studiot

Well to continue my bag of apples analogy, would you use the empty set to represent a bag of 5 pears as a set of zero apples? If so you would be ignoring 5 members of the set, S How would you perform arithmetic on this set so that for any (a,b) in S (a+b) is in S ? That would be adding apples to pears, which you could only do if you are considering fruit. There are equivalent situations in number theory and it depends upon what you are doing with your numbers to how you approach them. I call the set you refer to viz - {0,1,2,3,.......} - the set of orninal numbers. I call the set - {1,2,3,...} - the set of natural or counting numbers. You need both of these. The natural numbers allows the Dedekind route to the construction of the real number system, via the integers and rationals.

Well the solution of quadratic equations is something a high school student should be learning in algebra. However here is an interesting non algebraic method of solution by geometrical drawing (not by a graph). I have chosen to demonstrate using a specific example but the method may be used with any quadratic and, what is more interesting, extended to an algebraic polynomial of any degree.

Different ways to do algebra? Well in part at least it depends upon what you mean by algebra? Many of the traditional 'algebraic' processes can be accomplished by pure drafting techniques. These include addition, subtraction, multiplication, differentiation, integration, fractions, solution of equations and more. Some of these were embodied in the old fashioned slide rule. Again before calculators came, nomograms were popular. Then there are more specialised techniques mostly to do with vector and tensor algebra - link polygons, funicular polygons, Mohr diagrams, Mohr circles, Maxwell diagrams. So tell us what you mean by algebra?

Strain Energy: If you want a good geologists viewpoint of this subject the little book by professor J C Jaeger is brilliant Elasticity, Fracture and Flow, published by Methuen. Strictly there is a kinetic energy component to consider when a body suffers internal strain and this appears in the formal equations of continuum mechanics. However this is normally ignored as the movent is slow so the KE is small compared with the stored elastic or plastic strain energy.

The calculation is a standard one for eelctromagnetic lifting cranes in scrapyards, where the force is of course vertical, but the principle is the same. It is treated in electrical technology textbooks such as the stadard by Hughes. What more do you need to know? It would be useful to know some more details of the magnetic circuit and its generation.

Since the generated field is concentrated axially to each coil, the axis of the field will follow the axis of the coils. So if they are twisted into a helix then the field axis will take this shape on, if they are twisted into some other shape they will follow that.

Kinetic energy of (?thermal?) expansion? That is an unconventional way to deal with it. Using strain energy as an intermediate is more usual. Can you explain further?

Please explain this statement.

It matters because we need to differentiate between the number 'zero', which is a member of quite a few sets, and the situation where there is no member of a set involved in a statement. I have no apples in my bag ie the number of apples in my bag is zero. but If I consider a bag of apples it cannot contain zero apples or it is not a bag of apples. So the set of possible numbers of apples in my bag can contain zero, but the set of possible numbers of apples in a general bag of apples cannot. Student or otherwise, let me congratulate you the reasonableness of your discussion style - it will take you far.

One formal definition of sign is to use strictly greater than zero for positive and strictly less than zero for negative. Which makes it different from every other number in the set. Of course, one question is Which set? This is interesting because we require a set without a zero to be able to use proof by induction, which involves creating a sequence of conditions in one to one correspondence with the natural numbers, N. N does not include zero or we would not be able to do this.

OK that's parity, how about sign. Is zero positive or negative?

Not only are there number systems that include cardinals (infinities) - albeit at the expense of loosing certain arithmetic operations - there are also number systems without a zero, eg the natural or counting numbers. You might also like to consider whether zero should be counted as odd or even.

The force between two ferromagnetic surfaces is given by (B2A) / 2U0 It can be calculated by considering the virtual work of a small virtual displacement and equating to the energy stored in the magnetic field E = 0.5 HB

Perhaps it would help if you thought about what is meant by the phrase 'Thermodynamic State' and, if a system is in a given thermodynamic state, what causes it to remain in that state or change to a new one,

Hint : For the first fraction, what is (x2+x-6) divided by (x-2) ?? Can you not simplify it?

I recommend this Cambridge University book On Space and Time by Professor Majid.

In an otherwise empty room?

Sound waves are three dimensional pressure waves. They are three dimensional because the pressure in a fluid is the same in all directions. So the ruler moving up and down in the air generates regular pressure fluctuations in all directions. It is these fluctuations that are transmitted across the room as a sound wave.

Exeter Huh? There is a well established U3A geology group in nearby Taunton. You could contact the organiser to see what they do. PM me if you can't make contact through U3A. Just to add to Ophiolite's list (He is a pure scientist to looks at why and how and how did it get there etc) I see you are an engineer so you might like to also explore the question Of what use can we make of it? How does it affect the geography of the area and in turn the people of the area and their lifestyles?

There was more than this to your original question which I answered infinitely accurately (if concisely). As to your point about the size of the infinitesimals (delta x) for whole numebrs they it does not matter whether they are small or large for the proof. The sums in the integral are still exact.

Which is what I've been telling the OP. The only functions you can guarantee to integrate to 'infinite accuracy' are of the form y = an integer constant betwen limits that are integers.

I take exception to this since I was the first and only poster to give you a correct answer to your question as posed. The only number system that is 'infinitely accurate' is the whole number system and I councilled you to stay within it. If you do not it is physically impossible to achieve your stated goal.

Well a good start would be to understand mathematical limits properly. AJB has pointed you at a source for this, although I should allow Wikipedia is not always the most comprehensible. You should thank him for that. Did you want help understanding it?

But that was not all I said. You have not answered my comment on the core of this thread. The part you have replied to is of no value, taken out of context as you have.

Well the simplest way is to stick to whole numbers.