Rational number

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In mathematics, a rational number is a fraction (ratio)of two integers. A rational number can be written as one whole number divided by another whole number. The short way of writing this in math language is a/b or \frac{a}{b}, where b is not 0.

Most of the numbers you see every day are rational.

  • All the natural numbers are rational numbers.
  • All the whole numbers are rational numbers.
  • All fractions are rational numbers (unless they have square roots in them).

There are numbers which are not rational. For example: \sqrt{2} is irrational. When you write a rational number in a decimal system, the digits must repeat sooner or later, for example \frac{7}{990}=0.007070707070707070707070707070707.... But when you write an irrational number, it will not happen. For example: 0.12345678910111213141516171819202122... is not rational. Also, \sqrt{2}=1.414213562373095048801688724209.... The digits also do not repeat.

Every integer is a rational number, because it can be written as a / 1: a = a / 1. For example 3 = 3 / 1.

[edit] Arithmetic

  • The set of all rational numbers is written as Q, or \mathbb{Q}. \mathbb{Q} can be defined as the following:
\mathbb{Q} = \left\{\frac{m}{n} : m \in \mathbb{Z}, n \in \mathbb{Z}, n \ne 0 \right\}
  • You can add or subtract two rational numbers and you always get another rational number. We say that rational numbers are closed under addition and subtraction.
\frac{a}{b} + \frac{c}{d} = \frac{ad+bc}{bd}

 

  • You can multiply two rational numbers and you always get another rational number. We say that rational numbers are closed under multiplication.
\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}

 

  • You can divide two rational numbers and you always get another rational number, as long as you do not divide by zero. We say that rational numbers are closed under division.
  • Additive and multiplicative inverses exist in the rational numbers:
- \left( \frac{a}{b} \right) = \frac{-a}{b}

 

\left(\frac{a}{b}\right)^{-1} = \frac{b}{a} \mbox{ if } a \neq 0

[edit] Formal construction

Mathematically we can define them as an ordered pair of integers \left(a, b\right), with b not equal to zero.

We can define addition and multiplication of these pairs with the following rules:

\left(a, b\right) + \left(c, d\right) = \left(ad + bc, bd\right)
\left(a, b\right) \times \left(c, d\right) = \left(ac, bd\right)

To be sure, our expectation that 2 / 4 = 1 / 2 is right, we define an equivalence relation \sim upon these pairs with the following rule:

\left(a, b\right) \sim \left(c, d\right) \mbox{ iff } ad = bc

This equivalence relation does not change the addition and multiplication defined above, and we may define Q to be the quotient set of ~, it is we identify two pairs (a, b) and (c, d) if they are equivalent in the above sense.

We can also define a total order on Q by writing

\left(a, b\right) \le \left(c, d\right) \mbox{ iff } ad \le bc

[edit] Properties

The set of all rational numbers is countable. You can count the positive ones in the following way:

\begin{matrix} \frac{0}{1}      & \rightarrow & \frac{0}{2}  &             & \frac{0}{3}  & \rightarrow & \frac{0}{4}  &        \\            & \swarrow    &        & \nearrow    &        & \swarrow    &        &        \\ \frac{1}{1}      &             & \frac{1}{2}  &             & \frac{1}{3}  &             & \ddots &        \\ \downarrow & \nearrow    &        & \swarrow    &        &             &        &        \\ \frac{2}{1}      &             & \frac{2}{2}  &             & \ddots &             &        &        \\            & \swarrow    &        &             &        &             &        &        \\ \frac{3}{1}      &             & \ddots &             &        &             &        &        \\ \downarrow &             &        &             &        &             &        &        \\ \vdots     &             &        &             &        &             &        & \end{matrix}

That many numbers are counted several times here, is not a big problem. We could also imagine leaving them out. The important thing is that we have now a long list of positive rational numbers; so the (positive) rational numbers are countable. For using this method for all rational numbers, add after each step the same but negative number.

The rationals are a densely ordered set: between any two rationals, there sits another one, in fact infinitely many other ones.