TonyMcC

I know it's considered old fashioned today but the programming language known as BASIC is easy to learn. Also you can find a free download and much information on line. Just about all the computer instructions the language uses are common words which do as expected!

Hang on a minute! If you apply a voltage to a loop of wire that is free to rotate in a constant magnetic field it will move to a position at right angles to the magnetic field and then stop. You need a commutator to keep reversing the applied voltage at the correct positioning of the loop. Or is everyone assuming there is a commutator? Merged post follows: Consecutive posts mergedAnother aspect not considered is the phenomenon known as back e.m.f.. As the loop rotates it generates a voltage in opposition to the applied voltage. Thus the effective voltage is Applied Voltage minus Back e.m.f.. Since back e.m.f. increases with speed of rotation it has a lot to do with final maximum speed. As mentioned before, if you don't have flux density and loop resistance you can't really begin to consider making calculations. Merged post follows: Consecutive posts mergedI forgot to mention that if coil bearings are frictionless and there is no "windage" then maximum speed of rotation will be when back e.m.f. equals applied voltage giving zero current in the loop.

Kirchoff's current and voltage laws do apply to a.c. circuits but only instant by instant. You have to find the instantaneous values of voltage and current at a specific instant. This takes care of inductive and capacitive effects. Google " instantaneous voltage kirchoff " (without the inverted commas) for more information.

Thank you "the tree". If you look at my earlier entries for this subject you will see that you seem to have answered my earlier supposition. However I still am not really happy with your explanation concerning irrationality. As I understand it the square root of 2 is accepted as irrational not because its decimal expansion extends to infinity but for a completely different reason. It is easily proved that when expressed as a fraction and in its most simplified form both parts of the fraction must be even numbers. That really is irrational! Is it a proven fact that all square roots of whole numbers are either whole numbers or irrational? How is the irrationality proved?

Phase shifting of one component of an EM wave by 90 degrees results in circular polarisation. It is common practice in radar equipments. Google "circular polarisation radar" for more details.

Glad you found my check method interesting Ewmon. However I think you may have missed the main point; the number "2" is incidental - the METHOD works for all single digit numbers and usually works for multidigit numbers. Example showing the square root of 7 is infinitely long:- A X A =7 (making A the square root of 7) my calculator says the square root of 7 is 2.645751311, thus it says 2645751311 X 2645751311 = 7000000000000000000. (in fact my calculator says it is correct (7E18). However this must be incorrect as using the check:- 35 X 35 = 7 8 X 8 =7 64=7 1=7 Since adding digits can never make the LHS = (the square root of 7) X (the square root of 7), you will end up producing an infinitely long answer to the square root of 7. By the way the method works for cube roots etc.. Have fun. (Any bearing on Fermat's last theorem?!)

Thank you Physics Expert. I do realise that the formula fits the units and so could not be true for E=Mc^3. What I was trying to float was the idea that possibly the linkage between mass, energy and the speed of light was somehow of crucial importance. This just might be a means to proving that a cubic mass of a particular element when added to another cubic mass of the same element could not form another cubic mass of that element. I am thinking along the lines of the three imagined cubes being converted into energy. The three relevant amounts of energy would each have an amount cubed. It might not actually occur but you could imagine the raw energy as having shape - and each shape would be a cube. These "cubes" of energy would shrink to make the three cubes of matter. If either the raw energy "cubes" or the cubes of matter could not be defined (i.e. the side of any of the six cubes always proves to be irrational) then there is a situation which supports "Fermat". I like to imagine the three cubes of matter being transformed into three "cubes" of energy and back again. What does the mathematics say about the three "cubes"of energy and their relationship to the three cubes of matter? What would be the cube root of c^2? I do realise that Einstein was born much later than Fermat!

I have also worked at factorisation and just got more confused. My ideas are really more simple! Look up a list of formulae for any branch of physics or engineering and you will find that "squared" and "square root" appear regularly. Off hand I cannot think of a formula that uses "cubed" or "cube root" except in the case of three phase electricity. Three phase electricity I would suggest is not really part of the natural order of things. I wonder why "squared" and "square root" do feature so prominently? Do you think that it has anything to do with Einsteins formula E=Mc^2? What would be the effect if Einstein had proved E=Mc^3 for instance? Is Fermat's declaration more to do with how the universe "works" than it is to mathematics? I guess this just a passing thought.

