Question

How many irrational numbers are there between 1 and 6?

Answer 1

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.......`, `1.01001000200003.......`
`1.00002000200002.......`.
`1.00003000300003.......` and so on.

Similarly, between any two rational numbers `a` and `b`, we can have `(a+b)/2`, a rational number and then between `a` and `(a+b)/2` as well as `(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.

Answer 2

`bbc = 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 `bbc` 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.