Number Work
Ramanujam Number
Professor Hardy writes, " I remember going to see him once when Ramanujam was lying ill in Putney. I had ridden in taxi-cab No.1729 and remarked that number seemed to me rather a dull one, and that it was not an unfavourable omen. 'No", he replied," it is a very interesting number". it is the smallest number expressible as a sum of 2 cubes in 2 different ways.
9³ + 10³ = 1³ + 12³ where
9³ + 10³ = 1000 + 729 = 1729
1³ + 12³ = 1728 + 1 = 1729
Hence, it is known as Ramanujam Number.
An amazing number 65!
We have leamt about the great Mathematiolan Ramanujam's number, 1729.
We ask, is there a smallest natural number which can be expressed as a sum of two squares in two different ways?
yes, There is!
It is 65.
65 = 1² +8²
65 = 4²+7²
1 is a cubic number, 8 is a cube number.
That is 65 = 1³ + 4³
Again, 1 is a square number and 4 is also a square number.
Hence, 65 is the sum of cubes of two square numbers.
Another beautitul property!
65 + 56 = 121 = 11²
65 - 56 = 9 = 3²
Try to make every number your friend as Mathematician Ramanujan did.
Divisibility test for 2
Divisibility test for 2 is used in Table-Tennis game,
"After each 2 points have been scored,
the receiving player shall become the serving player and so on until the end of the game-
rule book
Activity1
(1) Take a 4 digit number. (All digits different)
(2) Arrange its digits in descending order. (a)
(3) Arrange its digits in ascending order. (b)
(4) Find (a)-(b).
(5) Repeat the process.
What do you see?
(1) 9265
(2) 9652
(3) 2569
(4) 9652-2569 = 7083
Repetition
(8730, 0378, 8352),
(8532-2358 = 6174),
(7641 - 1467 = 6174!)
Activity2
Write a five digit number.
Reverse it.
Subtract (smaller from the bigger)
score out any one or five digits in the difference. Tell the remaining digits in any order. TE
scored out digit is given immediately.
Example.
59264
(-) 46295
12969
Let the digit scored out be 6.
The remaining digits are 1, 2, 9, 9.
Find the total of 1, 2, 9 and 9. It is 21.
If we add 6, it is divisible by 9. Hence the scored out digit is 6.
{ if 0 or 9 is scored out, we can say either 0 or 9}
Activity3
Palindrome squares
Find the Square of 836.
It is 698896.
A Palindrome square - ( Reads left and right the same number).
Activity 4
Palindrome cubes.
11³ = 1331 = 11 x 11 x 11
101³ = 1030301 = 101 x 101 x 101
111³ = 1367631 = 111 x 111 x 111
11011³ = 1334996994331 = 11011 x 11011 x 11011
Activity5
All the five squares read alike across and downwards,
|
|
7 |
9 |
5 |
2 |
4 |
= 282² |
|
309² |
9 |
5 |
4 |
8 |
1 |
|
|
|
5 |
4 |
7 |
5 |
6 |
=234² |
|
169² |
2 |
8 |
5 |
6 |
1 |
|
|
|
4 |
1 |
6 |
1 |
6 |
=204² |
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