studiot

Why wouldn't they? Until John has spoken, no one knows what he knows or doesn't know. And John speaks last. And he immediately solves the problem. This is why I say something is missing in the presentation of the question.

Numbers are not lines and lines are not numbers. You can put them into one-to-one correspondence for some purposes, but they are not the same. Your question about positive and negative has no mathematical meaning. 3i is not a real number so why should it follow the rules for real numbers? Have you read the Wiki on imaginary numbers? It is quite good and offers a useful extended table of the values of powers of i compared to the one I offered earlier including negative powers. Note that some powers have real values, some have imaginary values. https://en.wikipedia.org/wiki/Imaginary_number Did you understand my point that a real number x a real number always yields another real number. But an imaginary number x an imaginary number does not yield an imaginary number.

Yes I agree. That would mean that the product is also zero. But James only knows the product and that any {abcd} with a zero in any position will yield a zero product. And there are many possibilities for this. So if James' product is zero he cannot know which {abcd} produced it. Further he also knows that the products are not {9999} = 6561 and {9998} = 5832 or he would have reported solving the problem since those are the only way to obtain those particular numbers. So he says 1) he does not know the answer. 2) He also says that John does not know the answer. When he says (1) he is not then lying about (2) because two of the proposed solutions John is considering, {0000} and {1000} ,have zero products. But they have different sums. So John can then ascertain whether his sum is 0 or 1, thus leading to unique solutions {0000} or {1000} What I can't see is how that allows him to say that James can now solve it because he did. I am assuming he does not pass on the sum or solution information.

How does that follow inevitably? I prefer to use list of potential solutions rather than range to avoid confusion.

It is interesting following you looking at this problem through the other end of telescope from me. You are starting with Jack and the range. I am working backwards from John and the sum. But John must know that {1111} is not an option since James would have already found that since it is the only {abcd} that yields a product of 1. So I am concentrating on which sum they all know can only be formed in one way. I still think these are 0, 1, 35 and 36.

Thank you for that useful information and link. +1

The range of the digits, not the range of the numbers abcd is specified. The largest possible digit is 9, the smallest is 0 so the range is either 0,1,2,3,4,5,6,7,8,or 9 The largest digit of these must appear in position a and the least in position d. I see no other possible interpretation of that part of the information, unless the question was incorrectly reproduced.

Given the greater than or equal to chain of relations, this means that 'a' must be the largest and 'd' the smallest, although the may actually be equal. That is how John can distinguish between 1000 and 0100 and 0010 and 0001. Only the first is allowable under this condition. a>=b>=c>=d. That is the assumption I am working on anyway.

+1. But I would put it more strongly than that. I would say it is entirely wrong to do so. Consider for instance x = 2 Then 1/(1-x) = -1 What does that tell you?

Thank you Eise for for those useful insights and clear way of putting things. +1 Michel, (and taeto) The whole point is that Imaginary numbers do not form a Field. The fail the Field axiom that the Field should be closed under multiplication in the biggest way possible. The axiom guarantees that the product of two members of the Field will be another member of that Field. That is if pi and qi are imaginary numbers then pi*qi should also be an imaginary number. But piqi = -pq, which is not imaginary. A great deal of work had been put in over the last 150 years to construct axioms systems in Mathematics that allow us to get on with the mathematics to achieve what we want to achieve, secure in the knowledge that the foundations support these requiements. The Field axioms are the ones that allow us to do arithmetic without the hiccups such as the ones Eise points to and you have found. Note that the natrual numbers do not form a field either. The simplest number system to form a field is the rational number system. This is, of course, why Mathematicians went to the bother of constructing the complex numbers, which are a Field. It is important to distinguish between imaginary numbers and complex numbers because the former do not comply with the normal rules of arithmetic. For instance none of the interior points in your red and blue squares are imaginary numbers. So you do not have imaginary squares.

