How many irrational numbers are there between 1 and 6?

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16
Jul 18, 2016

Answer:

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.

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6
Sep 24, 2017

Answer:

#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.

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