Take a look at the following example to see what happens when the restriction is violated.
Thursday, July 30, 2009
Common Math Mistake 9
Take a look at the following example to see what happens when the restriction is violated.
Common Math Mistake 8
Remember that if n is a positive integer then
The tan x^2 is actually not the best notation for this type of problem. We really should probably use tan( x^2) to make things clear.
Another thing to always keep in mind is that 1 in the cos^1(x) is NOT an exponent , it is there to denote the fact that we are dealing with an inverse trig function. ie.
Wednesday, July 29, 2009
What is a perfect number?
A perfect number is a whole number, an integer greater than zero; and when you add up all of the factors less than that number, you get that number. For example:
The factors of 6 are 1, 2, 3 and 6.
1 + 2 + 3 = 6
The factors of 28 are 1, 2, 4, 7, 14 and 28.
1 + 2 + 4 + 7 + 14 = 28
The factors of 496 are 1, 2, 4, 8, 16, 31, 62, 124, 248 and 496.
1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496
The factors of 8128 are 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064 and 8128.
The first four perfect numbers were known over 2,000 years ago. Some ancient cultures gave mystic interpretations to numbers that they thought were magic. Using a computer program, eventually ran for a billion of time the following 5 perfect numbers can be found:
28 = 1+2+4+7+14
496 = 1+2+4+8+16+31+62+124+248
8,128 = 1+2+4+8+16+32+64+127+254+508+1016+2032+4064
All of the perfect numbers that have been found so far fit the formula
2^(n1) * ( 2^n  1 )
where "n" is one of a very short list of prime numbers that can be used to create "Mersenne" prime numbers.
2^1 * ( 2^2  1 ) = 2 * 3 = 6
2^2 * ( 2^3  1 ) = 4 * 7 = 28
2^4 * ( 2^5  1 ) = 16 * 31 = 496
2^6 * ( 2^7  1 ) = 64 * 127 = 8,128
2^12 * (2^131) = 4096 * 8191 = 33,550,336
2^16 * (2^171) = 65536 * 131071 = 8,589,869,056
2^18 * (2^191) = 262144 * 524287 = 137,438,691,328
2^30 * (2^311) = 1073741824*2147483647=2,305,843,008,139,952,128
How many perfect numbers are there?
We do not know how many perfect numbers there are. We do know that there are an infinite number of prime numbers, which means there is a very high chance that there are an infinite number of perfect numbers. This is because there is a strong link between perfect numbers and a certain kind of prime number (the Mersenne primes).
How many perfect numbers are known?
So far, according to the Mersenne organization, there are 37 known Mersenne prime numbers. This means that there are 37 known "perfect" numbers. The newest prime number was discovered on
Are there any odd perfect numbers?
Nobody has found any odd perfect numbers, but we do not know if any odd ones exist. If any odd perfect numbers exist, they are not based on the "Mersenne" method of calculating perfect numbers.
What is pi?
A very brief history of pi:
Monday, July 27, 2009
Common Math Mistake 7
Most trig classes that I’ve seen taught tend to concentrate on doing things in degrees. I suppose that this is because it’s easier for the students to visualize, but the reality is that almost all of calculus is done in radians and students too often come out of a trig class ill prepared to deal with all the radians in a calculus class.
You simply must get used to doing everything in radians in a calculus class. If you are
asked to evaluate cos (x) at x =10 we are asking you to use 10 radians not 10 degrees!
The answers are very, very different! Consider the following,
cos (10) =  0.839071529076 in radians
cos (10) = 0.984807753012 in degrees
You’ll notice that they aren’t even the same sign!
So, be careful and make sure that you always use radians when dealing with trig functions in a trig class.
POLYOMINO
A polyomino is a polygon made up of squares joined edgetoedge. There is only one type of domino (two squares) and two types of trominoes (3 squares), but there are five different tetrominoes.
So... now take out your pen and paper to draw all the possible 35 Hexomino! Its really interesting...
Sunday, July 26, 2009
Convert 1 to 9 = 100
Convert the numbers 1 to 9 which will results to "100", by inserting any of arithmetic operations (+  * /) into the string of above numbers.
