The Nirmukta Freethought Community

Many of them especially in India do not understand the basic principles of mathematics or science while they are learning. They tend to mug the things up just like they mug the Shlokas believing it to be true without questioning "how". This will hinder the scientific thinking in child and the child will tend to believe in superstitions blindly. Scientific and mathematical theories can always be proved experimentally.

The best and simple example is negative number multiplied by a negative number. Teachers should demonstrate this in the class just like a lab experiment.


Consider multiplication of two integers a and b, i.e a x b.

Consider the number line with negative side as west and positive side as east and you standing at 0.

West <--- ___-3___-2___-1___|y0u_r_here|___+1___+2___+3___---> East

If a is positive then face yourself towards east,
If a is negative then face yourself towards west.

If b is positive, then move forward,
If b is negative, then move backward.

Now consider a x b and your initial position at 0,
you will face towards east (a is positive), move forwards (b is positive). You will end up on positive side.

Now consider -a x b and your initial position at 0,
you will face towards west (a is negative), move forwards (b is positive). You will end up on negative side.

Now consider a x -b and your initial position at 0,
you will face towards east (a is positive), move backwards (b is negative). You will end up on negative side.

Now consider -a x -b and your initial position at 0,
you will face towards west (a is negative), move backwards (b is negative). You will end up on positive side.

Thus minus times minus is a plus!

'a' tells you how big must be the steps while you move, 'b' tells how many steps you have to take, while the minus or plus sign tells you the direction.

Example:
-2 x -3 ,
This says face yourself west (-a) and move backwards (-b) by taking 3 steps of 2 feet each. You will end up at +6 feet towards east from 0.

(10-05-2010 12:05 AM)shrihara Wrote:  Many of them especially in India do not understand the basic principles of mathematics or science while they are learning. They tend to mug the things up just like they mug the Shlokas believing it to be true without questioning "how".

That is pretty much true of me. I learnt mathematics as a set of formulas and equations which need to be solved and not as a tool to understand Nature. But the sad part is the educational system here judged me as one the best amongst my peers when all I did was something that a computer can do a billion times faster. I came to know of mathematical logic only after I had completed my formal education, when it should have been one of first subjects that should be taught to a student of mathematics.

(10-05-2010 12:40 AM)Lije Wrote:  

(10-05-2010 12:05 AM)shrihara Wrote:  Many of them especially in India do not understand the basic principles of mathematics or science while they are learning. They tend to mug the things up just like they mug the Shlokas believing it to be true without questioning "how".

That is pretty much true of me. I learnt mathematics as a set of formulas and equations which need to be solved and not as a tool to understand Nature. But the sad part is the educational system here judged me as one the best amongst my peers when all I did was something that a computer can do a billion times faster. I came to know of mathematical logic only after I had completed my formal education, when it should have been one of first subjects that should be taught to a student of mathematics.

The sad thing is I used to be really good at math till around my 7th class. It was after this that it got too theoretical and I completely lost interest. And one thing educationists will tell you, a kid with no interest is a kid you cannot teach. I barely scraped through in mathematics until I didn't have to study it anymore. Except for a few areas where analytical logic could be extended to math, the entire discipline was meaningless for the most part. It had no significance as far as the real world was concerned. This is why in my latest article, the story begins with the little girl wondering how the rules of math relate to reality.

I completely agree with Lije. There is a good chance that if I had learnt mathematical logic before being force-fed formulaic techniques to solve incomprehensible problems (most of which, as Lije says, can be done by punching a few buttons today), I may have stayed interested in math through my higher education.

(17-05-2010 07:21 PM)Ajita Kamal Wrote:  This is why in my latest article, the story begins with the little girl wondering how the rules of math relate to reality.

I loved that article! Philosophy is another subject that should be taught to kids. If not as part of the curriculum, they should at least be encouraged to read up on it. Most people (including me at one point) think that philosophy is an esoteric, dead end subject.

(10-05-2010 12:05 AM)shrihara Wrote:  The best and simple example is negative number multiplied by a negative number. Teachers should demonstrate this in the class just like a lab experiment.


Consider multiplication of two integers a and b, i.e a x b.

Consider the number line with negative side as west and positive side as east and you standing at 0.

West <--- ___-3___-2___-1___|y0u_r_here|___+1___+2___+3___---> East

If a is positive then face yourself towards east,
If a is negative then face yourself towards west.

