1585  Stevin introduces decimal numbers.  (For example, writing 1/8 as 0.125)
1637  Cartesian geometry published by Fermat and Descartes
1684  Leibniz publishes The Calculus
1761  Lambert proves that Pi is irrational
1821  Cauchy publishes the "epsilon-delta" definition of a limit,
      which brought rigor to The Calculus.
1830  Galois publishes "Galois Theory", which explains why
      a general polynomial equation of order n can be solved in terms of radicals
      only if n <= 4.
1844  Louisville proves the existence of transcendental numbers
1851  Louisville constructs the first transcendental number
1854  Riemann publishes the Riemann Integral, the first rigorous definition
      of an integral.
1859  Riemann Hypothesis published
1860  Grassmann studies the question of the axiomatization of arithmetic.
1870  Heine defines "uniform continuity"
1872  Heine proves that a continuous function on an open interval need not be
      uniformly continuous.
1872  Weierstrass publishes the "Weierstrass function", the first example of
      a function that is continuous everywhere but differentiable nowhere.
1873  Hermite proves that "e" is transcendental
1874  Cantor proves that the algebraic numbers are countable and that
      the real numbers are uncountable, using the "diagonal slash" argument.
1874  Cantor publishes the first attempt at a rigorous set theory.
1878  Cantor proves that the transcendental numbers and the real numbers have
      the same cardinality, thus estabilishing the ubiquity of transcendental numbers
1878  Cantor publishes the "Continuum Hypothesis":
      "There is no set whose cardinality is strictly between that of the integers
      and the real numbers."
      In 1900, Hilbert included the question of the Continuum Hypothesis in his
      list of 23 unsolved problems.
1882  Lindemann proves that Pi is transcendental.  A corollary is the imposibility
      of squaring a circle with a compass and straightedge.
1883  Cantor publishes the Cantor Set, a rich source of counterexamples
1887  Poincare discovers the phenomenon of "Chaos" while studying celestial mechanics.
      There exist orbits that are neither unbounded nor limiting to a stable state.
1889  Peano publishes a set of axioms for arithmetic which are now the standard.
1898  Hadamard defines a dynamical system where all orbits exponential diverge from
      each other with a positive Lyapunov exponent.
1900  Hilbert publishes a list of 23 unsolved problems. They include:
      The Continuum Hypothesis           (proved independent of ZFC by Godel)
      Prove that the axioms of arithmetic are consistent.  (proved impossible by Godel)
      The Riemann Hypothesis     (still unresolved)
      What is the densest sphere packing?  (resolved in 1998)
1901  Russell publishes "Russell's Paradox", which shows that Cantor's set theory
      leads to a contradiction.  This was resolved in 1922 by the Zermelo-Fraenkel
      axioms of set theory.
1904  Lebesgue publishes the Lebesque Integral, a generalization of the
      Riemann Integral.
      "Lebesgue Measure" is the standard way of assigning a measure to subsets of
      n-dimensional Euclidean space.  For n = 1, 2, or 3, it coincides with the
      standard measure of length, area, or volume.
      The Lebesgue measure of the set of rational numbers in the interval [0,1]
      is 0, and the real numbers on this interval have measure 1.
      The Cantor set is an example of an uncountable set that has Lebesgue measure zero.
1904  Poincare Conjecture published
1904  Zermelo defines the Axiom of Choice.
      Previously, mathematicians had been using this axiom implicitly without realizing
      it.
      Kronecker's opposition to Cantor's theories became the inspiration for
      the mathematical outlook of "Constructivism", which asserts that it is
      necessary to construct a mathematical object to prove that it exists (proving
      its nonexistence does not imply its existence).
      Constructivism is at odds with the Axiom of Choice and the Law of the Excluded
      Middle.
1922  Zermelo-Fraenkel axioms of set theory developed (ZF).  This resolved
      Russell's Paradox.
1924  Banach-Tarsky paradox published, exhibiting a spooky consequence of the
      Axiom of Choice.
1931  Godel proves the Incompleteness Theorems.
      For any set of axioms that are nontrivial and consistent,
      there will exist statements about the natural numbers that are true but
      cannot be proven within the system.
      Also, the system cannot prove its own consistency.
      Cantor's "diagonal slash" argument was an inspiration for these theorems.
1935  Bourbaki textbooks published, with the aim of reformulating mathematics on
      an extremely abstract and formal but self-contained basis.  With the goal
      of grounding all of mathematics on set theory, the authors strove for rigour
      and generality.
1940  Godel proves that the Axiom of Choice and the Continuum Hypothesis cannot be
      disproved with the Zermelo-Fraenkel axioms (ZF).
      He also established that the Continuum Hypothesis cannot be disproved
      even if the Axiom of Choice is added to the Zermelo-Fraenkel axioms (ZFC).
1961  Lorenz finds that computer simulations of weather have extreme sensitivity to
      initial conditions.
1963  Cohen proves that the Axiom of Choice and the Continuum Hypothesis cannot be
      proved with the Zermelo-Fraenkel axioms, establishing that they are
      independent of ZF.
1967  Mandelbrot publishes examples of fractals from nature
1967  Bishop publishes "Foundations of Constructive Analysis", where he proved
      most of the important theorems in real analysis by constructive methods.
1982  Mandelbrot publishes "The Fractal Geometry of Nature"
1983  Langlands Program published
1994  Wiles proves Fermat's Last Theorem
2000  Millenium Prize problems published.  They include:
      The Riemann Hypothesis
      P versus NP
      The Poincare Conjecture
      Navier–Stokes existence and smoothness
2002  Perelman proves the Poincare Conjecture
2009  Chau proves the Fundamental Lemma for the Langlands Program
2013  Zhang and Maynard publish results that constitute progress toward resolving
      the twin prime conjecture.

