## Thursday, June 27, 2002

### Experience Mathematics #4 -- Division by zero!

Both Koonal Sharda (Class VI, DPS, Mathura Road) and Sarthak Parikh (Class VIII, Sardar Patel) gave the correct solutions with good explanations. What is creditable is that students are expected to answer these questions only when they reach Class XI. However, they say that Gauss found these patterns when he was only 10 years old! But anyway, well done.

Assuming the pattern that you observed continues forever—and it does—the sum $O$ of the first $n$ odd numbers is given by the formula: $O=1+3+…+2n-1=n^2$. This fact can be proved using Mathematical Induction. Adding $n$ is the same thing as adding $1$ $n$ times. So we obtain: $E=2+4+…+2n=n^2 +n$. From this it is easy to find the sum of the first $n$ numbers. We divide each term by $2$ to get $$1+2+3+…+n=(n^2 +n)/2 = n(n+1)/2.$$

Once again Sarthak asked some deep questions on which today’s experiments are based. He asked: What meaning can be given to division by 0?

At the outset, let me say that division by 0 is not allowed.

The reason is the confusion caused by the following argument: We know that $0=0.$ This implies that $0$ times $1$ is equal to $0$ times $2$. So $0 \times 1 = 0 \times 2$.  Now canceling $0$ from both sides (by dividing both sides with $0$) we obtain $1=2$, which clearly is false. And there is a theorem in mathematics that says that a false proposition implies any proposition. Which implies that Santa Claus exists. And also implies that Santa Claus does not exist.
Now you can see why mathematicians have very sensibly banned division by $0$.

However, this does not mean that mathematicians do not try to give division by $0$ some meaning. Try the following (you may need a calculator or a computer for some of these experiments):
1. Make a table of the function $y=1/x$. In the first column place numbers very close to $0$ that are positive. For example $0.1, 0.01, 0.001, \dots, 0.0000000001$. In the second column, replace $x$ in $1/x$ by this number and see what values come out for y. For example, when we replace $x$ by $.01$, we get $y=100.$ Describe the results of your experiments.

2. Now replace $x$ by negative numbers that come closer and closer to $0$ and find out what value $y$ takes. Describe the results of your experiments.

3. Try the above with the function $y=x^2/x$. Does the value come closer and closer to a number?

Describe the results of your experiments.

We say a function (depending on the variable $x$) approaches the limit $+\inf$  (read plus infinity) when $x$ approaches a number $a$ from the left, if given any large number $M$ (any large number you can think of), we can find a number a little smaller than $a$, so that the corresponding value of the function becomes larger than $M$. (The ‘left’ refers to the number being to the left of $a$ on the number line.) Similarly, we can define the limit approaching from the right, and the limit approaching minus infinity.
Intuitively, it is easier to understand the limit of a function if it approaches a number $q$, when $x$ approaches a number $a$. We replace $x$ by numbers close to $a$, and find the value of the function. If the value comes close to a number $q$, the limit is $q$. Try the following experiments and get a feel for the definition.

4. Multiply $(1-x)$ in turn by $(1+x),$ $(1+x+x^2),$ $(1+x^2+x^3)$ and simplify. Can you generalize the pattern?

5. What is the limit of $(1-x^2)/(1-x),$ $(1-x^3)/(1-x),$ $\dots$, $(1-x^n)/(1-x),$ as $x$ approaches $1$. (Hint: Replace $x$ by numbers close to $1$, like $1, 1.1, 0.9, 1.01, 0.99, 1.001, 0.999,$ $\dots,$ and guess the answer in each case.)

## Tuesday, June 18, 2002

### Experience Mathematics # 3 -- The sum of the first n odd numbers

Akarsh Gupta (Class VII, A.P.J., Noida), Sarthak Parikh (Class VIII, Sardar Patel) and Richa Sharma (Class IX, Swami Vivekananda Sarvodaya School) all gave excellent explanations/solutions to the questions asked in the previous column. The answers (very briefly) are:

Q1. The numbers that have $0, 2, 4, 6, 8$ in the last digit are divisible by $2$.

Q2. The numbers that have $0$ or $5$ in the last digit are divisible by $2$.

Q3. If the sum of the digits of a number is $9$ or a multiple of $9$, then that number is divisible by $9$.

Q4. If the sum of the digits of a number is a multiple of $3$ then that number is divisible by $3$.

Sarthak Parikh also asked an interesting question. Sarthak asked: How many lines of symmetry does a parallelogram have? What about the square? Rhombus? Rectangle? Octagon? Cut these shapes out of paper and label the vertices. Experiment by drawing various lines, and flipping the shape. If you get back the same shape (but with different labels) you have discovered a line of symmetry. For example, a square with vertices labeled ABCD, you may get a square labeled BACD (see the top two squares in the picture). If you notice, by flipping across a line you have rotated the square by 180 degree. You should also try to find other rotational symmetries. When you rotate a square ABCD clockwise by 90 degrees, you get the square DABC (bottom square in the picture). After doing some experiments, perhaps you would like to answer Sarthak’s query?

