Dear professors and students, greetings and courtesy. I wanted to know if the real numbers are the largest and the last set of numbers that exist, or if there are sets or sets of numbers that are larger than that, but maybe they have not been discovered yet? Which is true? If it is the last set of numbers that exists, what theorem proves the non-existence of a set of numbers greater than it? And if there is a larger set than that, in terms of the history of mathematics, by obtaining the answer to which mathematical problem, it was proved that the obtained answer is not closed with respect to the set of complex numbers and belongs to a larger set? Thank you very much
First, what do you mean by a number? If you mean a set of things that includes, say, rational numbers and extends the addition & multiplication operations in a way that is consistent with the usual rules (e.g., associativity: a+(b+c)=(a+b)+c & a*(b*c)=(a*b)*c, distributivity: a*(b+c)=a*b+b*c, zeros: 0+a=a, 0*a=0, commutativity: a+b=b+a, a*b=b*a, unit/identity: 1*a = a, inverses: every a has a b=-a where a+b=0, every a not zero has a b=1/a where a*b=1) then we can make a claim that the complex numbers are the biggest class of "numbers".
Actually, you can go further by constructing a non-Archimedean set of "real numbers" which extends real numbers and is still an ordered field, and then have an extended set of complex numbers of the form a+i.b where a, b are these extended real numbers.
Non-Archimedean means that there are "numbers" a, b > 0 where a/b is larger than any whole number 1, 2, 3, ...
But if you keep the regular real numbers, but are willing to lose something else like commutativity of multiplication (a*b=b*a) then there are the quaternions discovered/invented by William Rowan Hamilton in the 19th century. These have the form a+b.i +c.j +d.k where a, b, c, d are real numbers. The essential properties of the symbols i, j, k are that i*i = j*j = k*k = -1, and i*j = -j*i = k, j*k = -k*j = i, & k*i = -i*k = j. Quaternions form a division ring.
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