Yes it was a Microsoft product, intended for netbooks whcih were popular before smartphones. It came, for instance, with the Acer Aspire one and used to be advertised as Windows 7. However Microsoft made Windows 10 such that it was impossible to upgrade these to W10. (Probably because of the smaller hard drive and memory sizes in those days and also the popular processor for netbooks) But they still offered the W10 upgrade through the free offer scheme. This crashed the Aspire one so that it could neither be restored to W7, nor upgraded to W10. The net was full of complaints at the time.

Well yes, we know of four 'fundamental forces' in the Universe. All have different effects at different ranges or over different distances. Strong nuclear, weak nuclear, gravity and electromagnetic. For the first two the clue is in the name 'nuclear'. They are active within atoms or between sub atomic particles. So they do not answer your question here. They are basically too short range. That leaves gravity and the electromagnetic force. At ranges close enough to consider bonding of atoms to form molecules the elctromagnetic force outweighs gravity many times over. The electromagnetic force is entirely responsible for bonding atoms to form molecules. However it does this in conjunction with motion of charged particles, which can be very complicated. Ionic and covalent bonding is the result of the electrostatic part of the EM force The theory you are looking for is called the Lennard Jones potential https://www.google.co.uk/search?source=hp&ei=v9FrXMShJIneavuTofgC&q=lennard+jones+forces&btnK=Google+Search&oq=lennard+jones+forces&gs_l=psy-ab.3..0j0i22i30l5j0i22i10i30.820.12012..12582...0.0..0.108.1936.28j1......0....1..gws-wiz.....0..0i131j0i10.2W3nAzSDPBE Which leads to Lennard Jones forces. At greater ranges gravity also plays a part, for instance it stops our atmosphere flying away. Does this help?

As always with these the devil is in the detail Please post the exact question, exactly as written.

Your problem lies right here. It is indeed true that the power series expansion of 1/(1-x) is [math]\frac{1}{{1 - x}} = \sum\limits_{n = 0}^\infty {{x_n}} = 1 + x + {x^2} + ....[/math] But this is only true for |x| < 1 The series for |x| > 1 is divergent (That is the individual terms get larger and larger as the number of terms increases, so their sum gets larger and larger) For |x| < 1 the series is convergent (That is the individual terms get smaller and smaller as the number of terms increases, so their sum gets progressivly closer to some finite limit) You may only perform the term by term multiplication as in If the series is convergent. Yours is not. Congratulations, you have just found out the sort of nonsense that arises when you try this on a divergent series! You should always, but always, check the restrictions on the series variables before using them.

So John works out the answer, knowing that neither James nor Jack have enough information but knowing that the sum is 1. John can only do this if there is a unique combination that can only occur one way and can match his sum to that number. But since we are not told this sum and there are four possible unique sums that can only occur one way and also fit the information given to all. a>=b>=c>=d. That is a) 1000 b) 0000. c) 9998 d) 9999 Clearly if John knows that the sum = 1 he declares (a) as the only correct answer. But he would also solve it if he knew that the sum was zero and declare the correct answer to be (b) Or if he knew the digit sum to be 35 or 36 he could declare (c) or (d). He would not be able to solve this if his sum was greater than 1 and less than 35 since there are multiple ways to meet the abcd restriction for these sums But we do not know the sum so must look at his statement that James should also be able to solve it now. If James knows that the product is zero which means that either (a) or (b) will suit, but he cannot distinguish So the answer cannot be 1000 or 0000 or that the product is 5832 or 6561 when he can declare (c) or (d) So we must look again at what Jack says. Now Jack says I don't know all the digits but I know that the greatest minus the least is 1, which gives answer (c) or 0, which gives answer (d) I cannot see a way for us to distinguish between answers (c) and (d), all though each of Jack, John and James have enough to do this at the appropriate time.

