Tuesday, October 19, 2010
special properities of quadrilaterals
Diagonals are equal --------------------------------------Need not be rectangle or square
Diagonals are perpendicular ---------------------------Need not be a rhombus
Diagonals are perpendicular & equal------------------ Need not be a rhombus, rectangle or square
Diagonals bisect each other -----------------------------Parallelogram
Diagonals bisect & perpendicular----------------------- Rhombus
Diagonals bisect & equal ----------------------------------Rectangle
Diagonals bisect, perpendicular & equal ---------------Square
Name of the quadrilateral whose midpoints are joined-------------The Quadrilateral formed
Quadrilateral---------------------------------------------------------------------------- Parallelogram
Parallelogram ---------------------------------------------------------------------------Parallelogram
Trapezium ------------------------------------------------------------------------------Parallelogram
Rectangle ______________________________----------------------------------Rhombus
Rhombus ---------------------------------------------------------------------------------Rectangle
Square---------------------------------------------------------------------------------------- Square
A NEW CLASSIFICATION OF NUMBERS
P. G. RAJESWARA RAO
p_grrao@yahoo.co.nz
We can’t live without numbers. We need them in our daily activities – big and small.
I would like to share with you some special classification that I have observed with natural numbers
Take any natural number and find its digit sum square it and find the digit sum of the square. If we continue the procedure the outcome is as follows:
Take 22
Sum of digits = 4
Square of 4 = 16
Sum of digits = 7
Square of 7 = 49
Sum of digits = 13
Square of 13 = 169
Sum of digits = 16
Square of 16 = 256
Sum of digits = 13
Square of 13 = 169
22 , 4 , 16 , 7 , 49 , 13 , 169 , 16 , 256 , 13 , 169
The series repeats from 13 to 256 continuously
Take 35
35 , 8 , 64 , 10 , 100 , 1 , 1 , 1 , 1 , ……………………………….
It continues with one only
Take 75
75 , 12 , 144 , 9 , 81 , 9 , 81 , 9 , 81……………………………….
It continues with 9 and 81 alternatively
According to my observation all natural numbers came under any of these three types
So the numbers can be classified into three categories based on the special property
1) C TYPE : Cyclic type which will have repeating series of 13 – 256
2) N TYPE : These are the series of numbers which have 9 , 81 as repeating numbers
3) O TYPE : these are the series if numbers which will have only one recurring
From the above examples
22 is C TYPE
35 is N TYPE
75 is O TYPE
SOME SPECIAL OBSERVATIONS
1) Among the single digit numbers O type are 2, N type are 3, C type are 4.
2) Among the double digit numbers O type are 20, N type are 30, C type are 40.
3) N TYPE has no primes.
4) O TYPE has 7 primes among which 2 pairs are twin primes.
5) C TYPE numbers have 14 primes
6) Multiples of 3 , 6 , 9 belong to N TYPE
7) Multiples of 2 , 4 , 5 , 7 , 8 are distributed in the three categories
8) If a number belongs to one category then the sum of that number and nine multiples belongs to same category
9) This type of classification can also be done with cubes and digit sum there also three categories will exist.
Sunday, October 3, 2010
Saturday, October 2, 2010
NUMBER ZOO - 10
192 x 1 = 192
192 x 2 = 384
192 x 3 = 576
the 3 products contain nine different digits.same for the other 4 nos. try yourself.
NUMBER ZOO - 8
2519 ....is the smallest number ..
gives remainder 9 when divided by 10
gives reminder 8, when divided by 9
gives reminder 7, when divided by 8
gives reminder 6, when divided by 7
gives remainder 5, when divided by 6
gives reminder 4, when divided by 5..... so on...gives reminder 1 when divided by 2.
and gives reminder 0 when divided by 1.
NUMBER ZOO - 6
since 153=1^3 + 5^3 + 3^3.
370, 371, 407 are also Armstrong numbers
NUMBER ZOO -2
the no. itself is a set of counters for its digits.
e.g. 6 represents
6 0's in the no.then
2 1's,
1 2 & 1 6
& no 3,4,5,7,8,9
NUMBER ZOO - 1
the specialty of this number is
3816547280is divisible by 10
381654728 is divisible by 9
38165472is divisible by 8
3816547is divisible by 7
381654is divisible by 6
38165 is divisible by 5
3816 is divisible by 4
381 is divisible by 3
38 is divisible by 2
3 is divisible by 1
Special divisibility rule of 7
7 dividsibilty rule
step 1: divide the number pair wise from the right side
Step 2 : take the remainders with 7
step 3: take 1,2,4,8,16,32,........from right side
Step 4: multiply corresponding remainders with powers of two
step 5: find the sum of the products
step 6 : examine with 7 division
Illustration
N = 1234567
Step 1 : 1 / 23 / 45 / 67
Step 2 : 1 / 2 / 3 / 4
Step 3 : 8 / 4 / 2 / 1
Step 4 : 8 / 8 / 6 /4
Step 5 : 8 + 8 + 6 + 4 = 24
step 6: 24 is not divisible by 7there fore the given number is not divisble by 7
Primefriends
New topic this concept proposed by me:
Prime friend of a certain prime number is a numberwhich is exactly divisible by that prime number andleaves remainder 1, if it is divided by any naturalnumber from 2 to that number.
If N is the PRIME FRIEND of p (prime)N congruent 1 (mod n) where 2 less than or equals to n less than or equals to ( p – 1 )
N is exactly divisible by p
The least number which exhibits this property is taken as PRIME FRIEND
25201 is exactly divisible by 11 and It leaves remainder 1 when it isdivided by any natural number 2 to 10 including both .
Therefore 25201 is the prime friend of11. we
( I and two of my students Naga varun and deepak) have discovered prime friends upto 32 digited numberwe presented this paper in ALL INDIA MATH TEACHERS CONFERENCE