does $I(3)$ have?

Suppose you delete $3$ from each subset of $I(3)$. What do you get? Use this idea to do Q5.

Q5. List all the subsets of the set $I(4)$. How many subsets does $I(4)$ have?

Q6. How many subsets does $I(n)$ have? Guess the answer from your experiments above.

If you cannot find a pattern, you must have made a mistake in listing the sets earlier.

## Friday, August 16, 2002

### Experience Mathematics #9 -- Prime numbers

The prime numbers are numbers that only have $1$ and themselves as factors. The first few prime numbers are $2, 3, 5, 7, 11, 13, 17, 19, \dots$. One of the most important ideas in the theory of numbers is that a number can be written in a unique way as a product of prime powers. For example,
$4=2^2$, $15$ is $3$ times $5$, and $20$ is $2\times 5$.

The uniqueness of the prime factorization is used to show that the square root of $2$ is not a rational number. A rational number is a number in the form $p/q$, where both $p$ and $q$ are integers, and $q$ is not equal to zero. By doing the following questions, you can prove that the square root of two is an irrational number. The proof is by contradiction. We assume that the square root of $2$ can be written as a fraction, and then show our assumption implies a false statement.

Q1. Suppose the square root of $2$ equals $p/q$ for some integers $p$ and $q$. Show that $2q^2=p^2$.

Q2. Show that the power of $2$ in the prime factorization of the number $2q^2$ is an odd number.

Q3. Show that the power of $2$ in the prime factorization of the number $p^2$ is an even number.

The above two statements are contradictory. Thus our assumption, that the square root of $2$ is a rational number, must be false.

Can you generalize this proof to prove that the square root of $3$ and $5$ are also irrational numbers?

## Friday, August 09, 2002

### Experience Mathematics #8 -- Finding patterns

Mathematics is all about finding patterns. Begin learning to spot patterns in numbers today, and maybe one day you can solve a big mathematical problem—like Maninder Agarwal, Neeraj Kayal, and Nitin Saxena, computer scientists at IIT, Kanpur. These mathematical stars have just announced a new algorithm, by which they can tell whether a number is a prime number. An implementation of this algorithm will mean that everyone will have to re-look at computer programs used to keep messages transferred on the Internet confidential.

In today’s activity, find the next three terms in the following sequences.

Q1. $1, 3, 5, 7,\dots$

Q2. $2, 4, 6, 8, \dots$

Q3. $1, 4, 9, 16, \dots$

Q4. $2, 3, 5, 7, 11, 13, 17, \dots$

Q5. $1, 2, 4, 8, 16, \dots$

Q6. $1, 4, 1, 4, 2, \dots$

Q7. $3, 6, 9, 12, 15, 18, \dots$

Q8. $0, 3, 8, 15, 24, \dots$

Q9. $0, 1, 1, 2, 3, 5, 8, \dots$

Q10. $1, 3, 6, 10, 15, 21, \dots$

Q11. $2, 8, 20, 40, 70, \dots$

Q12. $1, 5, 14, 20, 55, \dots$

## Friday, August 02, 2002

### Experience Mathematics # 7 -- A continued fraction

We all know that the square root of $2$ is an irrational number. That is to say, it cannot be written as a fraction $p/q$. Here $p$ and $q$ are integers, and $q$ is not zero. But it can be written in the form of a continued fraction.

Here is how you can discover the continued fraction representation of the square root of $2$. You will need a calculator to do the calculations. Carefully understand the following calculations.
Note that the $.41421\dots$ starts repeating, and we get a fraction that looks like

You can chop off the fraction at any point and get a fraction that is approximately equal to the square root of $2$.

As for this week’s activity, do a similar calculation (using a calculator) for the square root of $3$ and the square root of $5$ and find the continued fraction representation of these irrational numbers.