Perhaps I should get nearer to the beginning of my thoughts. It seems to me that the implications of Fermat's Last Theorem are rather weird. He seems to imply that I could give you 2 cubes made of something malleable. You could roll them together to make a new cube. However you will not be able to measure the side of the new cube accurately. If you could do so then simple multiplication would provide an integer result. Example - given two cubes 6 units and 8 units the resultant cube might be measured as 8.99 units. If the result was accurate it would mean that 600^3+800^3=899^3. However many digits followed the decimal point would not matter, you would just end up with very large numbers. You could follow this procedure with any two cubes and thus there is ALWAYS a solution to A^3+B^3=C^3, but in all cases (according to Fermat) there will be an infinite number of digits following a decimal point. Fermat seems to imply that this would happen regardless of the number base used (binary, octal etc.etc.) This all seems so strange that perhaps we should look for irrationality. It would be nice if any irrationality found broke some simple mathematical rule or law. However perhaps an irrationality could be found in the laws of physics, mechanical principles, electrical principles, electronic principles, logical reasoning or even in the way the universe "works". Is it reasonable to think that irrationality might lie at the heart of any proof? I would expect any such irrationality, if found, to accommodate all powers above 2.

Well I guess this is "dying a death". People who might find a triangle a useful tool include navigators, electrical engineers and even the people who hang pictures on walls! (triangle of forces adapted from the parallelogram of forces)

Thanks Maths Expert. Actually the fact that Root 2 is irrational was the reason I chose it to "test my test". Not all infinitely long decimal representations are irrational and I suppose some have quite long repeating patterns. What I am wondering is :- a) If a whole number does not have a whole number square root does it always result in an infinitely long square root? b) If it does result in an infinitely long square root does that indicate probable irrationality for that square root? Please note - I am wondering, not claiming. I would value your (or anybody's) opinion. I don't want to end up with silly logic similar to:- Animals have 4 legs, dogs have 4 legs so all animals are dogs!

Many years ago I was shown a simple check for multiplication. The check can indicate a definite mistake; but cannot indicate definite accuracy. The check is performed by adding the individual digits of each part of the calculation. If this results in a number with more than one digit then add the digits of the new number and repeat as necessary until you have a single digit representing each part of the calculation. Example:- 39 leads to 3+9=12, and 1+2=3 58 leads to 5+8=13, and 1+3=4 39 x 58=2262 leads to 2+2+6+2=12, and 1+2=3 If the single digit representing 39 (i.e.3) is multiplied by the single digit representing 58 (i.e.4) we get 12 which leads to 1+2=3. The fact that this agrees with the single digit representing the answer indicates that the calculation might be correct, but more importantly if the single digit representing the answer had been any other number the calculation would have been performed incorrectly. Another example:- 58764 (5+8+7+6+4=30 ; 3+0=3) 3271 (3+2+7+1=13 ; 1+3=4) ------------ 192217044 (1+9+2+2+1+7+0+4+4=30 ; 3+0=3) Performing the check, 3 x 4=12 and 1+2=3 since this agrees with the answers single digit no error has been detected. Although this description might seem complicated it is really very simple and can be done "in your head" with very little practice. Examining the square root 0f 2 _________________________ Think of the calculation A x A= 2 So A must be the square root of 2 Someone suggests that the square root of 2 is 1.4 If they are correct then 14 x 14 =200 So using the test 14 (1+4=5) 14 ( 1+4=5) and 5x5 = 25, and 2+5=7 ---- 200 (2+0+0=2) Since "7" and "2" do not have the same value this calculation must be incorrect. Someone then suggests 1.41 So using the test 141 (1+4+1=6) 141 (1+4+1=6) and 6x6=36, and 3+6=9 --------- 20000 (2+0+0+0+0 =2) Since"9" is not "2" this calculation is incorrect. My calculator gives the square root of 2 as 1.414213562 therefore 1414213562^2=2 x 10^18. If you do the test the two relevant digits are "4" and "2" indicating an error. The fact is you can add digits beyond 1.414213562 all the way to infinity without getting a "correct" indication! This argument holds good for many (probably most) numbers. Perhaps any whole number falls in to one of two categories. Perhaps it either has an easily found whole number square root or it is an infinitely long decimal number. Perhaps this is true of any number base you might use (octal, hexadecimal etc.)??

Thanks Uncool for your message. I'm a bit rusty on geometry and having trouble with your construction. Any chance of a diagram or reference I can look up. You are probably right and as long as your solution only involves ratios and right angled triangles thats OK. I was a bit hasty by not stopping at powers of 3. I think you just may have introduced a "wobbly" by ending up using a right angled triangle where the required (i.e. unknown) output or a fraction of it finds its way into more than one side. In any case I am still quite happy with further work I have thought out. I would like to follow your argument so if you can make it clearer in any way I should be grateful - Tony Merged post follows: Consecutive posts mergedIs there anybody out there who can explain more clearly the construction used by "uncool" to obtain a rod C in response to nothing except rods A and B that gives the incorrect sum A+B=C which can be corrected by raising each term by the power of 4? It would be nice if the necessary steps could be written down as an algorithm which I could incorporate in a simple computer program - but I would enjoy doing that for myself. I feel that "uncool" has probably made a mistake somewhere, or perhaps is "pulling my leg". Merged post follows: Consecutive posts mergedOK let us try a different tack. If you have read this far you will know that I think a proof of FLT should emerge from analysis of INCORRECT sums of A+B=C. One way of attempting this is to form triangles from A,B and C. For example a triagle of sides 6,8,and 9 represents the incorrect sum 6+8=9 which when all terms are cubed becomes the (almost) correct sum 6^3 + 8^3 = 9^3. Next simple thought:- To many people the triangle described above is a very useful (almost) CORRECT sum! If you, dear reader, are intersted enough perhaps you would like to give me an example and how the triangle might be used in the "real world".

