The Magic of “Pi”
Let’s start with the circle – can there be a more simple shape. Just take any point, and map out all the other points on a plane that are at some fixed distance from it. And you get a circle.
The word itself is from the Greek for circle – kirkos (Oddly enough, the word “circus” comes from the same root). Of course, our ancestors were familiar with the circle much before they had any language. In nature, they could see the moon, and the sun. When they cut a tree, they could see that the trunk had a circular cross-section. They invented the wheel – so they knew about the shape. And they believed it to be a symbol of perfection.
So there you have it – a simple shape, nothing seemingly complicated about it. A point. Another set of points that are equidistant from it. What could be simpler?
Ah, but in this seeming simplicity lie some wonderful things. Let’s just take one of them – the number “pi”.
What is pi, you ask. Well, it’s just the ratio of the circle’s circumference to it’s diameter. This ratio is the same for all circles – no matter how large or how small.
So what is the value of this number?
This is where the story starts to get strange. When you were in 3rd grade, you were probably told that the value of this number, this “pi” is 3 (which makes sense, you really didn’t know much about decimal numbers at that point). Then in grade 4 or so, when you got used to fractions and decimals, you were probably told to use the value 22/7 or 3.14 for this value to solve problems.
Here’s the funny part – although we can define what this number is (remember – ratio of the circumference to diameter of any circle) we can’t tell the exact value of the number. Sure, we know that goes something like “3.1415926535,,,” but there is no precise value. The part after the decimal point never ends. It goes on and on into infinity, and the numbers don’t repeat. What this means is, unlike a fraction like 1/3, which is also infinite (0.333333…) we can’t get a repeating pattern that lets us predict what, for example, the billionth digit after the decimal is (for 1/3, the billionth, or trillionth digit after the decimal point is always going to be “3″). It must be said that mathematicians recently have come up with some clever tricks that let you compute the value of any digit of the value of pi very fast (and without knowing any of the previous digits), but that involves rather advanced mathematics, and we won’t go into that here.
Another fascinating fact about pi is that it can be proven (some very clever German mathematicians showed this in the 19th century) that you can never get an equation with a finite number of operations on integers to give you the value of “pi”.
Mathematicians have been trying to find the value of pi for over 4000 years. By 1900 BC, Egyptian and Babylonian mathematicians knew the value to within 1%. Indian mathematicians also knew a very good approximation – the Shatapatha Brahmana , a 6th century BC work from India, gave the value as 339/108. Of course, today we know the value to trillions of digits.
More strangeness – “pi” appears everywhere in physics, maths and engineering. Even in places that seemingly have no connection to the circle. For example, if you were to try to compute the average height of all the people in the country, the formula for that would have a “pi” in it. In advanced physics, some of Einsteins equations trying to describe the nature of the universe have “pi”, as does Heisenberg’s equation governing the behavior of really small particles. Strange isn’t it? And all the more reason to learn maths. The secrets of the world around can only be understood through mathematics.
And yes, dont forget to celebrate National PI day (March 14, at 1:59. (3/14 1:59))
Let’s start with the circle – can there be a more simple shape. Just take any point, and map out all the other points on a plane that are at some fixed distance from it. And you get a circle.
The word itself is from the Greek for circle – kirkos (Oddly enough, the word “circus” comes from the same root). Of course, our ancestors were familiar with the circle much before they had any language. In nature, they could see the moon, and the sun. When they cut a tree, they could see that the trunk had a circular cross-section. They invented the wheel – so they knew about the shape. And they believed it to be a symbol of perfection.
So there you have it – a simple shape, nothing seemingly complicated about it. A point. Another set of points that are equidistant from it. What could be simpler?
Ah, but in this seeming simplicity lie some wonderful things. Let’s just take one of them – the number “pi”.
What is pi, you ask. Well, it’s just the ratio of the circle’s circumference to it’s diameter. This ratio is the same for all circles – no matter how large or how small.