And one more interesting thing is by keeping the numbers in the above order i.e. 123456789..
One is: 1+2+34+5+6+78+9
Some others are:
123+4567+89=100
12+3+456/7+89=100
1+2+3+4+5+6+7+8*9=100
1 + (2*3*4) + (5*6)+(7*8)  9 =100
1*2+3+45+6789=100
(1+2+3+4)*5+67(8+9)=100
If u have any other such expression, please write in the comment ....
TRICK TO SUBSTRACT NUMBER EASILY
To subtract a large number from 1,000
subtract all but the last number from 9, then subtract the last number from 10
1000
547
step1: subtract 5 from 9 = 4
step2: subtract 4 from 9 = 5
step3: subtract 7 from 10 = 3
answer: 453
BEAUTY OF MATHS
12 x 8 + 2 = 98
123 x 8 + 3 = 987
1234 x 8 + 4 = 9876
12345 x 8 + 5 = 98765
123456 x 8 + 6 = 987654
1234567 x 8 + 7 = 9876543
12345678 x 8 + 8 = 98765432
123456789 x 8 + 9 = 987654321
1 x 9 + 2 = 11
12 x 9 + 3 = 111
123 x 9 + 4 = 1111
1234 x 9 + 5 = 11111
12345 x 9 + 6 = 111111
123456 x 9 + 7 = 1111111
1234567 x 9 + 8 = 11111111
12345678 x 9 + 9 = 111111111
123456789 x 9 + 10 = 1111111111
9 x 9 + 7 = 88
98 x 9 + 6 = 888
987 x 9 + 5 = 8888
9876 x 9 + 4 = 88888
98765 x 9 + 3 = 888888
987654 x 9 + 2 = 8888888
9876543 x 9 + 1 = 88888888
98765432 x 9 + 0 = 888888888
1 x 1 = 1
11 x 11 = 121
111 x 111 = 12321
1111 x 1111 = 1234321
11111 x 11111 = 123454321
111111 x 111111 = 12345654321
1111111 x 1111111 = 1234567654321
11111111 x 11111111 = 123456787654321
111111111 x 111111111 = 12345678987654321
12345679 x 9 = 111111111
12345679 x 18 = 222222222
12345679 x 27 = 333333333
12345679 x 36 = 444444444
12345679 x 45 = 555555555
12345679 x 54 = 666666666
12345679 x 63 = 777777777
12345679 x 72 = 888888888
12345679 x 81 = 999999999
9 x 9 = 81
99 x 99 = 9801
999 x 999 = 998001
9999 x 9999 = 99980001
99999 x 99999 = 9999800001
999999 x 999999 = 999998000001
9999999 x 9999999 = 99999980000001
99999999 x 99999999 = 9999999800000001
999999999 x 999999999 = 999999998000000001
6 x 7 = 42
66 x 67 = 4422
666 x 667 = 444222
6666 x 6667 = 44442222
66666 x 66667 = 4444422222
666666 x 666667 = 444444222222
6666666 x 6666667 = 44444442222222
66666666 x 66666667 = 4444444422222222
666666666 x 666666667 = 444444444222222222
111/1+1+1 = 37
222/2+2+2 = 37
333/3+3+3 = 37
444/4+4+4 = 37
555/5+5+5=37
666/6+6+6=37
777/7+7+7=37
888/8+8+8=37
999/9+9+9=37
PLATONIC SOLIDS
A 3dimensional object bounded by regular polygons.
The five regular polyhedrons are Tetrahedron, Cube, Octahedron, Icosahedron and Dodecahedron.
Ramanujan Number 1729
One favorite story about Ramanujan
One favorite story about Ramanujan revolves around a visit that Hardy paid to him in hospital. Hardy and Ramanujan had a habit of discussing the properties of different numbers. On this particular visit, Hardy commented to Ramanujan that the number of the taxi that he had just arrived in was 1729  a very uninteresting number. Ramanujan quickly replied that it was in fact a very interesting number as it was the smallest number that could be represented as the sum of two cubes in two ways:
1729 = 10^{3} + 9^{3}
and
1729 = 12^{3} + 1^{3}
Common Math Mistake 6
Many a times while solving some question, we have division of two fractions.