For Andrew and any other interested party. As I have said I believe there is something peculiar about all whole number incorrect sums (A+B=C) and that peculiarity precludes the possibility of that sum being corrected by raising the three terms by any power other than 2. This simple exersize in logic convinced me that this is an idea that was worth developing:- You stand outside a room with an open window. Inside is a person with a normal schoolboys geometry set. This person will follow a sequence of instructions that you have provided beforehand (i.e. An algorithm). You want to pass 2 rods of any length through the window (A and B), have the person inside follow your algorithm and pass out to you a single rod equal to the sum of A and B, called C). This of course is very easy:- 1) Accept A and B 2) Draw a straight line 3) Place A and B on the straight line so that they touch 4) Cut rod C such that it extends from the open ends of A and B 5) Pass rod C out of the window. Next, and only slightly more difficult, you want to pass rods A and B into the room and have rod C returned such that A^2+B^2=C^2. (e.g. If A=3, B=4, then C=5) 1) Accept A and B 2) Draw a straight line 3) Draw another line that intersects this line at 90 degrees.( Use compasses and you don't even need numbers!) 4) Use a right angle from the intersection to align A and B so that they touch. 5) Cut rod C so that it extends from the open ends of A and B. 6) Pass rod C out of the window. Next you want to pass rods A and B through the window and have rod C returned such that A^3+B^3=C^3( eg A=6, B=8, C=9(approx)) SUDDENLY WE ARE PRESENTED WITH AN IMPOSSIBLE SITUATION. The reason for this is that each pair of input rods A and B of differing ratios requires that the person inside the room sets a different angle with which to align those rods. What you want the person inside the room to do is construct a triangle of a particular shape. Basic geometry requires 3 parameters in order to do this ( 3 sides or 2 sides and included angle). However you are only inputting 2 of the required parameters. This argument also applies to all powers >2. O.K. - This doesn't prove that that the required triangle doesn't exist - only that the method described would demand 3 inputs. Perhaps any method of calculating (A^n+B^n)^(1/n) (where n>2) would require 3 inputs. For powers n=1 and n=2 one of the parameters needed to form the required triangle is a constant ( 180 degrees and 90 degrees respectively) and a constant can be accomodated in the algorithm. (For n=1 the "triangle" is very very flat, its angles being 0, 0, and 180 degrees!) Perhaps this realisation gave Fermat food for thought. If my simple ideas interest anyone then there is more where this came from. I would welcome any comments

I have visited your homepage Andrew (If I may call you that) and am very impressed. Perhaps more importantly we are very different in our levels of formal qualifications and life experiences. Much of my thoughts concerning "Fermat" come from pondering the various jobs I have undertaken and the training required to perform them. My experiences have all been of a practical nature and I'm afraid my highest qualification is Higher National Certificate in Electrical and Electronic Engineering. Much of my ponderings have more to do with physics, science and computer programming than pure mathematics. If you have the patience I would value your opinion if I float my ideas bit by bit until you "shoot me down". Out of interest my career involved troubleshooting and teaching others to troubleshoot military radar systems and troubleshooting mainframe computer systems. My final position was as a College Lecturer teaching basic principles of Mechanical, Electrical and Electronic Engineering. As a hobby in the 70's I taught myself computer programming (whilst working as a computer Engineer) and ended up teaching it as a secondary subject at my College. I mention these facts so that you will realise my limitations. For what it is worth Fermat may quite possibly have been more like me than you - or perhaps not !

I saw the television programme at the time. That was what grabbed my interest. I don't remember anyone suggesting that a simple solution might be found from analysing incorrect sums. I have a transcript of the programme which I will read again. By the way can you see why it should not be possible to obtain the incorrect sum A+B=C which relates to A^n+B^n=C^n where n>2? I would like to quote Prof. Wiles "sometimes you realise that nothing that's ever been done before is any use at all and you just have to find something that is completely new, and it's a mytery where it comes from"

I am getting on in years and since retirement I have been giving Fermat's Last Theorem some thought. I believe most people are attempting to solve this puzzle in the wrong way. There may be NO good reason why (say) A^3 + B^3 = c^3 should not have a whole number solution. However there is a very simple reason why the relevant "wrong sum" A+B=C cannot logically be deduced. Please note, I am not saying it cannot exist - only that there is a simple reason that it cannot be deduced. The reason applies to all powers>2. Perhaps this was Fermat's starting point!? I should be interested if this thought gets any response. More later - perhaps a conjecture if not quite a proof.