Banach-Tarski Theorem(interesting to know)

This must be one of the strangest theorems ever proved. In 1924 Banach and Tarski showed that you could cut up a solid sphere into six pieces and then reassemble them into two solid spheres of exactly the same size. In fact, the sphere could be cut and reassembled into any shape or size object whatsoever. However, the shape of the pieces stretch your understanding of volume and area

Divisibility Rules!

To find if some number X is divisible by a certain number, test the number by using the information in the table below.

By 2	If the last digit divisible by two, then X is too
By 3	If the sum of the digits of the number X is divisible by three, then X is too
By 4	If the last two digits are divisible by four, then X is too
By 5	If the last digit is 5 or 0, then X is divisible by 5
By 6	If X is divisible by 2 and by 3, then X is divisible by 6
By 7	This rule is called L-2M. What you do is to double the last digit of the number X and subtract it from X without its last digit. For instance, if the number X you are testing is345678, you would subtract 16 from 34567. Repeat this procedure until you get a number that you know for sure is or is not divisible by seven. Then the X's divisibility will be the same.
By 8	If the last three digits are divisible by 8, then X is too
By 9	If the sum of the digits of the number X is divisible by nine, then X is too
By 10	If the last digit of X is 0, then X is divisible by 10
By 11	What you do here is to make two sums of digits and subtract them. The first sum is the sum of the first, third, fifth, seventh, etc. digits and the other sum is the sum of the second, fourth, sixth, eighth, etc. digits. If, when you subtract the sums from each other, the difference is divisible by 11, then the number X is too
By 12	If X is divisible by 4 and by 3, then X is divisible by 12
By 13	This rule is called L+4M. What you do is to quadruple the last digit of the number X and add it from X without its last digit. For instance, if the number X you are testing is345678, you would add 32 to 34567. Repeat this procedure until you get a number that you know for sure is or is not divisible by thirteen. Then the X's divisibility will be the same.
By 14	If X is divisible by 7 and by 2, then X is divisible by 14
By 15	If X is divisible by 5 and by 3, then X is divisible by 15
By 16	If the last four digits are divisible by 16, then X is too
By 17*	This rule is called L-5M. See rules for 7 and 13 on how to apply.
By 18	If X is divisible by 9 and by 2, then X is divisible by 18
By 19*	This rule is called L+2M. See rules for 7 and 13 on how to apply.
By 20	If X is divisible by 5 and by 4, then X is divisible by 20
By 21	If X is divisible by 7 and by 3, then X is divisible by 21
By 22	If X is divisible by 11 and by 2, then X is divisible by 22
By 24	If X is divisible by 8 and by 3, then X is divisible by 24
By 25	If the last two digits of X are divisible by 25, then X is too
Higher	You can use multiple rules for multiple divisors...for instance, to check if a number is divisible by 57, check to see if it is divisible by 19 and 3, etc., since 57 = 19 x 3...

Guess Your Birthday!

Here's a fun trick to show a friend, a group, or an entire class of people. I have used this fun mathematical trick on thousands of people since 1963 when I learned it. Tell the person (or class) to think of their birthday...and that you are going to guess it.

Step 1) Have them take the month number from their birthday: January = 1, Feb = 2 etc.
Step 2) Multiply that by 5
Step 3) Then add 6
Step 4) Then multiply that total by 4
Step 5) Then add 9
Step 6) Then multiply this total by 5 once again
Step 7) Finally, have them add to that total the day they were born on. If they were born on the 18th, they add 18, etc.

Have them give you the total. In your head, subtract 165, and you will have the month and day they were born on!

How It Works: Let M be the month number and D will be the day number. After the seven steps the expression for their calculation is:

5 (4 (5 M + 6 ) + 9 ) + D = 100 M + D + 165

Thus, if you subtract off the 165, what will remain will be the month in hundreds plus the day!

Did You Know That?? (Contd.)

