# theory of rational and irrational numbers

## 1 Basic Definitions

1. 1.

A number is said to be rational if it can be expressed as a quotient of integers (with non-zero denominator). The set of all rational numbers forms a field, denoted by $\mathbb{Q}$.

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The real numbers are the set of all possible decimal expansions (where we don’t allow any expansion to end in all $9$’s). For the formal definition please see the entry real number (http://planetmath.org/RealNumber). The real numbers form a field, usually denoted by $\mathbb{R}$.

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A real number is said to be irrational (http://planetmath.org/IrrationalNumber) if it is not rational, i.e. it cannot be expressed as a quotient of integers.  The decimal expansion is non-periodic for any irrational, but periodic for any rational number.

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For example, $\sqrt{2}$ is irrational.

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## 2 Small Results

1. 1.
2. 2.

$\sqrt{2}$ is irrational (http://planetmath.org/SquareRootOf2IsIrrationalProof). Similarly $\sqrt{d}$ is irrational as long as $d\in\mathbb{N}$ is not a perfect square  .

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Every irrational (and also rational) number is a limit of sequence of rational numbers (see real numbers (http://planetmath.org/RealNumber)).  An example:  the sequence  $(1+\frac{1}{1})^{1},\,(1+\frac{1}{2})^{2},\,(1+\frac{1}{3})^{3},\,...$converges to the number $e$.

7. 7.

The number e is irrational (this is not as difficult to prove as it is to show that e is transcendental  ). In fact, if $r\in\mathbb{Q}\setminus\{0\}$ then $e^{r}$ is also irrational (http://planetmath.org/ErIsIrrationalForRinmathbbQsetminus0). There is an easier way to show that e is not a quadratic irrational.

8. 8.

Every real transcendental number  (such as $e$) is irrational, but not all irrational numbers are transcendental — some (such as $\sqrt{2}$) are algebraic.

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10. 10.

If $a^{n}$ is irrational then $a$ is irrational (see here (http://planetmath.org/IfAnIsIrrationalThenAIsIrrational)).

11. 11.

A surprising fact: an irrational to an irrational power can be rational.

12. 12.

$\pi$ and $\pi^{2}$ are irrational (http://planetmath.org/PiAndPi2AreIrrational).

## 3 BIG Results

The irrational numbers are, in general, “easily” understood. The BIG theorems appear in the theory of transcendental numbers. Still, there are some open problems:  is Euler’s constant irrational? is $\pi+e$ rational?

Title theory of rational and irrational numbers TheoryOfRationalAndIrrationalNumbers 2013-03-22 15:14:10 2013-03-22 15:14:10 alozano (2414) alozano (2414) 15 alozano (2414) Topic msc 11R04 TheoryOfAlgebraicNumbers AlgebraicNumberTheory