About numbers in higher level math, theoretical math?

Question

Im just curious. I am a calculus student and am well versed in mathematics. I have a high degree of conceptual understanding of mathematics that, not to sound arrogant, stem well beyond most people who perform just as well in the same courses as me.

It just dawned on me a minute ago, however, that there might exist numbers that I have yet to learn about. I mean, come on, how long was it before I learned about imaginary numbers, and then about complex numbers?

My question is simple, and I hope that someone well more versed than me can answer this thoroughly.

Simply put, do numbers exist that are not complex? If real and imaginary numbers lie on two number lines, and complex numbers form the complex plain, where both purely real and purely imaginary numbers lie on the axes. What lies on the z-axis? Are there numbers that do not lie strictly on the complex plain?

Answer 1

It depends on what you mean by "numbers". The complex numbers are a pretty "complete" set of numbers, in the following sense:

- algebraically complete: a polynomial with complex coefficients can be completely factored in linear factors.

- topologically complete: a sequence of numbers which lie ever closer together (Cauchy sequence) has a complex number as its limit.



Your idea of introducing new "axes" is useful. However, adding just one axis (let's say, mutiples of a "number" z) does not work. Here is a proof: any number of this form could be written as

... a + b.i + c.z,

with a, b and c real numbers. Suppose that we calculate i * z. This is also a number of this form, say

... i * z = a + b.i + c.z.

However,

... (i * z) * i = a.i - b + c.(z * i)

... (i * z) * (i - c) = a.i - b

... i * z = (a.i - b) / (i - c)

... z = (a + b i) / (i - c),

which proves that z would be a regular complex number.



In this proof I used two special properties of numbers, namely associativity and commutativity (i *z = z * i). Without these properties, the proof becomes invalid.



That is the inspiration for a new sort of "numbers", called Quaternions. They are of the form

... a + b.i + c.j + d.k

and defined by the rules

... i^2 = -1, j^2 = -1, k^2 = -1 (like complex numbers)

... i.j = k, j.k = i, k.i = j.

From these definitions, it follows immediately that

... i.k = i.(i.j) = (i.i).j = (-1).j = -j,

and in the same way j.i = -k and k.j = -i.

As you see, the order of multiplication matters. For instance,

... (3i + 2k) . (i - 4j) = -3 + 8i - 12k + 2j , but

... (i - 4j) . (3i + 2k) = -3 - 8i + 12k - 2j.



A further development are the Octonions, which are an eight-dimensional extension of the real numbers. These numbers are non-associative; that is, in general

... (a . b) . c =/= a . (b . c).



Note: the idea of adding more directions defines a *vector space*. Vectors are extremely useful in mathematics, but they are not considered numbers. In general, vectors cannot be multiplied to give another vector (except in 3 and 7 dimensions, but that is basically the case of Quaternions and Octonions), and you cannot divide by vectors.

Answer 2

Not really a problem. If somebody came out and proved the math to be wrong, he'd re-write astrophysics (cosmology specifically) and experience a career slingshot rarely seen. Plus, I can always learn the math and discover for myself if I was right; I'm just a little too busy learning things that are more relevant to my career (I'm also trying to learn things like "a good place for chicken wings in my city," but I accept that that may be a lost cause). But when it comes to religion, you can't test the claims by building your knowledge. The proof is never shown. All you get are assertions, and evidence to the contrary doesn't "rewrite the books." Instead, you either get "atheistic agendas/vast scientific conspiracies/Christ said we'd be persecuted" sort of things that completely ignore the point at hand, or strawmen are destroyed to rally the flock. It's really two different things. The math can be learned and examined; religious works are merely to be accepted.

Answer 3

"Square root of ( -1)" based computation is not easily concievable to ordinary users of computing!



It is not a need of usual computing!



I do not even appreciate using numbers less than zero!



You may fix that I am not having exposure to the type of computing you have explained in paragraphs 1,2 and 3



By a study of Vedic Mathematics I have succeeded to control and use some more-digits numbers, which is a manner of number application anybody can do!



A simple example is....



0 1 2 3 4 5 6 7 9 is 1/81 recurring



9 8 7 6 5 4 3 2 1 is 80/81 recurring

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00 01 02 03 04.......95 96 97 99 is 1/ 9801 recurring



99 98 87 96 95.......04 03 02 01 is 9800/9801 recurring

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or any higher order number groups of 3, 4, 5 or any higher



When we merge each pair as one number they are either...



(9 times 1)^2 or



(99 times 01)^2 or



(999 times 001)^2 or



(9999 times 0001)^2 or a higher order similar number^2!



Only matter to be remembered is 1n 1/81, 1/9801, 1/998001 etc a concerned second last group (8, or 98 or 998 ...) has to be skipped!



I also know why said jump takes place!



It is interesting to note that simple logic like these has been used in Ancient India itself which effectively computed any number of digits squares, cubes, value of pi etc which is now regarded as not-scientific?



Your last para "simply put it, do numbers exist...." is intersting. Answer to it is YES, but not when you regard a supreme knowledge like Vedic Mathematics as not-scientific" (though it is not a fault of yours)!





Regards

Answer 4

yes and no. As the complex plane can be viewed as E2, Euclidean 2-space, there can be sn srbitrary number of dimensions. En is the generaliation to n mutually perpendicular axes. And in fact, there's Hilbert Space, which has infinitely many dimensions, but with the twist that for any point in it, all but a finite number of coordinates are 0.



You might say, but these (the points of these spaces) aren't numbers. Well, they generalize the concept of number. In fact, most of "higher" math consists of axiomatic systems and their theorems. groups, rings, algebras, R-=modules, topologies, categories, to name a few.