Math Magic.. :D

Mathematics: Interesting facts and numbers

Some beautifull examples with multiplication
;
12345679 x 9 = 111111111;
12345679 x 8 = 98765432;

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An interesting number 2519
Let us look at 2519 Mod n (n = 2, ... 10).
2519 Mod n means reminder portion of (2519\n), where "\" is the integer division.

2519 Mod 2 = 1; 2519 Mod 3 = 2; 2519 Mod 4 = 3; 2519 Mod 5 = 4; 2519 Mod 6 = 5; 2519 Mod 7 = 6; 2519 Mod 8 = 7; 2519 Mod 9 = 8; 2519 Mod 10 = 9.

2519 = 1259 x 2 + 1; 2519 = 839 x 3 + 2; 2519 = 629 x 4 + 3; 2519 = 503 x 5 + 4; 2519 = 419 x 6 + 5; 2519 = 359 x 7 + 6; 2519 = 314 x 8 + 7; 2519 = 279 x 9 + 8; 2519 = 251 x 10 + 9.

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Example of wrong proof
Find a mistake in the following chain of arguments, pretending to prove that 2=1

1)	Let a = b
2)	Multiply 1) by a a2 = ab
3)	Add a2 – 2ab to both parts of 2) a2 + a2 – 2ab = ab + a2 – 2ab
4)	3) could be simplified: 2a2 – 2ab = a2 – ab
5)	It is the same as 2(a2 – ab) = 1(a2 – ab)
6)	Reduce 5) by (a2 – ab). 2=1

Where is a mistake?
Mistake is in the 6^th step.
We can not divide by (a² – ab) because a²– ab = 0.
a = b, so a²– ab = 0.

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An interesting fact about primes
Mathematicians of XVIII^th century proved that numbers 31; 331; 3331; 33331; 333331; 3333331; 33333331 are primes. It was a big temptation to think that all numbers of such kind are primes. But the next number is not a prime.
333333331 = 17 * 19607843

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An elegant proof that
It is obvious that 1 = (2 -1).
= * (2 -1) = (1 + 2 + 2² + ... + 2ⁿ) * (2 -1) =
(2 + 2² + 2³ ... + 2ⁿ⁺¹) - (1 + 2 + 2² + ... + 2ⁿ) = 2ⁿ⁺¹ - 1.

Order of Operations - BODMAS

"Operations" mean things like add, subtract, multiply, divide, squaring, etc. If it isn't a number it is probably an operation.

But, when you see something like...

7 + (6 × 5² + 3)

... what part should you calculate first?

Start at the left and go to the right?
Or go from right to left?

Calculate them in the wrong order, and you will get a wrong answer !

So, long ago people agreed to always follow certain rules when doing calculations, and they are:

Order of Operations

Do things in Brackets First. Example:

		6 × (5 + 3)	=	6 × 8	=	48
		6 × (5 + 3)	=	30 + 3	=	33	(wrong)

Exponents (Powers, Roots) before Multiply, Divide, Add or Subtract. Example:

		5 × 22	=	5 × 4	=	20
		5 × 22	=	102	=	100	(wrong)

Multiply or Divide before you Add or Subtract. Example:

		2 + 5 × 3	=	2 + 15	=	17
		2 + 5 × 3	=	7 × 3	=	21	(wrong)

Otherwise just go left to right. Example:

		30 ÷ 5 × 3	=	6 × 3	=	18
		30 ÷ 5 × 3	=	30 ÷ 15	=	2	(wrong)

How Do I Remember ? BODMAS !

B	Brackets first
O	Orders (ie Powers and Square Roots, etc.)
DM	Division and Multiplication (left-to-right)
AS	Addition and Subtraction (left-to-right)

The only strange name is "Orders". "Exponents" is used in Canada, and so you might prefer "BEDMAS". There is also "Indices" so that makes it "BIDMAS". In the US they say "Parenthesis" instead of Brackets, so they say "PEMDAS"

Divide and Multiply rank equally (and go left to right).

Add and Subtract rank equally (and go left to right)

After you have done "B" and "O", just go from left to right doing any "D" or "M" as you find them. Then go from left to right doing any "A" or "S" as you find them.

Examples

Example: How do you work out 3 + 6 × 2 ?
Multiplication before Addition:
First 6 × 2 = 12, then 3 + 12 = 15

Example: How do you work out (3 + 6) × 2 ?
Brackets first:
First (3 + 6) = 9, then 9 × 2 = 18

Example: How do you work out 12 / 6 × 3 ?
Division and Multiplication rank equally, so just go left to right:
First 12 / 6 = 2, then 2 × 3 = 6

Oh, yes, and what about 7 + (6 × 5² + 3) ?

7 + (6 × 52 + 3)
7 + (6 × 25 + 3)	Start inside Brackets, and then use "Orders" First
7 + (150 + 3)	Then Multiply
7 + (153)	Then Add
7 + 153	Brackets completed, last operation is add
160	DONE !