True, but I am not sure what you are referring to. Was it this? I was simply using the distributive property to rearrange the brackets. Some interesting powers i2 = -1 (definition) i3 = -i since i3 = i2 * i = -1*i = -i i4 =( i2)2 = (-1)2 = +1 and so on for higher powers. The complex numbers C form a field with addition and multiplication as defined for complex numbers. In particular (C,+) is a group. In an additive group, the inverse of an element x is written −x. Minus something denotes the additive inverse to that thing. Yes they form the field of complex numbers. This is not a fully ordered field, as are the reals. Further i by itself is not a complex number and therefore needs no additive inverse. The problem with saying that i is positive or -i negative is it leads to a contradiction. This is a direct consequence of the fact that a complex number, in whatever format, is a two part entity. So addition and its inverse (and multiplication) needs to consider both parts. If i > 0 then i2 > 0 so -1 > 0 which is a contradiction. If i = 0 then i2 = 0 so -1 = 0 which is a contradiction. If i < 0 then i2 > 0 so -1 > 0 which is a contradiction since the ordered field axioms imply that the square of any nonzero number is positive.

Why? You can invent it. What is wrong with this concept? Can you? What does minus something mean or plus something? It means that you can order the magnitude of something -(something) < 0 < +(something). now consider the complex numbers 4+3i and 4-3i Can you do this with these complex numbers? They both have the same magnitude = square root (42 + 32) = 5 But what about 4+3i and 4-12i ? Which comes first now? You cannot divide complex numbers into positive and negative, as you can for the real numbers. Purely imaginary numbers (those with no real part ) are really complex numbers of the form 0+ai or 0-ai , where a is some real number. And as shown above they have the same magnitude, although you can place -a < 0 < +a along the real line.

Thank you. That is not what I get from a search, even not in Wolfram Alpha. All numbers, including -1, have two square roots. -i does not exist but -1 * i does So (-1 * i) *(-1 * i) = (-1 * -1) *(i * i) = (+1)*(-1) = -1 So the square roots of -1 are +1i and -1i.

So you want to teach a bit of basic maths. My posts were directed towards this in all sincerity. That was why is suggested introducing infinity through 'eternity' Also you could introduce infinity through the counting route one, two, many, (so many it doesn't matter how many = infinity) This is the basis of the optics I was talking about. Infinity is so far away that rays are parallel and you are considering what happens when you put [math]1/\infty [/math] into the lens formula (which I assume you know). This happens a lot in Science and Engineering for instance the idea of infinite dilution in Chemistry. By 5 years they should be doing some counting and you could introduce the idea of putting in corrrespondence. In truth they play (and love) games based on this from about 3. If you can get over the idea of one-to-one correspondence and remainders that would be a real bonus in basics.

My apologies, I didn't read your elucubration properly, but I understand now. My midnight oil must be dimmer than yours. No not complex numbers, just imaginary numbers, they are different. There is no positive and negative with complex numbers. What you have is two real axes, scaled by an imaginary constant, i. The positive and negative attach to the real numbers along each axis so, as you correctly observe +5 * +10 = +50 but +5(i) * +10(i) = +50i2 = -50 This situation occurs in nature and mathematics. For instance it could be the vibration mode of a two dimensional membrane (drumskin) with the positive/negative denoting direction of travel so one part of the drumskin is travellong forwards and the other backwards. Also we have integration above and below the line so the integral of a sine curve from zero to 360o is half positive and half negative (above and below the line) an sums to zero. In order to calculate the area under that graph you have to split into to and reverse the sign of the negative part, before adding.

I am confused. Are we preparing ideas to put to a 5 year old or helping a 35 year old with some maths?

Nice. +1

Not a problem, infinity is a big subject. I would suggest that convergent infinity getting closer and closer to some finite object (it doesn't have to be a number, in fact I wouldn't recommend it at that age) is too subtle for a 5 year old. But the idea of going on and on (for ever), or divergent infinity is much easier to grasp. How about asking about divergent infinity? How many steps to grandma's or to the corner shop or whatever? Then ask how many to the next village / town? Then ask how many to Copenhagen etc Then how many to the Moon. I think there are infinity/eternity stories in Viking mythology (about Loki's punishment) and Hans Anderson that you might like to invoke (simplified) as well. Physicists use infinity when discussing lens theory in real world optics. Have you seen this?

You have shown i and -i appearing on both the real and imaginary axes. There are no such numbers on either lines.

I think my answer fits all the statements in the order given.