Guys, quit playing with small numbers...1. This is the most amazing formula of all and nearly all mathematicians put it at the top of their list of favourite formulae. It's known as Euler's Formula, although so is another formula below. The formula contains three constants, all with different origins. e is used for logarithms and is connected with exponential growth and radioactive decay, i is the basis for complex numbers and comes, of course, from circles. Put these three together and bingo! you get -1. I have seen this formula so many times but I never fail to be amazed by it. Rearranging giveswhich now includes five of the most important mathematical constants. 0 and 1 are the foundation stones of arithmetic and their importance is perhaps best appreciated by studying group theory. Even more amazingly, this formula is a special case of which links e with trigonometric functions, another surprise. Keith Devlin's book Mathematics: the Science of Patterns sums things up very well, and there's an excerpt at Euler's Formula. Read Keith Devlin's speech to graduating students making the invisible visible 2. Fermat's Little Theorem There are two theorems linked to Fermal's name and this is the other one. It says that if p is a prime number and a is an integer then a^p-1 - 1 is divisible by p or, equivalently, if p is a prime number and a is an integer then a^p - a is divisible by p For example, if p = 13 then 6¹² - 1 = 2 176 782 335 which is divisible by 13. This formula is very useful. For example, you can use it to show a number is not prime. 6³ - 1 = 215 isn't divisible by 4, so 4 can't be prime. This is blindingly obvious in this case, but is useful for larger numbers. If a^p-1 - 1 is divisible by p for all integers a with no factor in common with p, then it is highly probable that p is prime. However, it needn't be and then p is called a Carmichael number (561 is the smallest). For a proof of the theorem try Fermat's Little Theorem. For more on the use of this result see Probable Prime. 3. Euler's formula V - E + F = 1 This is an important formula in graph theory. Draw any two-dimensional graph, that is, a set of a points called vertices, and some line segments called edges which connect the vertices. Make sure it is in one piece (connected in mathematical language). Then count up the number of vertices V, the number of edges E, and the number of faces (regions) F that it encloses. Then no matter how you do this you will find V - E + F = 1, For example, (vertices are numbered in black, edges in blue and faces in red) thus V = 14, E = 21, F = 8 so V - E + F = 14 - 21 + 8 = 1. Draw your own graph and check it. You will sometimes see the formula written as V - E + F = 2 because the face outside the graph has been included. What happens to this formula if you draw it on the surface of a sphere, a torus or just in three dimensions? Euler's Formula in Higher Dimensions. Try to use this formula to show there are only 5 regular solids (known as Platonic solids) Euler's Formula, The Five Platonic Solids and to show there is no solution to an old riddle: There are three houses each of which must be connected to the water, gas and electricity mains. Can this be done in such a way that the cables and pipes do not cross over or under each other and do not pass through the houses or the mains supply? Euler developed graph theory to solve the Königsberg bridge problem. There were seven bridges over the River Pregel at Königsberg in Prussia. Euler solved the problem of whether there is a way of traversing each bridge once and only once starting and returning to the same point in the town. Picture of Königsberg bridges. For more information see Topology enters mathematics. 4. This little formula, called the difference of two squares formula, crops up everywhere, often without warning. For example, the following problem leads to an outrageous polynomial unless you use this formula: A geometric series has first term p, and eight times the sum of the first six terms is equal to nine times the sum of the first three terms. Find the common ratio of the series. Fermat used it to factorise large numbers (no calculator or computer was available!) into the product of two numbers roughly equal in size. For example to factorize the number 119143 try the smallest number x so that x² - 119143 is positive and keep increasing x by 1 until x² - 119143 is a perfect square, y² say. Then x² - 119143 = y² and so it follows that 119143 = x² - y² = (x - y)(x + y) thus factorising 119143 (and what do you get?). This method is known, not surprisingly, as Fermat's method of factoring. You can use the difference of two squares formula to amaze your friends with your mathematical ability:

• Find 35² - 34² easily. The formula tells you it's 69 in an instant.
• You can find squares easily. To find 56² take the two numbers 6 less and 6 more than 56 and multiply them together as 50 × 62 = 3100 (you do know how to multiply by 50 easily don't you?), then add 6² to get 3136 which is 56². Use the formula to see how this works.

Alain, now is it + or - ??

Fascinating!You forgot to mention that Euler's Theorem, in its discrete form, is the basis of the Fast Fourier Transform, which in turn makes digital sound filtering possible. And digital sound is what FSX uses to render aircraft sounds. And speaking of FSX...Whew, I think we're back on topic!Cheers,- jahman.

For all that drivel, the answers are still -1, 0, 1 and 2, hardly big numbers.

Fascinating!You forgot to mention that Euler's Theorem, in its discrete form, is the basis of the Fast Fourier Transform, which in turn makes digital sound filtering possible. And digital sound is what FSX uses to render aircraft sounds. And speaking of FSX...Whew, I think we're back on topic!Cheers,- jahman.

