Prove that there are infinitely many irrational numbers!

Jamie Medina

Jamie Medina

Answered question

2022-11-25

Prove that there are infinitely many irrational numbers!

Answer & Explanation

driwant9HB

driwant9HB

Beginner2022-11-26Added 11 answers

HINT: Show that there is an injection from an infinite set, e.g. N , into the set of irrational numbers. That is, send every natural number n to some irrational number. Also, recall that if we take a non-zero rational number, its sum and product with any irrational number is irrational.
phumzaRdY

phumzaRdY

Beginner2022-11-27Added 1 answers

Based on the fact that the set of natural numbers is infinite:
Take your favourite irrational number, say π , and now look at the set { n π ; n N } . Prove this set is infinite and all its elements are irrational
Based on the fact that | R | = 2 0 > 0 = | Q | = | N | : By difference of cardinalities , if I r r = the set of irrational numbers, then
| I r r | = | R Q | = 2 0 0 = 2 0 (yes, that difference is ugly...)

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