Maths Diary of a Nutter: 2008

Monday, December 8, 2008

Pascal triangle and number 11

When I was playing with number 11, I find a rather familiar number pattern for 11^x
we all know that 11^2 = 121 and 11^3 = 1331
121 and 1331 remind me of the polynomial (a+b)^x which leads me to pascal triangle.

Look at the number rows above. The first represent 11^0 which is 1, the second represent 11^1 which is 11 and soon on.
does this mean that 11^5 = 15101051? of course not. It should be calculated this way.

So, the answer is 161051

This method could be applied for 11^x for any value of x due to the nature of no.11

11 = 10+1

Just replace the x with 10 and the y with 1, we will get

$(10+1)^n = \sum_{k=0}^{n}{10^{n-k}1^k }$

since 1^k

will always be one,

$(10+1)^n = (_{0}^{n}) 10^n + (_{1}^{n}) 10^{n-1} + (_{2}^{n}) 10^{n-2} + (_{3}^{n}) 10^{n-3} + ... +(_{n-2}^{n}) 10^2 + (_{n-1}^{n}) 10^{1} + 1$

notice that $(_{k}^{n})$ can be derived from pascal triangle.

The last digit of 11^x will always be 1 when the second last will always be the last digit of x

edit: I just realised that the long equation is being cut. Since it does not affect that much, I'll just leave it that way.

last edited on : 15th of May 2010

Tuesday, November 25, 2008

Subtraction problem

Consider this subtraction problem

1241
- 587

Most people won't like to do this in their head (or even paper)

This subtraction problem could be simplified by subtracting 600 instead of 587

1241-600 = 641

But, we have subtracted too much, 13 (600-587) too much.
So we add the "missing" 13 to the 641 we have. And we get the answer which is 654.

End of Chapter 0

* If there is a positive feedback, I'll post more of this.

Tuesday, November 18, 2008

Tips on fast mental calculation

Hey guys, I just bought a book on mental calculation. It is called Think Like A Maths Genius, written by Arthur Benjamin and Michael Shermer. Some of the tricks are damn obvious, but I'll type it here anyway.

Chapter 0
Quick Tricks:
Easy (and impressive) Calculation

Instant Multiplication of 11

71 x 11
To solve this,, add the digits 7 + 1 = 8 in the middle of 7 and 1
71 x 11 = 781

How about
76 x 11
Similar way, 7 + 6 = 13
BUT the answer is not 7136
It should be done this way

1
736+
836

How about 3 digit numbers?
Simple. It can be done in similar way.
Eg. 123 x 11
1 + 2 = 3
2 + 3 = 5
123 x 11 = 1353

Multipying 2 2-digit number with the same first digit and second digit that sums to 10.

71 x 79
The first 2 digit will be 7 x 8 = 56
and The last 2 digit will be 1 x 9 = 9
so the answer is 5609.

*more will be added soon.