Rational number

From Wikipedia, the free encyclopedia

In mathematics, a rational number is a number that 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 people 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: $\sqrt{2}$ (the square root of 2) is irrational(not rational). When people write a rational number in a decimal system, the digits must repeat sooner or later, for example $\frac{7}{99}=0.07070707070707070707070707070707...$ .

In the case of a rational number, we can always make it into a fraction in the following way:

Let´s say

$x = 0.070707...$

then

$100 x = 7.070707...$ (x multiplied by 100)

So, since the decimal part (the part to the right of the decimal point) is the same as x, we can always subtract x from both sides to eliminate all the decimals to get,

$100 x - x = 7.070707 - 0.070707070$

or $99 x = 7$

We can then divide both sides by 99.

That is, $x = 7 / 99$

Since x = 0.070707, we can finally rewrite the previous equation to show that 0.070707 can be written as the ratio of two integers, 7 and 99.

$0.07070707... = 7 / 99$

$x= \frac{7}{99}$

or

$0.0707070707...= \frac{7}{99}$

When people write an irrational number, the digits never repeat. An example of an irrational number is: $\sqrt{2}=1.414213562373095048801688724209...$ .

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

[change] Arithmetic

The set of all rational numbers is written as Q, or $\mathbb{Q}$ . $\mathbb{Q}$ can be defined as the following:

$\mathbb{Q} = \left\{\frac{m}{n} : m \in \mathbb{Z}, n \in \mathbb{Z}, n \ne 0 \right\}$

People can add or subtract two rational numbers and they always get another rational number. We say that rational numbers are closed under addition and subtraction.

$\frac{a}{b} + \frac{c}{d} = \frac{ad+bc}{bd}$

People can multiply two rational numbers and will always get another rational number. We say that rational numbers are closed under multiplication.

$\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$

People can divide two rational numbers and they always get another rational number, as long as they do not divide by zero. We say that rational numbers are closed under division.

Two rational numbers $\frac{a}{b}$ and $\frac{c}{d}$ are equal if and only if $a d = b c$

Additive and multiplicative inverses exist in the rational numbers:

$- \left( \frac{a}{b} \right) = \frac{-a}{b}$

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

[change] 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

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

[change] Properties

The set of all rational numbers is countable. People 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.

Rational number

From Wikipedia, the free encyclopedia

[change] Arithmetic

[change] Formal construction

[change] Properties

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