What is pi, you ask. Well, it’s just the ratio of the circle’s circumference to it’s diameter. This ratio is the same for all circles – no matter how large or how small.
So what is the value of this number?
This is where the story starts to get strange. When you were in 3rd grade, you were probably told that the value of this number, this “pi” is 3 (which makes sense, you really didn’t know much about decimal numbers at that point). Then in grade 4 or so, when you got used to fractions and decimals, you were probably told to use the value 22/7 or 3.14 for this value to solve problems.
Here’s the funny part – although we can define what this number is (remember – ratio of the circumference to diameter of any circle) we can’t tell the exact value of the number. Sure, we know that goes something like “3.1415926535,,,” but there is no precise value. The part after the decimal point never ends. It goes on and on into infinity, and the numbers don’t repeat. What this means is, unlike a fraction like 1/3, which is also infinite (0.333333…) we can’t get a repeating pattern that lets us predict what, for example, the billionth digit after the decimal is (for 1/3, the billionth, or trillionth digit after the decimal point is always going to be “3″). It must be said that mathematicians recently have come up with some clever tricks that let you compute the value of any digit of the value of pi very fast (and without knowing any of the previous digits), but that involves rather advanced mathematics, and we won’t go into that here.
Here’s the funny part – although we can define what this number is (remember – ratio of the circumference to diameter of any circle) we can’t tell the exact value of the number. Sure, we know that goes something like “3.1415926535,,,” but there is no precise value. The part after the decimal point never ends. It goes on and on into infinity, and the numbers don’t repeat. What this means is, unlike a fraction like 1/3, which is also infinite (0.333333…) we can’t get a repeating pattern that lets us predict what, for example, the billionth digit after the decimal is (for 1/3, the billionth, or trillionth digit after the decimal point is always going to be “3″). It must be said that mathematicians recently have come up with some clever tricks that let you compute the value of any digit of the value of pi very fast (and without knowing any of the previous digits), but that involves rather advanced mathematics, and we won’t go into that here.
Another fascinating fact about pi is that it can be proven (some very clever German mathematicians showed this in the 19th century) that you can never get an equation with a finite number of operations on integers to give you the value of “pi”.
Mathematicians have been trying to find the value of pi for over 4000 years. By 1900 BC, Egyptian and Babylonian mathematicians knew the value to within 1%. Indian mathematicians also knew a very good approximation – the Shatapatha Brahmana , a 6th century BC work from India, gave the value as 339/108. Of course, today we know the value to trillions of digits.
More strangeness – “pi” appears everywhere in physics, maths and engineering. Even in places that seemingly have no connection to the circle. For example, if you were to try to compute the average height of all the people in the country, the formula for that would have a “pi” in it. In advanced physics, some of Einsteins equations trying to describe the nature of the universe have “pi”, as does Heisenberg’s equation governing the behavior of really small particles. Strange isn’t it? And all the more reason to learn maths. The secrets of the world around can only be understood through mathematics.
And yes, dont forget to celebrate National PI day (March 14, at 1:59. (3/14 1:59))
Algebra – A fascinating subject
Algebra is when most children get introduced to the concepts of “abstract” mathematics. Suddenly their world gets involved in quantities like “x” and “y”s. And if not introduced properly it can lead to fear of the subject. But if taught properly they can see that the “x” and “y” are like words that form a beautiful poem and they’ll delight themselves when they learn this secret language and understand the beauty of algebra.
Understanding and practice can go a long way in helping children here. Sites like EduGain(www.edugain.com) can help children in doing this.
Let’s look at a simple problem that can illustrate the beauty of this language (adults may like it too). Note that this article is meant only for children studying in grade 6 and above.
Suppose you take a long, really long string. You then cut out enough of it so that it can go all the way around the earth (I told you it was really long).
So now you have this string that’s on the ground and encircling the earth along the equator.Now suppose you add another 30 centimeters to that string (that’s about the length of your long ruler). We then smoothen it out by pulling it up from the ground all over the equator so that it forms a proper circle again.
The question is “How high do you think the string will be above the ground now?”
