Joyful Gurukul Days

This lesson every parent/teacher must teach their students.


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.punyani

Making division by 5 easy!

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That's the thing about numbers :)
Come join us, to experience the beauty of numbers 🌸
Joyful Gurukul Days, starting with fresh batches tomorrow.
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Ever wondered about TAXI NUMBERS???
If not, m sure you'll start doing it after reading our mathematical story of the week.
Enjoy!

The British mathematician G. H. Hardy went to the hospital to see the Indian mathematician Srinivasa Ramanujan. In Hardy's words,
"I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."

For those who dont know,
1729=1^3+12^3=9^3+10^3
1729 is known as the Hardy–Ramanujan number.

Courtesy: google.

Wait for another story untill next wednesday.:)

I always felt that any invention in this world came from a very simple idea, which make people eager to know what's in store next and they take up and start contributing by building up techniques in various different directions, which then help the society at large.

Yesterday A BIG WAOW came on my face when i was reading Gauss story about his trick to sum the numbers from 1 to 100 and i decided to share.

Human beings are curious species indeed, and to broaden this and enlighten you with some more on the same, I am going to post Different ways of finding Sum of Natural Numbers. I hope you enjoy it.

1. Sum to n of the Natural Numbers Using Differences.

We can find the formula for a set of numbers using differences.

For instance, the following table shows the sum of some natural numbers, but we have also used zero, for convenience:
n 0 1 2 3 4 5
Sn 0 1 3 6 10 15
Δ1 1 2 3 4 5
Δ2 1 1 1 1
Sn is the sum of the numbers to n.

Because we find that Δ2 produces constant values, we assume the formula for the sum of the natural numbers is a quadratic, of the form an^2+bn+c.

Using our values, we substitute 0, 1, and 3 in the Equation:
n Equation Equation Number
0 c =0 1
1 a+b=1 2
2 4a+2b=3 3
In Equations 2 and 3, we have noted that c=0.
By subtracting twice Equation 2 from Equation 3, we get:
2a=1,
So
a=1/2
Substituting the value for a in Equation 2, we find that b is also 1/2,
So the sum of the first n natural numbers, Sn,

Sn = n^2/2 + n/2 = n(n+1)/2

Bravo!
Wait for more on it till tomorrow.😁

Have u noticed we end up saying *why does bad happen with every time?* or *everybody is the same* even if the event had happen with you just once.
Ever wondered why?? hmmm?

Relax My dear, its not your fault, you r normal. Such is a nature of mind. Very peculiar. It's Always in look out to generalise a behaviour / pattern be it situational /personal or mathematical.

When we read Gauss's story from yesterday, it's natural to ask that how we can generalise it to any natural number ? is there a formula by which we can find it?
isn't it?

Guys, your wait ends here. n there you go with the derivation of formula of same for any set of n natural numbers. I hope you enjoy it.

Let S

S = 1 + 2 + 3 + ......... + n

S= n + (n-1) + (n-2) + ......... + 1

________________________

2S = (n+1) + (n+1) + (n+1) + .......... + (n+1) [n times (n+1)]

2S = n(n+1)

S = n(n+1)/2

And all the credit of this wonderful discovery goes to Sir Gauss which gave us a powerful chapter named "Arithmetic Progressions."

And WHY? led to AHA !! so happy.😀😀

Stay tuned for the next one today.

Mathematical Story of the week :

1. The one of Gauss

There was a boy in a class studying math with, of course, a math teacher. This boy's name is Carl Friedrich Gauss (1777 - 1855). One day this math teacher presented a challenging mathematical problem to the class where Gauss is in.

The math problem is to add up all the numbers starting from 1 and ending with 100.

Every students picked up a piece of paper and started to add up the numbers one after another from number 1 onwards.

Within a short span of time, while his fellow students were still struggling, Gauss went forward to the teacher and submitted his answer.

That action surprised not only his math teacher but the whole class. But that is not all.....

The interesting thing is that his answer is correct.

How did he do that so fast?

He came out a different way of analysing the mathematical problem. Instead of the normal way of adding the first numbers onwards, Gauss looked at the problem with a different angle.

What he did was to split the range of number from 1 to 100 into two equal halves, 1 to 50 and 51 to 100. He noticed that if he flipped the last half to start from 100, and adding it the two ranges together, he will get something stunting.

He discovered that by adding the first pair, 1 + 100, he got an answer of 101. For the second pair, 2 + 99, he again got the same answer 101.

This answer of 101 was still valid for the rest of the number pair addition. And since there were 50 pairs of numbers, the final total is 101 x 50 which gave Gauss an answer of 5050.

Courtesy: Google

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B.tech: Can anyone tell what is the nth differential coefficient? And calculate for 1/(ax+b) ?

CLASS 11: CONIC SECTIONS
The equation of hyperbola with vertices (0,6), (0,-6) and eccentricity 5/3 is ______ and its foci are ______.