Ten Ways of Looking at Real Numbers

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#4. Real Numbers as a Completion of the Rational Numbers

Ostrowski’s theorem: The inequivalent valuations on the rational numbers are absolute value, the trivial valuation, and the p-adic valuations for every prime p.

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#4. Real Numbers as a Completion of the Rational Numbers

The metric completions of the rationals defined by a valuation are:

discrete topology on Q (with the trivial valuation)

R (with absolute value)

Rp (the p-adic reals, with the p-adic valuation).

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#5. The Real Numbers as a Field

The real numbers form an ordered field.

Subfields of the real numbers:

rational numbers

real algebraic number fields

computable real numbers

constructible real numbers

The algebraic completion of the real numbers is the field of complex numbers

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#5. The Real Numbers as a Field

The field of real numbers is the prototypical real-closed field: its algebraic closure is a finite extension.

The Artin-Schreier theorem characterizes a real-closed field:

it has characteristic 0

algebraic closure by adjoining i, where i2 = -1

it has a linear order

every positive number has a square root

-1 is not a sum of squares

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#5. The Real Numbers as a Field

Any field is a vector space over a subfield.

The real numbers form a vector space over the rational numbers.

A basis for this vector space is called a Hamel space.

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#6. The Real Numbers as an Algebra

To what extent can the operations on the reals extend to finite-dimensional algebras over the reals?

Here we list a few results.

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#6. The Real Numbers as an Algebra

The finite-dimensional associative real division algebras are the real numbers, complex numbers, and the quaternions. (Frobenius)

The finite-dimensional real commutative division algebras with unit are the real numbers and the complex numbers. (Hopf)

The finite-dimensional real division algebras have dimension 1, 2, 4, or 8. (Kervaire, Milnor)

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#7. The Cardinal of the Real Numbers

Cantor showed that the real numbers are not equinumerous with the integers.

Write as the cardinal of the set of real numbers, the cardinal of the continuum.

The Continuum Hypothesis: Does ?

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