How many irrational numbers are there between 1 and 6?

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16
Jul 18, 2016

Infinite irrational numbers.

Explanation:

Between any two numbers, however large or small the difference between them may be, we have infinite rational as well as irrational numbers. As such between $1$ and $6$ too we have infinite irrational numbers.

Irrational numbers in their decimal form are non-repeating and non-terminating numbers. Hence say between $1$ and $1.01$, we can construct infinite irrational numbers like $1.00001000100001 \ldots \ldots .$, $1.01001000200003 \ldots \ldots .$
$1.00002000200002 \ldots \ldots .$.
$1.00003000300003 \ldots \ldots .$ and so on.

Similarly, between any two rational numbers $a$ and $b$, we can have $\frac{a + b}{2}$, a rational number and then between $a$ and $\frac{a + b}{2}$ as well as $\frac{a + b}{2}$ and $b$, we can construct more rational numbers. And repeating this we can have infinite rational numbers between any two rational numbers.

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6
Sep 24, 2017

$\boldsymbol{c} = {2}^{\omega}$

Explanation:

The rational numbers between $1$ and $6$ are infinitely many but countable. That is, you could write a rule for a sequence which would eventually list any particular rational number between $1$ and $6$.

The irrational numbers between $1$ and $6$ are uncountably infinite. No sequence indexed by natural numbers can list all of them. If the (transfinite) number of rationals is written as $\omega$, then the number of irrationals can be written as ${2}^{\omega}$. Sometimes the symbol $\boldsymbol{c}$ is used, denoting the cardinality of the continuum (i.e. how many real numbers there are in total).

Despite there being far more irrational numbers than rational numbers, we have the properties that:

• Between any two distinct rational numbers is an irrational number.

• Between any two distinct irrational numbers is a rational number.

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