Maths Mania

 Mathematics is made of two Greek words 'Mathema' and 'Mathematikos'.
Mathema means science, knowledge or learning.
Mathematikos means found of learning.
So MATHEMATICS means found of learning of science or knowledge.

Approximation of π

Aryabhata worked on the approximation for pi (), and may have come to the conclusion that  is irrational. In the second part of the Aryabhatiyam (gaṇitapāda 10), he writes:

caturadhikam śatamaṣṭaguṇam dvāṣaṣṭistathā sahasrāṇāmayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.

"Add four to 100, multiply by eight, and then add 62,000. By this rule the circumference of a circle with a diameter of 20,000 can be approached." 

This implies that the ratio of the circumference to the diameter is ((4 + 100) × 8 + 62000)/20000 = 62832/20000 = 3.1416, which is accurate to five significant figures.

It is speculated that Aryabhata used the word āsanna (approaching), to mean that not only is this an approximation but that the value is incommensurable (or irrational). If this is correct, it is quite a sophisticated insight, because the irrationality of pi was proved in Europe only in 1761 by Lambert.

After Aryabhatiya was translated into Arabic (c. 820 CE) this approximation was mentioned in Al-Khwarizmi's book on algebra

 Train Your Brain

Q1. Consider three buckets of capacity 3L,5L,8L respectively. If 8L bucket is filled completely with water and other two buckets are given empty. Then what will you  do to get  exact  4L water in 8L bucket atlast. Condition is that you have to use all the three buckets ?

Q2. Consider an general equation of ellipse, having a circle inside it. Such that a ray passing through center of circle and falls on a point where circle and ellipse touches each other. After it, the ray reflected  back and passes through center of ellipse. find the radius of circle?

Q3. Any number (x) raise to power zero  equals to one .Why?

Q4. Any number raise to power infinite equals to infinite but one raise to power infinite is one. Why ?

         
Q5. A line parallel to axis of parabola suffers reflection.Find equation of line after reflection if slope of line is 2.(Taking standard equation of parabola).Also find the points where reflected line touches parabola.

Q6. If equation of parabola is y^2=4x.find equation of normal if normal passes thrugh (1,1).How many normals passes through this curve?

Q7.                                  1^2         2^2         3^2          4^2        5^2          6^2       ................................ .
1st successive difference        3              5              7             9             11   ....................... .
 2nd successive difference              2              2              2            2     .................... .
 
                                         1^3              2^3            3^3             4^3               5^3         6^3................... .
1st succesive difference              7                   19              37               61                91............. .
2nd succesive difference                       12                18             24                 30........... .
3rd successive difference                                6                6               6................ .

Is this true for 5th power    1^5      2^5       3^5         4^5       5^5        6^5..............?
Is  this   true for nth power  1^n     2^n        3^n        4^n        5^n         6^n..............?

 Maths Magic

1. Square and cube of number 69 includes all numbers (i.e from 0 to 9).

2. As we know that square of 13 is 169. An interesting fact regarding this is, if digit 13 is reversed i.e 31 and we square the reversed number, we get number 961 which is the reverse of 169.
3. Think of a five digit number and multiply it with 11 and then multiply the result with 9091.You should get an interesting fact.
4. Curves of infinite length may cover finite area.
5. A continous function may be differentiable but a differentiable function is always continous.
6. In a group of 23 people atleast two have the same birthday with the probability greater than 1/2.
7.One can cut

π into 8 pieces with three movements.

8. A continuous function may grow considerably virtually without changing.
9. Equilateral triangle is the only triangle having rational sides and and angles.
10. We cannot add different things but their shapes are added(We cannot add apples abd oranges but we can add their shapes).
11. An interesting fact about no.s which makes some different series and all are prime.
          31
          331
          3331
          33331
          333331
          3333331
all these are prime numbers.
* But 33333331 is not a prime number.
12. Note the fact :
       here (a/b=c); c is remainder.
       2519/2=1,
       2519/3=2,
       2519/4=3,
       2519/5=4,
       2519/6=5,
       2519/7=6,
       2519/8=7,
       2519/9=8,
       2519/10=9,
13. A straight line has 1 dimension ,a plane has two and fractals have almost fractal dimensions.
14. 1/1089=0.0009182736455463637281.....................................................Note that when we are dividing 1 by 1089 we get value increasing pattern of 9 table.
15. Four is the only word in english language that is spelt with same amount of letters as the number itself.

16. If 1089*9=9801.The digits are reversed.Also 10989,109989,1099989 and so on when multiplied with 9 reverses its digits.

17. When 21978*4=87912.The number 21978 reverses its digits.
18. If u divide square of any number by 8 u always get remainder 0,1,4.
19. In english language "FORTY" is the only number that has all its letter in alphabetical order.
20. 1/37=0.0270270270............... and 1/27=0.0370370370.................... .Note the result.
21. A circle circumscribing the triangle formed by three co-normal points passes through the vertex of parabola.
22. A circle circumscribing the triangle formed by any three tangent to a parabola passes through the focus.
23. The radical axis of two circles is the locus of a point which moves so that the length of tangent drawn from it to two circles are equal.
24.