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.

Pizza Slices

Figure 1

Take a pizza and pick an arbitrary point in it. Suppose you cut the pizza into 8 slices by cutting at 45 degree angles through that point, and color the alternate pieces red and green.

Surprising theorem: the total area of the red slices and the total area of the green slices will always be the same!

In fact, this theorem is true if the number of slices is any multiple of 4 except for 4, and the slices are cut by using equal angles through a fixed arbitrary point in the pizza.

Alternatively, if instead of equal angles, you use equal-length arcs on the circumference and slice from a fixed arbitrary point in the pizza, the conclusion still holds if the number of slices is even and greater than 2.

Interesting and Little-Known Algebra and Geometry Facts

Here are a few helpful and neat little facts that evade most students and teachers of algebra and geometry:

a) Though everyone can factor a^2 - b^2, a3 - b^3, a^3 + b^3 and a^4 - b^4, most folks do not know that:

a^4 + b^4 = (a^2 + ab(sqrt(2)) + b^2) (a^2 - ab(sqrt(2)) + b^2

b) An approximation exists for the factorial function (for large n) which seems hardly related but works:

Stirling's Formula ---> n! ~~ e^(-n) n^n sqrt(2 (pi) n)

c) The area of any regular polygon of n sides, each of length x, is given by:

Area = (1/4)nx^2 (cot(180°/n))

d) The radii of circumscribed (R) and inscribed (r) circles within such regular polygons are given by:

R = (x/2) csc (180°/n) and r = (x/2) cot (180°/n)

e) The radius of a circle inscribed within any triangle of sides a, b, and c with semi-perimeter s is given by:

r = (sqrt (s (s-a) (s-b) (s-c)) / s

e) The radius of a circle circumscribed about any triangle of sides a, b, and c with semi-perimeter s is given by:

R = abc / 4 (sqrt (s (s-a) (s-b) (s-c))

f) The perimeter P and area A of polygons (of n sides) inscribed in a circle of radius r is given by:

P = 2nr sin(pi/n) and A = (1/2) nr^2 sin (2pi/n)

g) The perimeter P and area A of polygons (of n sides) circumscribed about a circle of radius r is given by:

P = 2nr tan (pi/n) and A = nr^2 tan (pi/n)

h) Factoring the seemingly prime expression a^4 + 4b^4 becomes (and there are a family of these)

a^4 + 4b^4 = (a^2 + 2b^2)^2 - (2ab)^2 (thanks to N. Hobson)

500 Digits of e

Named after the world famous mathematician and extreme child prodigy Leonhard Euler, the natural logarithmic base has innumerable applications in all fields of science, business, and mathematics...here is just the first 500 digits or so...

2.71828 18284 59045 23536 02874 71352 66249
77572 47093 69995 95749 66967 62772 40766 30353 54759 45713
82178 52516 64274 27466 39193 20030 59921 81741 35966 29043
57290 03342 95260 59563 07381 32328 62794 34907 63233 82988
07531 95251 01901 15738 34187 93070 21540 89149 93488 41675
09244 76146 06680 82264 80016 84774 11853 74234 54424 37107
53907 77449 92069 55170 27618 38606 26133 13845 83000 75204
49338 26560 29760 67371 13200 70932 87091 27443 74704 72306
96977 20931 01416 92836 81902 55151 08657 46377 21112 52389
78442 50569 53696 77078 54499 69967 94686 44549 05987 93163
68892 30098 79312 77361 78215 42499 92295 76351 48220 82698
95193 66803 31825 28869 39849 64651 05820 93923 98294 88793
32036 25094 43117 30123 81970 68416 14039 70198 37679 32068
32823 76464 80429 53118 02328 78250 98194 55815 30175 67173

Chisenbop Multiplying by 9

Hold out your hands in front of you so that your thumbs point toward one another.

Visualize that your left pinky finger represents 1, the next finger 2, and so on left to right, until your right pinky finger represents 10. Those fingers represent the number you wish to multiply by 9. To do so, simply put the finger down you wish to multiply by 9. All fingers to the left of the down finger represent the tens digit of the answer while all fingers to the right represent the ones digit.

Example: 6 x 9. Put the finger representing 6 down (the right hand thumb). To the left of the down finger, you have 5 fingers up. That's your tens digit, 5. To the right, you have 4 fingers up. There's your ones digit, 4. Put those together and you have your answer: 54. Pretty cute.

Arithmetic Curiosities

- Here are just a few interesting patterns in arithmetic that you or your students may explore. Verify these results with paper and pencil or with calculator (if you must):

1 x 9 + 2 = 11	9 x 9 + 7 = 88	9 x 9 = 81	6 x 7 = 42
12 x 9 + 3 = 111	98 x 9 + 6 = 888	99 x 99 = 9801	66 x 67 = 4422
123 x 9 + 4 = 1111	987 x 9 + 5 = 8888	999 x 999 = 998001	666 x 667 = 444222
1234 x 9 + 3 = 11111	9876 x 9 + 4 = 88888	9999 x 9999 = 99980001	6666 x 6667 = 44442222
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	1 x 8 + 1 = 9 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	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