For example: simplify
This has got two possible values… HOW? Check this out!
Noticed the difference in the way of solving! Well, which is the correct answer? BOTH...
In case (i) it is divide 2 by (3/5) and in case (ii) it is divide (2/3) by 5 .
In doubt? Do comment!
Saturday, July 25, 2009
Common Math Mistake 5
Distributive Property of multiplication over addition
Let me consider a very common mistake committed by many students. Simplify the following:
Many students usually simplify this as Do not forget to apply the distributive property ie. 3 will get multiplied with both 2x and 5. So, the correct answer is:
Hope you will not be one of them …
Friday, July 24, 2009
Common Math Mistake 4
There seems to be a very large misconception about the use of square roots out there.
Students seem to be under the misconception that
This is not correct however. Square roots are ALWAYS positive! So the correct value is
This is the ONLY value of the square root! If we want the 4 then we do the following
Here is the proper solution technique for this problem.
Thursday, July 23, 2009
Common Math Mistake 3
Consider this question: Solve the following for x :Too many students get used to just canceling (i.e. simplifying) things to make their life easier. So, the biggest mistake in solving this kind of equation is to cancel an x from both sides to get,While,x = 1 is a solution, there is another solution that we’ve missed. Can you see what it is?
Wednesday, July 22, 2009
22 July 2009 is Pi Approximation Day
Dear all
Common Math Mistake 2
One common error that many students make is based on what they assume.
Since 2(x + y) = 2x + 2y then everything works like this. However, here is a whole list in which this doesn’t work.
It’s not hard to convince yourself that any of these aren’t true. Just pick a couple of numbers and plug them in! For instance,
I hope that you will surly not commit this error!
Tuesday, July 21, 2009
Bihar cop in league of top mathematicians
The website is jointly edited by wellknown maths wizards Michael Roby Wetherfield of the UK and Prof Hwang Chienlih of National Taiwan University, Taipei.
A collection of identities for computing the value of `pi', discovered by Nimbran during 200708, were sent to the two mathematicians on June 1, 2009. Four of these made their way to the website on June 7, 2009. In a subsequent email to Nimbran, Wetherfield acknowledged that "stimulated by your (Nimbran's) results, I generated new identities myself within a week".
Nimbran came up with two more formulae this month. These two too Wetherfield and Chienlih uploaded on their website. "I am surprised by your super ability about finding new excellent identities!" Chienlih wrote to Nimbran in a congratulatory mail.
`Pi' is a mysterious number that denotes the ratio of the circumference of a circle to its diameter. In general parlance, its value is taken as 22/7 which is just an approximation. Mathematicians across the world have been fascinated by this number and research has been going on to find a better approximation. Therefore, any formula which gives a better value of the `pi' in the shortest possible time is welcome in the mathematical world.
The Bihar cop has been engaged in mathematical research actively for over a decade and some of his research papers have been published in reputed journals brought out by Bihar Mathematical Society, Indian Mathematical Society and National Council of Educational Research and Training. His papers were also sent to the journals of London Mathematical Society and Fib Quarterly of The Fibonacci Association (USA) which commented favourably on the works of Nimbran.
A BA with economics and maths as optionals, Nimbran was the first nonmathematician to give an elementary proof of Fermat's last theorem for a cube in which he had proved that no integral cube number can be broken into two integral cube numbers. Prof K C Prasad of Ranchi University and Prof Emeritus Tej Narain Sinha of Bhagalpur University, who are regarded as the authority on TarryEscott problem (on numbers), expressed surprise on seeing Nimbran's proof.
"I was inspired by reading the works of great Indian mathematician S Ramanujan," Nimbran told TOI. He said he evaluates the value of `pi' through different approaches using Inverse Tangent Function which was first used by John Machin in 1706 and has since been used till date to find the value of `pi'.
"Now there are only three leading authorities on `pi' in the world Wetherfield, Lih and Jorg Arndt of Australian National University," Nimbran said. He was modest enough not to include his name in the league.
Common Math Mistake 1
Everyone knows that 0/2 = 0 , the problem is that far too many people also say that 2/0 = 0 or 2/0= 2! Remember that division by zero is undefined! You simply cannot divide by zero so don’t do it!
