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Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

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Section
BuyFindarrow_forward

Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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Show that the set of all irrational numbers is dense along the number line by showing that given any two real numbers, there is am irrational number in between.

To determine

To prove:

Along the number line, the set of all irrational numbers is dense by showing that, for any two real numbers, there is always an irrational number in between.

Explanation

Given information:

The set of all irrational numbers.

Concept used:

Rational number can be written in the form of pq, where p,q and q0.

Proof:

To show that the set of all irrational numbers is dense along the number line, it has to be shown that given any two real numbers no matter how close to each other, there is an irrational number in between.

Let two real (rational or irrational) numbers be r1 and r2 such that r2>r1.

The objective is to find an irrational number x such that r1<x<r2.

r1 and r2 are real numbers such that r2r1>0.

x is the irrational number such that r1<x<r2.

It is known that 2 is an irrational number.

Hence, 12 is also an irrational number because the product of a rational number and an irrational number is always an irrational number.

2<1 so, 0<12<1

Multiply r2r1 on each side of the above inequality.

Since r2r1>0, so, the sign of the inequality doesn’t change.

(r2r1).0<(r2r1)12<(r2r1)0<( r 2 r 1 )2+r1<(r2r1)

Add r1 to each side of the above inequality

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