See The Numbers
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See the Numbers
All you guys and gals, you are warned not to repeat what you have done till now, and for which you have repented and cried your hearts out in the past. It would not be respecting yourself and giving recognition to the gift lying within you. So you better pull your socks up and pledge to yourself that from now on you shall daringly explore the hidden facts embedded in mathematics. Before moving forward, I must say that I assume the following about you: 1. 2. 3. 4. 5. 6. 7. 8.
You are aware of basic family tree of natural numbers, whole numbers, integers, fractions, perfect numbers, real numbers and complex numbers. You are aware of simple equations such as linear and quadratic and the basic nuisances encountered in it. You don’t get scared when you see triangles, circles, lines and four sided figures. You don’t mind if the side of any triangle becomes an equation or some equations vomit some numbers as their solutions. You give “zero†the right status just like you give to other numbers though I would never like to see this number in your report card. You never lose the confidence and become the victim of “imagining yourself to be a failure.†You will respect every element of the mathematical fact coming across to you as much as you respect your parents and friends. You are not scared of ghosts, at least the one who teaches you mathematics.
Ok, let us start from the building blocks of mathematics i.e. the natural numbers, or the numbers which we use for counting. My main emphasis here would be tell you how closely you have to look at numbers What happens when we add 1 + 3? The addition gives the number 4 which is a square. What happens when we add 1 + 3 + 5? Again, our addition gives us a perfect square- the number 9. If I take 4 balls and arrange them, I can easily make a square like figure. Same goes with 9 and with all the numbers which are perfect squares Similarly, there are some numbers which when arranged geometrically can give a triangular figure. These numbers are 1, 3, 6, 10, 15...etc. If you are not getting me clearly, following figures would definitely help you out
Observe the triangular behavior of the numbers given as follows
So what’s my point? It’s this- use pictures as much as you can because your imagination is your best friend. What else can you observe here? You added 1 + 3 in the figure and got 4, you added 1 + 3 + 5 and got 9. Therefore, when you add 1 + 3 + 5 + 7 +…up to n numbers in all, you would get n 2 . Moreover if you find the area of the square having dimensions n, it gives you an area n 2 . What else can be derived from these figures? If we put two triangular numbers of the side n together, they form a rectangle, n + 1 by n, whose area is n(n+1). What? What did you say...you don’t agree with me? Ok, let’s check
it out; just observe the following figures.
If you just look at the rectangle at the extreme right corner you would see that its area is n multiplied by n + 1. See how easily we can derive these relations from the numbers diagram? Also, you can say easily that
I shall cover some more extension of this concept on the next page What do you get if you add two consecutive triangular numbers?
That’s a perfect square!! So we saw that numbers gain their importance in our mind only when we explore how they can help us in understanding the relation between them. The ghost shall be happy if the reader understands what he has tried and responds to it. The reader should bear in mind that the ghost is really concerned about them. The reader is advised to explore more relations as it would help him know the numbers in a better way!!
All you guys and gals, you are warned not to repeat what you have done till now, and for which you have repented and cried your hearts out in the past. It would not be respecting yourself and giving recognition to the gift lying within you. So you better pull your socks up and pledge to yourself that from now on you shall daringly explore the hidden facts embedded in mathematics. Before moving forward, I must say that I assume the following about you: 1. 2. 3. 4. 5. 6. 7. 8.
You are aware of basic family tree of natural numbers, whole numbers, integers, fractions, perfect numbers, real numbers and complex numbers. You are aware of simple equations such as linear and quadratic and the basic nuisances encountered in it. You don’t get scared when you see triangles, circles, lines and four sided figures. You don’t mind if the side of any triangle becomes an equation or some equations vomit some numbers as their solutions. You give “zero†the right status just like you give to other numbers though I would never like to see this number in your report card. You never lose the confidence and become the victim of “imagining yourself to be a failure.†You will respect every element of the mathematical fact coming across to you as much as you respect your parents and friends. You are not scared of ghosts, at least the one who teaches you mathematics.
Ok, let us start from the building blocks of mathematics i.e. the natural numbers, or the numbers which we use for counting. My main emphasis here would be tell you how closely you have to look at numbers What happens when we add 1 + 3? The addition gives the number 4 which is a square. What happens when we add 1 + 3 + 5? Again, our addition gives us a perfect square- the number 9. If I take 4 balls and arrange them, I can easily make a square like figure. Same goes with 9 and with all the numbers which are perfect squares Similarly, there are some numbers which when arranged geometrically can give a triangular figure. These numbers are 1, 3, 6, 10, 15...etc. If you are not getting me clearly, following figures would definitely help you out
Observe the triangular behavior of the numbers given as follows
So what’s my point? It’s this- use pictures as much as you can because your imagination is your best friend. What else can you observe here? You added 1 + 3 in the figure and got 4, you added 1 + 3 + 5 and got 9. Therefore, when you add 1 + 3 + 5 + 7 +…up to n numbers in all, you would get n 2 . Moreover if you find the area of the square having dimensions n, it gives you an area n 2 . What else can be derived from these figures? If we put two triangular numbers of the side n together, they form a rectangle, n + 1 by n, whose area is n(n+1). What? What did you say...you don’t agree with me? Ok, let’s check
it out; just observe the following figures.
If you just look at the rectangle at the extreme right corner you would see that its area is n multiplied by n + 1. See how easily we can derive these relations from the numbers diagram? Also, you can say easily that
I shall cover some more extension of this concept on the next page What do you get if you add two consecutive triangular numbers?
That’s a perfect square!! So we saw that numbers gain their importance in our mind only when we explore how they can help us in understanding the relation between them. The ghost shall be happy if the reader understands what he has tried and responds to it. The reader should bear in mind that the ghost is really concerned about them. The reader is advised to explore more relations as it would help him know the numbers in a better way!!