I did not want to paste to much of it, as you can see Rimshot is already confused but here's more... The Prime Number Theorem This is a difficult to prove formula which gives the number of prime numbers less than a given number. It says: What this means is that if x is any positive real number then , which is the number of primes less than x, is approximately (where log is to base e, sometimes called the natural log or ln). The ~ means that the approximation is such that divided by gets closer and closer to 1 as x gets larger. Here are some values to see that this formula is reasonable:

x

÷

1000

168

145

1.159

1 000 000

78 498

72 382

1.084

1 000 000 000

50 847 478

48 254 942

1.053

Source: An Introduction To The Theory Of Numbers Perhaps the power of this formula can only be realised when you discover that the primes don't occur in any pattern so that their occurrence is quite hard to pin down. There's lots more on this formula at How Many Primes Are There? See also Bertrand's postulate. 6. Number of divisors Most people are unable to calculate how many divisors that a number has without writing them all down, and doubtless wandering whether they've missed some. Yet it can be done very easily provided you can factorize the number into powers of prime numbers. For example, 108 = 2² × 3³. Then you can list all the divisors, including 1 and 108, as made up by multiplying each number in 1, 2, 2² with a number in 1, 3, 3², 3³ giving the 12 combinations 1 × 1, 1 × 3, 1 × 3², 1 × 3³ 2 × 1, 2 × 3, 2 × 3², 2 × 3³ 2² × 1, 2² × 3, 2² × 3², 2² × 3³ Thus there are 12 divisors which comes from adding one to each of the powers in 2² × 3³ and then multiplying them together. You should be able to see how this generalizes to the formula: If then the number of divisors of n isNumber of divisors Divisor Counting Chimbumu and Chikwama get out of jail 7. This is a special case (s = 2) of the Zeta Function, first proved by Euler after defeating the best efforts of famous mathematicians like Jacob Bernoulli, Johann Bernoulli and Daniel Bernoulli (all members of the same family). It is yet another formula that involves the ubiquitous which seems to turn up in all sorts of surprising places. You can find fourteen different proofs of the formula at Robin Chapman's Home Page, though they are mostly non-elementary. The proofs Evaluating zeta(2) are in DVI, postscript (PS) or PDF file formats. What is even stranger is that if you pick two positive integers at random, the odds of them having no common divisor are 1 in which is 1 in 1.644934... . This result, along with others, appears at Notable Properties of Specific Numbers. 8. Wallis's Product Yet another formula that gives , and it was discovered by John Wallis. You'll find a simple program that will run on Casio 9x50 calculators which uses Wallis's product at Sanoy's Casio Programs and is repeated here: 0->A~Z 1->C 1->B Locate 1,1,"TRIES:" Locate 1,3,"PI:" Lbl 1 (4B^2)/(4(B^2)-1)*C->C B+1->B Locate 4,3,2C Locate 7,1,B Goto 1 so that you may be able to convert it to run elsewhere. A much more well-known formula for is to use the series for tan^-1 and the fact that to get Gregory's Formula This is discussed in How Pi is calculated.

WOW! Where is this thread gonna go next? Are we going to discuss how MSFlight will become the next "Star Trek" trend and bring the universe together and bring peace and security to all who gets 300 hours in a Cessna C172 then fly over New York Times Square at midnight January 1 2054 at 100' alt while you drop confetti out your secret bay doors Ctrl+F8 and you must humm the tune Yankee Doodle Dandy and tap your feet but watch that you don't hit your rudder pedals! Then your Cessna becomes a spaceship that will take you to distant planets and learn the secrets of life. Then and only then peace will come to you and the earth! and to think this tread started as a FSX and Flight screenshot comparison

WOW! Where is this thread gonna go next? Are we going to discuss how MSFlight will become the next "Star Trek" trend and bring the universe together and bring peace and security to all who gets 300 hours in a Cessna C172 then fly over New York Times Square at midnight January 1 2054 at 100' alt while you drop confetti out your secret bay doors Ctrl+F8 and you must humm the tune Yankee Doodle Dandy and tap your feet but watch that you don't hit your rudder pedals! Then your Cessna becomes a spaceship that will take you to distant planets and learn the secrets of life. Then and only then peace will come to you and the earth! and to think this tread started as a FSX and Flight screenshot comparison

Hey! don't blame me if Rimshot is confused.....

WOW! Where is this thread gonna go next? Are we going to discuss how MSFlight will become the next "Star Trek" trend...

BTDT: See here.Cheers,- jahman.

Another stupid speculation by Mr Know-It-All, Congratulations on amounting to nothing.

Speculations are not stupid by definition.Speculations then , from your point of view that you explained as a gentlemen btw, but honestly I couldn't care less about your wet dreams.

Speculations are not stupid by definition.Speculations then , from your point of view that you explained as a gentlemen btw, but honestly I couldn't care less about your wet dreams.

I never stated that all speculations are stupid, I said the speculation you made was stupid ...

Hey! don't blame me if Rimshot is confused.....

Not as confused as me, I was miles off the mark. In fact I didn't even know what the mark was.

Hey! don't blame me if Rimshot is confused.....

I must indeed say pictures of pretty bottoms (or even fish) confuse me a lot less! :D

I must indeed say pictures of pretty bottoms (or even fish) confuse me a lot less! :D

You can't beat a good ' Figure'