It wasn't until quite a bit later that mathematicians formalized and proved a rigorous mathematical statement which justified this intuition called invariance of domain, which says you can't do this in a continuous way. More generally, people were pretty sure you couldn't fit $\mathbb^m$ if $n$ was greater than $m$. Many people thought that this was impossible before he did it they had an intuition that you couldn't possibly "fit" the plane into the line. Before people explicitly constructed the real numbers and used them to define and prove things about other concepts, it was never totally clear what was true or what was false, and everybody was very confused.įor example, Cantor proved that the number of points in the plane is the same as the number of points on a line. Mathematicians need these kind of sophisticated constructions because they are what is required for rigorous proofs. The truth is that the real numbers are a sophisticated mathematical construction and that any explanation of what they "are" which pretends otherwise is a convenient fiction. I think you were being a little too hard on Isaac. ![]() The transcendental numbers are complex numbers (sometimes limited to real numbers) that are not algebraic. The algebraic numbers are numbers that are solutions to polynomial equations with integer coefficients. The imaginary numbers are sometimes defined to be the "pure imaginary" numbers-complex numbers for which the "real part" a = 0, sometimes with the added restriction that b is not zero-and are sometimes defined to be the non-real complex numbers. The complex numbers are the numbers that can be expressed as a b * i where a and b are real numbers and i behaves like a real number under addition/multiplication/distribution/etc., with the added rule that i 2 = -1. The irrational numbers are the real numbers that are not rational numbers. The real numbers are the set of numbers that are limits of Cauchy sequences of rational numbers. The rational numbers are numbers that can be expressed as a ratio of an integer to a non-zero integer. The integers are the whole numbers and their additive inverses. The whole numbers are the natural numbers with the additive identity element called 0. Whether or not 0 is a natural number varies in various texts.) (This definition includes 0 in the natural numbers altering rules 1, 3, and 5 to refer to one instead of zero excludes 0 from the natural numbers.
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