Sounds very hard to solve, right. But let’s see…
We know that the circumference of a circle is C = 2 x π x R, where R is the radius.
(don’t worry about π – let’s just use the value of “3″ for now)
So, C = 2 x 3 x R = 6 x R
Now we don’t know the exact radius of the earth here, but we don’t need to. Let’s leave it as a symbol R. So the length of the original string = 6 x R. We’ll just call it 6R (and know it means the value of R, whatever it is, multiplied by 6).
So C = 6R
Now we added 30 cm to C. The length of the new string is now C+30.
What we need to find out is how much R increases.
Let’s say that R increased by some length Z. The value of Z is what we want to find
So for the string, we can say it has a new circumference (Cnew) a new radius (Rnew)
We also know that since it’s a circle
Cnew = 6Rnew
And we know that the new circumference Cnew = C+30,
and that
Rnew = R + Z (remember we already assumed R increased by a length Z which we are trying to find out).
So we get
C + 30 = 6(R + Z).
Expanding this, we get,
C + 30 = 6R + 6Z
But we already know that 6R = C
Let’s replace the 6R by C on the right side
C + 30 = C + 6Z
The two “C”s cancel out, leaving us with
30 = 6Z, or
Z = 30/6 = 5
Amazing isn’t it. The new string would be 5 cm above the ground all everywhere. Just by adding 30 cm extra.
And what’s more amazing – we solved it through Algebra
Algebra is when most children get introduced to the concepts of “abstract” mathematics. Suddenly their world gets involved in quantities like “x” and “y”s. And if not introduced properly it can lead to fear of the subject. But if taught properly they can see that the “x” and “y” are like words that form a beautiful poem and they’ll delight themselves when they learn this secret language and understand the beauty of algebra.
Understanding and practice can go a long way in helping children here. Sites like EduGain(www.edugain.com) can help children in doing this.
Let’s look at a simple problem that can illustrate the beauty of this language (adults may like it too). Note that this article is meant only for children studying in grade 6 and above.
Suppose you take a long, really long string. You then cut out enough of it so that it can go all the way around the earth (I told you it was really long).
So now you have this string that’s on the ground and encircling the earth along the equator.Now suppose you add another 30 centimeters to that string (that’s about the length of your long ruler). We then smoothen it out by pulling it up from the ground all over the equator so that it forms a proper circle again.
The question is “How high do you think the string will be above the ground now?”
Sounds very hard to solve, right. But let’s see…
We know that the circumference of a circle is C = 2 x π x R, where R is the radius.
(don’t worry about π – let’s just use the value of “3″ for now)
(don’t worry about π – let’s just use the value of “3″ for now)
So, C = 2 x 3 x R = 6 x R
Now we don’t know the exact radius of the earth here, but we don’t need to. Let’s leave it as a symbol R. So the length of the original string = 6 x R. We’ll just call it 6R (and know it means the value of R, whatever it is, multiplied by 6).
So C = 6R
Now we added 30 cm to C. The length of the new string is now C+30.
What we need to find out is how much R increases.
Let’s say that R increased by some length Z. The value of Z is what we want to find
So for the string, we can say it has a new circumference (Cnew) a new radius (Rnew)
We also know that since it’s a circle
Cnew = 6Rnew
And we know that the new circumference Cnew = C+30,
and that
Rnew = R + Z (remember we already assumed R increased by a length Z which we are trying to find out).
and that
Rnew = R + Z (remember we already assumed R increased by a length Z which we are trying to find out).
So we get
C + 30 = 6(R + Z).
Expanding this, we get,
C + 30 = 6R + 6Z
But we already know that 6R = C
Let’s replace the 6R by C on the right side
C + 30 = C + 6Z
The two “C”s cancel out, leaving us with
30 = 6Z, or
Z = 30/6 = 5
Amazing isn’t it. The new string would be 5 cm above the ground all everywhere. Just by adding 30 cm extra.
And what’s more amazing – we solved it through Algebra
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