Explain the density of rational numbers with exampl

1.	Explain the density of rational numbers with exampl
Answer» The Density of the Rational/Irrational Numbers We will now look at a theorem regarding the density of rational numbers in the real numbers, namely that between any two real numbers there exists a rational number. Theorem 1 (The Density of the Rational Numbers):Letbe any two real numbers where. Then there exists a rational numbersuch that. Proof:Suppose that. Sincewe have thatand furthermore we have that. Now we know by the Archimedean properties that since, then there exists a natural numbersuch that. If we multiply this out we get thator rather. Now we know that sinceand since, and by the Archimedean properties that sincethen there exists a natural number, call itsuch thator equivalently.

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The Density of the Rational/Irrational Numbers

We will now look at a theorem regarding the density of rational numbers in the real numbers, namely that between any two real numbers there exists a rational number.

Theorem 1 (The Density of the Rational Numbers):Letbe any two real numbers where. Then there exists a rational numbersuch that.

Proof:Suppose that. Sincewe have thatand furthermore we have that. Now we know by the Archimedean properties that since, then there exists a natural numbersuch that.

If we multiply this out we get thator rather. Now we know that sinceand since, and by the Archimedean properties that sincethen there exists a natural number, call itsuch thator equivalently.

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