Here is a very good example that can arise when you divide by zero. See if you can find the mistake in the work below.
So, we’ve managed to prove that 1 = 2! Now, we know that’s not true so clearly we made a mistake somewhere. Can you see where the mistake was made?
However, if this is true then we have a b = 0 ! So, in step 5 we are really dividing by zero!
That simple mistake led us to something that we knew wasn’t true, however, in most cases your answer will not obviously be wrong. It will not always be clear that you are dividing by zero, as was the case in this example. You need to be on the lookout for this kind of thing.
Remember that you CAN’T divide by zero!
Math is NOT a spectator sport!
You should always try to get the best grade possible! You might be surprised and do better than you expected. At the very least you will lessen the chances of underestimating the amount of work required and getting behind.
I will now write few posts about the common math mistakes which students do commit.I’m convinced that many of the mistakes are caused by student getting lazy or getting in a hurry and not paying attention to what they’re doing.
I hope that the students will be able to understand this and will assure not to repeat such mistakes in future.
Monday, July 20, 2009
NUMBER OF ZEROES
Let me share a very interesting question with all of you.
If you multiply first 100 natural numbers, how many zeroes will come at the end of this product?
Do we really need to multiply all the numbers from 1 to 100 or else you have some logic?
Please read below for further details and enjoy solving mathematics!
well, class XI and XII students know well about the greatest integer function. Junior class students can also find out a simple logic to solve this problem.
Sunday, July 19, 2009
Kaprekar Constant
Given by D.R. Kaprekar (19051988) : Fond of numbers. Well known for "Kaprekar Constant" 6174.
Procedure to arrive Kaprekar Constant:
1. Take any four digit number in which all digits are not alike.
2. Arrange its digits in descending order and subtract from it the number formed by arranging the digits in ascending order.
3. If this process is repeated with reminders, ultimately number 6174 is obtained, which then generates itself.
click here to know more.
Try it! its very interesting...
Friday, July 17, 2009
Which is greater: pi or 22/7?
We know that the approximate value of pi is 22/7. Have you ever thought that which is actually greater, pi or 22/7?
Here is a result from calculus (don't worry, if you have not studied that... just enjoy it!)
Here, the given function is positive and since it is integrated in positive limits, so its value is positive.
\frac{22}{7}\pi> 0
i.e
\frac{22}{7} > \pi
Divisibility rules for 7, 13, etc.
When i was studying in the junior classes, I was taught about the divisibility rules for 2, 3, 4, 5, 6, 8, 9, 10 etc... I asked my teacher that may be by mistake you have skipped the rule for number 7. He told that there is no divisibility rule for 7. I was surprised, but no solution at that time.
Well, after many years... I came across rule for not only 7, but for given any prime number using vedic mathematics... it sounds good!
I am sharing that with all the readers.
Thursday, July 16, 2009
No. of Squares and Rectangles
Wednesday, July 15, 2009
I witnessed Basketball match
Tuesday, July 14, 2009
XII C teaching
Monday, July 13, 2009
Fermat's Last Theorem
Fermat's Last Theorem states that
x^n + y^n = z^n
has no nonzero integer solutions for x, y and z when n > 2.
That is to say, there are no integers x, y, z such that x^3 + y^3 = z^3, or integers x, y, z such that x^7 + y^7 = z^7.
Inspiring Math Quotations
"Perfect numbers like perfect men are very rare." – Descartes
"The reading of all good books is like a conversation with the finest people of past centuries." – Descartes
"I hear and I forget. I see and I remember. I do and I understand."  Chinese Proverb.
Small minds discuss persons. Average minds discuss events. Great minds discuss ideas. Really great minds discuss mathematics."
"The advancement and perfection of mathematics are intimately connected with the prosperity of the state." – Napoleon
"A man is like a fraction whose numerator is what he is and whose denominator is what he thinks of himself. The larger the denominator, the smaller the fraction." – Tolstoy
"The mathematical sciences particularly exhibit order, symmetry, and limitations; and these are the greatest forms of the beautiful." – Aristotle
NUMBER OF TRAILING ZEROES IN A PRODUCT (PART 2)
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