If b is positive, then move forward,
If b is negative, then move backward.

Now consider a x b and your initial position at 0,
you will face towards east (a is positive), move forwards (b is positive). You will end up on positive side.

Now consider -a x b and your initial position at 0,
you will face towards west (a is negative), move forwards (b is positive). You will end up on negative side.

Now consider a x -b and your initial position at 0,
you will face towards east (a is positive), move backwards (b is negative). You will end up on negative side.

Now consider -a x -b and your initial position at 0,
you will face towards west (a is negative), move backwards (b is negative). You will end up on positive side.

Thus minus times minus is a plus!

'a' tells you how big must be the steps while you move, 'b' tells how many steps you have to take, while the minus or plus sign tells you the direction.

I agree that math often appears like religion to little kids. They appear to be things they just have to agree to like you've illustrated with your example. Yes, it can be taught better.

However, I don't see how your mechanical procedure to do multiplication, really explains why the product of two negative quantities is positive. I suspect that kids will just it to find the sign of the answer and then do the multiplication of the absolute value of the numbers and then just append the sign. I will venture the following explanation:

Think of a positive number as a quantity of something that you will get, and a negative number as something you have to give. If I give you 3 candies you have plus 3 candies = 3 or +3. If you give me 3 candies, you have minus 3 candies = -3.

Multiplication is repeated addition (or subtraction). If I give you 3 candies seven times, you would have 3 + 3 + 3 + 3 + 3 + 3 + 3 = 21 candies in the end. We write this as 3 x 7. If you give me 3 candies seven times, then you will have -3 x 7 = -21 candies in the end. If I give you 3 candies -7 times, it means that you give me 3 candies 7 times. This may be the hard part, but I think because taking becomes giving (with some careful thought it can be deduced that there can be no other possibility). So you end up with -21 candies again, i.e. 3 x (-7) = -21.

Finally, if you give me 3 candies -7 times, it is the same as I giving you 3 candies 7 times. This may seem like a leap of faith, but it is the same logic as for 3 x (-7). Therefore, -3 x -7 = 3 x 7 = +21.

It may be a little hard, but I think it should work for a fairly creative child. What do you all think?

@donatello
your idea of multiplication as repeated addition works only for integers. i wonder if you can show Pi*Pi as an addition or subtraction?

The geometric picture the OP is giving is a better analogy for vectors as he/she is associating a direction with the above mentioned example.

Interesting article. I never thought of thinking about why product of two negative numbers is positive. I agree with Srihara that maths is taught just like religion in schools, without any real detailed explanations and its really a pity.

But I agree with donatello that Srihara's explanation doesn't really explain the why part. It just proposes a convention that obeys those rules. The explanation given by donatello looks far more mathematical. I don't agree with Alan D'Souza's objection. Integer multiplication can indeed be defined as repeated addition or subtraction. The reason it doesn't apply to fractions doesn't really invalidate it. The multiplication operator for fractions has to be accordingly defined.

Folks, I would like to direct you to this wonderful article by Steven Strogatz in NY Times. I also urge you to peruse the entire series of articles on mathematics by Strogatz.

http://opinionator.blogs.nytimes.com/201...-my-enemy/

He puts it awesomely. Consider multiplying -1 with a decreasing sequence of numbers. Logically, the product should also have a particular sequence, which, empirically turns out to be an increasing order. This is an inductive approach to understanding multiplication of negative integers.

-1 X 3 = -3
-1 X 2 = -2 = -3 + 1
-1 X 1 = -1 = -2 +1
-1 X 0 = 0 = -1 + 1
-1 X -1 = ? = 0+1 = 1

@Alan Once the mathematics of integers is understood, it can be generalized to the mathematics of real numbers. A major breakthrough, historically, was made by Georg Cantor who described two orders of infinities- one applying to rational numbers and another to irrational numbers. Before we jump from integers to pi, we have to first describe rational numbers. Rational numbers can simply be written in a direct (one-to-one) correspondence with integers. This was first articulated by Georg Cantor. Now, rational numbers would suffice to understand most of the world. But you routinely encounter numbers which cannot be expressed as rational numbers, starting with sqrt(2).


