infinite rational numbers Archives – Mathemafia http://sh016.global.temp.domains/~mathemaf/tag/infinite-rational-numbers/ Mathematics is like music - Enjoy it!!! Sat, 08 Mar 2025 10:27:32 +0000 en-US hourly 1 https://wordpress.org/?v=7.0 https://mathemafia.com/wp-content/uploads/2020/08/cropped-M__Logo__SM-1-32x32.png infinite rational numbers Archives – Mathemafia http://sh016.global.temp.domains/~mathemaf/tag/infinite-rational-numbers/ 32 32 Mean method : Infinitely many rational numbers between any two rational numbers https://mathemafia.com/mean-method-infinitely-many-rational-numbers-between-any-two-rational-numbers/ Tue, 22 Sep 2020 13:36:47 +0000 http://sh016.global.temp.domains/~mathemaf/?p=2080 Concept: In this tutorial, you will learn that there are infinitely many rational numbers between any two rational numbers. There are multiple ways to learn it and I have taught by calculating the mean (average) of two numbers. Rational numbers…

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Concept: In this tutorial, you will learn that there are infinitely many rational numbers between any two rational numbers. There are multiple ways to learn it and I have taught by calculating the mean (average) of two numbers.

Rational numbers are infinitely many
Integers are infinitely many

Number System

Polynomials

Important Concepts – Miscellaneous

Surface area and Volumes

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]]> Proof – Rational numbers are infinitely many https://mathemafia.com/rational-numbers-are-infinitely-many/ Tue, 18 Aug 2020 14:17:26 +0000 http://sh016.global.temp.domains/~mathemaf/?p=1455 Theorem: Prove that there are infinitely many rational numbers. Proof is quite easy. You can watch video tutorial of proof as well.

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Rational Number - Definition

Before we prove that there are infinitely many rational numbers, let us recall the definition of Rational Numbers.

A number that is of the form p/q where p and q are integers and q≠0 is called a rational number.

\( \begin{aligned} \frac{p}{q}\ \ \ \ where \ \ p,q\in\mathbb{Z}, \ \ q\ne0 \end{aligned}\)

Proof - Rational Numbers are Infinitely Many

Theorem: Prove that there are infinitely many rational numbers.

Proof: Let us assume that there are finitely many Rational Numbers.

\( \begin{aligned} &\Rightarrow \text{∃ a greatest rational number, say } x\\\\ &\Rightarrow x = \frac{p}{q} \ \ where \ \ p,q\in\mathbb{Z}, \ \ q\ne0  \end{aligned}\)

Let us get a new number say y by adding 1 to x.

\( \begin{aligned} y&=x+1\\\\ &=\frac{p}{q}+1\\\\ &=\frac{p+q}{q} \end{aligned}\)
\( \begin{aligned} &Now \ \ p+q \in\mathbb{Z}\\\\ &\Rightarrow y = \frac{p+q}{q} \ \ \text{is also a rational number}\\\\ &Now \ \ y > x \end{aligned}\)

This is a contradiction that x is the greatest rational number.

This means our assumption is wrong that there are finitely many rational numbers.

Hence we can conclude that there are infinitely many rational numbers.

Watch on YouTube

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