Here is another activity involving squares—the squares of natural numbers.
Complete the following table. For n going from 1 to 20 write the square of n in the second column, and the sum of the first n odd numbers in the third column. The odd numbers are (1, 3, 5, …).

 $n$ $n^2$ Sum of first $n$ odd numbers $1$ $1$ $1=1$ $2$ $4$ $1+3= 4$ $3$ $9$ $1+3+5=9$ … $20$

Make a picture showing the pattern above. (Hint: see the picture below)

1. Find a formula for the sum $O$ of the first $n$ odd numbers: $O = 1+3+5+7+\cdots +(2n-1).$
2. Use the formula for $O$ to find a formula for the sum $E$ of the first $n$ even numbers: $E=2+4+\cdots+2n .$
3. Use the formula for $E$ to find a formula for the sum of the first $n$ natural numbers.

Check your work by putting $n=1, 2, 3, 4, 5$ for each formula you find.

## Thursday, June 13, 2002

### Experience Mathematics # 2 - Divisibility

Among all the students who tried to answer the questions presented last week, Anirudh Matange (Class VI, Ahlcons School) gave the best explanations. The answers are:

Q1. The Commutative Law for multiplication.

Q2. The Prime Numbers $(2, 3, 5, 7, 11, 13, 17, 19)$ give rise to only two rectangles:

Q3. The largest number of rectangles arise from $12, 18,$ and $20.$

Q4. The rectangles with side $2$ can be formed by the Even Numbers: $2, 4, 6, \dots, 20.$

Q5. All the Multiples of $3$ (namely: $3, 6, \dots, 18$) can form rectangles with side $3.$

Here is another mathematical activity on the concept of divisibility. When a number $n$ divides a number $m$ evenly, we say that $m$ is divisible by $n$, or that $n$ is a factor of $m$. There are a number of tests that determine whether a given number is divisible by $2, 3, 5,$ or $9.$ By doing the following experiments, you can discover these tests for yourself.

1. Consider the numbers: $2030, 4201, 89782, 129083, 124, 5435, 67656, 9087, 8888, 90919.$ For each number, you have to tell the last digit when the given number is multiplied by $2$.

2. For each of the numbers in Activity 1, you have to tell the last digit when the given number is multiplied by $5$.

3. Make a table with 2 columns. In the first column, place the multiples of $9$ less than $100$ ($9, 18, 27, 36,\dots , 81, 90$). In the second column, note the sum of their digits.

4. Make a table with two columns. In the first column, put any $10$ multiples of $3$ less than $100$—such as $12, 15, 63, 51$—that are not all multiples of $9$. In the second column, note the sum of their digits.

Q1. Which of the following numbers is divisible by $2$: $100000, 1201, 2342, 9083, 2124, 21245, 1906, 6757, 1978, 9879.$

Q2. Which of the following numbers is divisible by $5$: $100000, 1201, 2342, 9083, 2124, 21245, 1906, 6757, 1978, 9879.$

Q3. What is the rule for checking whether a number is divisible by $9$?

Q4. Which of the following numbers is divisible by $3$: $101010, 1201, 20112, 2124, 21223, 1906, 6757, 1978, 9879, 1080.$

Bright students should verify, and then prove that their guesses are correct. But the proofs, while easy, are not covered in school syllabi. For the moment, it is enough to know the tests of divisibility without knowing why they work.

## Sunday, June 02, 2002

### Experience Mathematics # 1 -- Mathematical activities

Introduction

Children learn by doing. By doing these activities, they will experience interesting mathematical ideas. They will also gain experience in thinking mathematically. This will help them understand concepts easily, and better their performance in exams.
It is also very important to remember that the encouragement of parents and grandparents motivates a child a lot. Praise them when they show their intelligence by doing mathematical activities successfully. This will make the children work hard to become better at mathematics.
Why do children find mathematics difficult? The most important reason is that they find mathematics removed from their daily existence. However, it is not too difficult to give mathematical experiences to children. In this column, we will give an activity for children. It is a good idea for parents to help the child with the activity, if the child is studying in class I-V. For older children, show them this column and challenge them to explain the answers of the questions below.
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This activity requires 20 round pegs. The round pieces of a Carrom Board set will do nicely. You may also use 20 buttons of the same size.

Pick a number—say 15. Take 15 pegs and make as many rectangles as you can out of them. Each time you are able to make a rectangle, reproduce the rectangle in a drawing book and note the dimensions. (You can use dots to make rectangular arrays.) For example, using 15 pegs, you can draw the following rectangles.

Repeat this activity for each number from 1 to 20. Now try to answer the following questions:
1. When you rotate a rectangle by 90 degrees, you get another rectangle with the same number of dots. What is this law called?
2. List the numbers that give rise to only 2 rectangles?
3. Which number (or numbers) lead to the largest number of rectangles?
4. What are the dimensions of all the rectangles with one side consisting of 2 dots?
5. What are the dimensions of all the rectangles with one side consisting of 3 dots?
It is not necessary that you will be able to answer all the questions. But it is important to try hard. In the next column, we will give explanations of the mathematics underlying this activity.