Let us suppose, with p/q being the SIMPLEST form of the fraction (e.g. 2/4 is not the simplest, 1/2 is),
sqrt(2) = p/q
2 = p^2/q^2
2q^2 = p^2

Since only an even number can square to an even number, this means that p must be even, so we can write p = 2n
2q^2 = 4n^2
q^2 = 2n^2

This means that q is also even. If p and q are even, then p/q = 2n/2m = n/m, which contradicts our claim that p/q is the simplest form. Therefore, sqrt(2) cannot be written as p/q.

These numbers fill up the spaces on the number line that are not occupied by rational numbers or integers. And they are infinitely more numerous too! And cannot be counted, because counting requires you to put them on your fingers, or map them on to the integers, which is impossible to do. Hence, such infinities are called "uncountable" infinities.

How do you define pi*pi? You really cannot multiply it in the traditional sense of adding a finite number of objects. Hence, one needs to resort to tricks and use various properties of pi, like its approximation. Multiplying irrational numbers like pi and 1/pi or sqrt(2)*sqrt(2) are done quite simply.

Breaking down multiplication as adding numbers is only a *computational* tool. So, computationally, one cannot achieve pi*pi to full precision, just as you cannot achieve pi to full precision. "Full precision" doesn't make sense. However due to interesting properties you can achieve pi*(1/pi) and (sqrt(2)*sqrt(2)) to full precision.

(17-05-2010 07:21 PM)Ajita Kamal Wrote:  I completely agree with Lije. There is a good chance that if I had learnt mathematical logic before being force-fed formulaic techniques to solve incomprehensible problems (most of which, as Lije says, can be done by punching a few buttons today), I may have stayed interested in math through my higher education.

I am assuming you, like me were educated in India. It is a sad fact that the foundations of mathematics are not taught in school. I strongly recommend the book "What is Mathematics?" by Courant and Robbins for an awesome introduction to the foundations of mathematics. The logic behind concepts such as natural numbers, negative numbers, irrational numbers, etc. are not taught sufficiently in our education system.

The book by Courant is a very nice one. There are also some very good books by J.H Hardy which are meant for a layman to see the world of mathematics as a piece of art and not as a sum of formulae to cram up.

(10-05-2010 12:05 AM)shrihara Wrote:  Many of them especially in India do not understand the basic principles of mathematics or science while they are learning. They tend to mug the things up just like they mug the Shlokas believing it to be true without questioning "how". This will hinder the scientific thinking in child and the child will tend to believe in superstitions blindly. Scientific and mathematical theories can always be proved experimentally.

The best and simple example is negative number multiplied by a negative number. Teachers should demonstrate this in the class just like a lab experiment.


Consider multiplication of two integers a and b, i.e a x b.

Consider the number line with negative side as west and positive side as east and you standing at 0.

West <--- ___-3___-2___-1___|y0u_r_here|___+1___+2___+3___---> East

If a is positive then face yourself towards east,
If a is negative then face yourself towards west.

If b is positive, then move forward,
If b is negative, then move backward.

Now consider a x b and your initial position at 0,
you will face towards east (a is positive), move forwards (b is positive). You will end up on positive side.

Now consider -a x b and your initial position at 0,
you will face towards west (a is negative), move forwards (b is positive). You will end up on negative side.

Now consider a x -b and your initial position at 0,
you will face towards east (a is positive), move backwards (b is negative). You will end up on negative side.

Now consider -a x -b and your initial position at 0,
you will face towards west (a is negative), move backwards (b is negative). You will end up on positive side.

Thus minus times minus is a plus!

'a' tells you how big must be the steps while you move, 'b' tells how many steps you have to take, while the minus or plus sign tells you the direction.

Example:
-2 x -3 ,
This says face yourself west (-a) and move backwards (-b) by taking 3 steps of 2 feet each. You will end up at +6 feet towards east from 0.

Yes, the concept is explained well in physical terms. There is another approach to understand the concept.

a x b means add a's b times.

+ means addition

- means substraction

- also means do the opposite

therefore

a x -b does not allow us to add a's -b times; therefore we subtract a's b times and get a "-" result.

eg

5 x -4 = - (5) - (5) - (5) - (5) = -20

-a x b means add -a's b times.

Now,

-a x -b will mean subtract -a's b times

eg

-5 x -4 = -(-5) - (-5) - (-5) - (-5) = 5 + 5 + 5 + 5 = + 20.

I do not remember very clearly, but my school teacher back in 5th or 6th standard explained something like this.

Two negatives cancel out each other guys. It's as simple as that!

Why so you ask?

This might help..