f(1)   =    0
f(2)   =    1
f(3)   =   1/2
f(4)   =   1/3
f(5)   =   2/3
f(6)   =   1/4
f(7)   =   3/4
f(8)   =   1/5
f(9)   =   2/5
f(10)  =   3/5
f(11)  =   4/5
f(12)  =   1/6
f(13)  =   5/6
f(14)  =   1/7
f(15)  =   2/7
f(16)  =   3/7
etc.

If a set can be bijected with the integers we say it is "Countable". The rational numbers are countable.

"Algebraic numbers" are numbers that can be expressed as the root of a non-zero n-degree polynomial with integer coefficients. The rational numbers correspond to roots of polynomials of degree 1. The algebraic numbers are countable.

Rational number          Expressible as A/B, where A and B are integers.
Irrational number        Not a rational number.
Algebraic number         Expressible as the root of a non-zero polynomial
                         with integer coefficients.
Transcendental number    Not an algebraic number.

This is Cantor's "diagonal slash" argument that he used to establish that the real numbers are more numerous than the integers. Godel's theorems are inspired by the diagonal slash argument.

In terms of subsets,

Integers  <  Rational numbers  <  Algebraic numbers  <  Transcendental numbers

0  =  Rational numbers  =  Algebraic numbers  <  Transcendental numbers  =  Real numbers  =  1

Integers  =  Rational numbers  =  Algebraic numbers  <  Transcendental numbers  =  Real numbers

The Continuum Hypothesis conjectures that there exists no set whose cardinality is strictly between that of the integers and the real numbers. If such a set S existed, its cardinality would be such that

Integers  <  S  <  Real numbers

Hermann Weyl, 1949: "Mathematics with Brouwer gains its highest intuitive clarity. He succeeds in developing the beginnings of analysis in a natural manner, all the time preserving the contact with intuition much more closely than had been done before. It cannot be denied, however, that in advancing to higher and more general theories the inapplicability of the simple laws of classical logic eventually results in an almost unbearable awkwardness. And the mathematician watches with pain the greater part of his towering edifice which he believed to be built of concrete blocks dissolve into mist before his eyes."

Hermann Weyl, 1939: "In these days the angel of topology and the devil of abstract algebra fight for the soul of each individual mathematical domain."

Poincare: "There is no actual infinite; the Cantorians have forgotten this, and that is why they have fallen into contradiction."