Why negative number multiplied by a negative number is positive number?
Different branches and form of mathematics (statisitcs, mathematical physics, boolen algebra) found practical utility in modern world and blurred the boundary line between that areas of maths and other subject. Ok call it mathematics of utility. But some areas remained as abstract. Fine this is all about the situation in modern era. The quoted mathematicians are from medieval or modern era. My point is that when the mathematics originated in the world, it had originated as practical human need. Even art too originated in the same way. But slowly it (both art & literature) detached from its origin and took more abstract form of development. Nothing wrong in that. Each branch of knowledge develop with its own dynamic and characteristics. We can make intellectual arguments on whether the concept of sphere represent any physical quantity or not. True. You can argue in both way. But the fact is that the geometrical concept of sphere originated from human observations on similar shape objects in world. The simple concept of “projectile motion” in physics (which is taught to children in school classes) too can be challenged saying that there is no real projectile (parabolic motion) in earth, if we consider all parameters involved. True. As long as Newton’s gravitation law is valid, Newton first law of motion is not a practical reality in the world, because no object will be in the state of motion or in strait line motion. But we conceive the physical world using such models and abstractions. In most branch of knowledge there exists, some element of abstraction, without which we cannot generalize. But as I mentioned, mathematics is highly abstract. You cannot deny the validity and correlation of projectile motion concept in physics with the real world motions, only because it is not perfect but a generalized abstraction. Similarly fundamental Mathematics concepts too have a link with physical reality. In physics too, there is theoretical physicist and practical physicist. So the importance of abstractions in the growth of human knowledge cannot be denied. Since mathematics is more abstract, it is difficult to link it with practical reality. During the centuries after its origin the concepts had grown to different levels of abstraction. I feel it is the duty of mathematician to make it more accessible and understandable to the world, so that world can benefit from it. Most of the researches, heavily funded in many universities in the world go futile without such a movement. It is my wish to go the effort of mathematicians in the progress and benefit of humanity. No meaning to argue for generations whether mathematics is platonic or utilitarian? It will degrade to the level of an argument “whether art is for art or for human being”. Any attempt to artificially find utility of any subject or channel it to extract benefit out of it will be detrimental. Because it kills the natural dynamics of that subject and it development with its own originality. You should be creative to extract benefit out of it.

While we accept the importance of abstraction, one should be clear in mind that the test of all theorems are the practical life (experiments) in this physical world. That is why Euclidean geometry found its limits in modern times. That no way deny the greatness and validity of Euclid. But remind us the importance physical world – Judgment day is here in physical world, not how logical my mathematical deduction or induction was. Nothing sacrosanct about mathematical explanation of most fundamental concept “multiplication of negative numbers”, it is a philosophical question – physicst, mathematicians, logicians, linguist, .. all branch of knowledge has say. Because it is fundamental concept.

(Even now it is a debate in many Universities, whether dynamics is a branch of mathematics or physics). But mathematical approach and explanation has validity in advanced mathematics. An artistic or physicst approach is not suitable. Similarly for modern literature, approach logician will be inappropriate. When a branch of knowledge advances and attain its own characteristic and its growth dynamics is matured, it will naturally bring out its own approaches or methodologies. But on fundamental questions, each no approach is sacrosanct. If you cannot explain basic physical reality with mathematics, it does not deny the physical reality but the validity of that mathematical argument. This is applicable for all knowledge. Similarly, the validity of advanced branch of any knowledge need to be questioned and subject matter to be rebased, if it goes against reality. This is what happened when non-Euclidean geometry or relativity theory was build up.

(Fourier series when applied to any wave in real world gives its constituent components as harmonics of a fundemental. But real world waveforms are not so simple and strait forward to fit in the model of ‘non-periodic’ or ‘periodic’. It is very difficult in many cases to specify the time period of a waveform or to ascertain it is non-periodic/periodic. It practical observation some waveform behaves as both. Based on the time period chosen you will get different frequency contents. You may feel to use fourier transform, but practical results show different. But again, series will not explain all observed phenomenon. When analyzing the noise effect of some complex electronics circuits which are sensitive to specific frequencies for specific time duration, this issue comes up. No mathematical predictions based on Fourier theorem works. (Spectrum analyzers are not useful because they are designed based on Fourier theorem). You have no other option, but fall back to first principle. Observe the circuit electric parameter practically and decide. It is too technical to explain here. If any specific interest, please contact me at mihraj2012@gmail.com. I can share the experimental details)

Mihiraj, your posts are not very relevant to this thread, so it will be good if you can start a new one to discuss your concerns.

Also, Mathematics is historically never been developed in CONJUNCTION with its applications as you seem to claim.Geometry is pretty old and it was developed and appreciated for its logical consistency. Infact almost all major mathematics concepts have been discovered in era which had no use of idea.

I am in no opposition of making Mathematics easier to understand by explaining its uses. But I am opposed to the idea that progress in Mathematics MUST be done only as and when its applications are clear. This logic is seriously flawed and its like ordering discoveries which rarely happens.

Finally, number multiplication IS defined by the laws that I mentioned. So there is no question of "needing it". And you most certainly cannot use scaling because as I had already pointed out, it doesn't lead you to any answer.
I have been asked this question on multiple occasions (along with 'why is 1 not a prime number?'). I would just like to add that any ring theorist worth his salt would point out that (-1) x (-1) = 1 is a consequence of distributive law (along with the other basic laws of course). The notion of 'Additive inverse of one squared is equal to one' is a consequence of the marriage of the addition and multiplication, in a ring, in the form of a distributive law. It is nothing special to integers (For instance, its true for matrices!).

Here I will claim that the number line way of teaching multiplication is not just a style, but *must be* the preferred methodology. The geometry of complex numbers and it's algebra are deeply intertwined, and we should allow kids to relish this beautiful fact. Let me explain:

I use the number line analogy to teach multiplying integers. However, I ask them to turn by 180 degrees to counter a negative sign. One might wonder how does such a simple restatement of the OP's claim give us something new.

Now here's a remarkable observation which I tell kids: "If rotating by 180 degrees is equivalent to multiplying by -1, then what would rotating by 90 degrees correspond to?". Well then you tell kids, "we don't know what it is multiplying by, so let's call it 'i'. So multiplying by i twice corresponds to rotating by 180 degrees, so i^2 = -1 !!"

I taught 'n'th roots of unity to my 6th std cousin this way.

Multiplying a complex number by e^(ix) is simply rotating by an angle x. It is a simple, brilliant observation. However one can hold off defining e and Euler's identity until kids reach 10th.

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