1. The next sentence is true but you must not believe it
2. The previous sentence was false
3. 12+3-4+5+67+8+9=100 and there exists at least one other representation of 100 with 9 digits in the right order and math operations in between
4. One can cut a pie into 8 pieces with three movements
5. Program=Algorithms+Data Structures
6. There is something the dead eat but if the living eat it, they die.
7. A clock never showing right time might be preferable to the one showing right time twice a day
8. Among all shapes with the same area circle has the shortest perimeter
9. Curves of infinite length may enclose finite areas.
10. Falsity implies anything.
11. There is order in chaos.
12. To get cafe au lait one should carry coffee to milk and not milk to coffee.
13. Sets may be thick, thin and normal.
14. In some circumstances index equals the content.
15. In other circumstances, an index may have a content of its own.
16. There are things distant yet near. There are others that are near yet distant.
17. There are three plane regions that share exactly the same boundary.
18. A continuous linear function must have the form f(x)=ax. Discontinuous linear functions look dreadful.
19. A continuous function may grow considerably virtually without changing.
20. You can't add apples and oranges but you can add their shapes.

Interesting Historical Facts

1. The number 10 is used as a convenient base to count with, but the Gauls of ancient France, the Mayas of Central America, and other peoples used a base of 20. The Sumerians, the Babylonians, and others after them used a base of 60—convenient because 60 can be evenly divided by 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30. The use of base 60 survives in the division of hours into minutes and minutes into seconds, and the division of the circle into 360 (60 × 6) degrees.

2. The earliest known unit of length was used around 2,300 B.C. by megalithic tomb builders in ancient Britain. The name of the unit is not known, but its length was about 2.72 feet

3. Euclid is the most successful textbook writer of all time. His Elements, written around 300 B.C., has gone through more than 1,000 editions since the invention of printing.

4. The modern decimal position system, in which the placing of numerals indicates their value (units, tens, hundreds etc.), was the invention of the Hindus, around 800 A.D. Their invention of the sign for zero greatly simplified arithmetic computation. By comparison, the Roman numeral system containing no zero was awkward.

5.A billion in America is different from a billion in Great Britain. An American billion is a thousand million (1,000,000,000), but a British billion is a million million (1,000,000,000,000). Most of the other names for large numbers are different in the U.S. and the U.K. I typically use the American names in this site.

Did You Know That?

1. =3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 82148 08651 32823 ...
2. A sphere has two sides. However, there are one-sided surfaces.
3. There are shapes of constant width other than the circle. One can even drill square holes.
4. There are just five regular polyhedra
5. In a group of 23 people, at least two have the same birthday with the probability greater than 1/2
6. Everything you can do with a ruler and a compass you can do with the compass alone
7. Among all shapes with the same perimeter a circle has the largest area.
8. There are curves that fill a plane without holes
9. Much as with people, there are irrational, perfect, complex numbers
10. As in philosophy, there are transcendental numbers
11. As in the art, there are imaginary and surreal numbers
12. A straight line has dimension 1, a plane - 2. Fractals have mostly fractional dimension
13. You are wrong if you think Mathematics is not fun
14. Mathematics studies neighborhoods, groups and free groups, rings, ideals, holes, poles and removable poles, trees, growth ...
15. Mathematics also studies models, shapes, curves, cardinals, similarity, consistency, completeness, space ...
16. Among objects of mathematical study are heredity, continuity, jumps, infinity, infinitesimals, paradoxes...
17. Last but not the least, Mathematics studies stability, projections and values, values are often absolute but may also be extreme, local or global.
18. Trigonometry aside, Mathematics comprises fields like Game Theory, Braids Theory, Knot Theory and more
19. One is morally obligated not to do anything impossible
20. Some numbers are square, yet others are triangular

Whats Special About First 20 Numbers

0 is the additive identity.
1 is the multiplicative identity.
2 is the only even prime.
3 is the number of spatial dimensions we live in.
4 is the smallest number of colors sufficient to color all planar maps.
5 is the number of Platonic solids.
6 is the smallest perfect number.
7 is the smallest number of faces of a regular polygon that is not constructible by straightedge and compass.
8 is the largest cube in the Fibonacci sequence.
9 is the maximum number of cubes that are needed to sum to any positive integer.
10 is the base of our number system.
11 is the largest known multiplicative persistence.
12 is the smallest abundant number.
13 is the number of Archimedian solids.
14 is the smallest number n with the property that there are no numbers relatively prime to n smaller numbers.
15 is the smallest composite number n with the property that there is only one group of order n.
16 is the only number of the form x^y = y^x with x and y different integers.
17 is the number of wallpaper groups.
18 is the only number that is twice the sum of its digits.
19 is the maximum number of 4^th powers needed to sum to any number.
20 is the number of rooted trees with 6 vertices.