Question

Irrafind irrational number between two irrational numbers

Answer 1

Answer:

Step-by-step explanation:

Let us consider an example

√2 and √3 are irrational numbers

√2 = 1.4142 (nearly)

√3 = 1.7321 (nealry)

Now we have to find an irrational number which should lie between 1.4142 and 1.7321

The irrational number is 1.50500500050000.....

Answer 2

Suppose we have two rational numbers a and b, then the irrational numbers between those two will be, √ab. Now let us find two irrational numbers between two given rational numbers.

1. Find an irrational number between two rational numbers 2 – √3 and 5 – √3

Let x be the irrational number between two rational numbers 2 – √3 and 5 – √3. Then we get,

2 – √3 < x < 5 – √3

⇒ 2 < x + < √3 < 5

We see that x + √3 is an irrational number between 2 – √3 and 5 – √3 where 2 – √3 < x < 5 – √3.

2. Find two irrational numbers between two given rational numbers.

Now let us take any two numbers, say a and b. Let x be any number between a and b. Then,

We have a < x < b….. let this be equation (1)

Now, subtract √2 from both the sides of equation (1)

So, a – √2 < x < b – √2……equation (2)

= a < x + √2 < b

Addition of irrational number with any number results into an irrational number. So, x + √2 is an irrational number which exists between two rational numbers a and b.