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#1 2005-08-30 18:11:07
- Jai Ganesh
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- Registered: 2005-06-28
- Posts: 48,423
2n ones and n twos
I noticed this when I was browsing the net for interesting Mathematics.
I liked this proof, maybe you like it too!
Write, side by side, the numeral 1 an even number of times. Subtract from the number thus formed the number obtained by writing, side by side, a series of 2s half the length of the first number. You will always get a perfect square. For instance,
1111 - 22 = 1089 = 33²
Can you say why this is?
11...1 - 22...2 = 11...1 11...1 - 2(11...1)
------ ------ ------ ------ ------
2n times n times n times n times n times
= 11...1 00...0 - 11...1
------ ------ ------
n times n times n times
= 11...1 x (100...0 - 1)
------ ------
n times n times
= 11...1 x 99...9
------ ------
n times n times
= 11...1 x 9 x 11...1
------ ------
n times n times
= 3² x 11...1²
------
n times
= 33...3²
------
n times
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
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#2 2005-08-31 19:35:46
- wcy
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- Registered: 2005-08-04
- Posts: 117
Re: 2n ones and n twos
wow this is amazing
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#3 2005-09-01 06:54:18
- Roraborealis
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- Registered: 2005-03-17
- Posts: 1,594
Re: 2n ones and n twos
Why does that work?
School is practice for the future. Practice makes perfect. But - nobody's perfect, so why practice?
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#4 2005-09-01 20:05:43
- Jai Ganesh
- Administrator
- Registered: 2005-06-28
- Posts: 48,423
Re: 2n ones and n twos
Because, the resultant is always 3² or 33² or 333² or 3333² etc.
Follow every step of the proof carefully, you can understand the reasoning 
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
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