Question

How to insert irrational numbers between two given rational numbers.
please give answer step by step please ​

Answer 1

Answer:

How can I find an irrational number which is between two rational numbers?

Why do toppers choose BYJU'S for JEE preparation?

In order to find numbers in general between a and b we can go like this

c=m⋅a+n⋅bm+n

It is easy to see that

a<m⋅a+n⋅bm+n<b

if you plug rational numbers into m and n you get a number c that is rational and lies between a and b

But if you plug in irrational numbers for m and n you probably (not always, i will come to that later on ) end up with irrational numbers between a and b

for example let’s say

a=3, b=5 and let’s use m=π and n=φ

( π=pi and φ=phi=1+5√2 )

we get

c=3⋅π+5⋅φπ+φ

which looks pretty irrational to me :)

c=3⋅π+5⋅φπ+φ

You can go on trying different values for m and n and you will end up with different irrational numbers between a and b.

Now we can go on proving the following

If a and b rational ( a<b ) then c=m⋅a+n⋅bm+n is irrational when mn is irrational

Since a<b there is a k=b−a and b=a+k ( k positive rational number)

c=m⋅a+n⋅bm+n=

m⋅a+n⋅(a+k)m+n=

m⋅a+n⋅a+n⋅km+n=

a⋅(m+n)+n⋅km+n=

a⋅(m+n)m+n+n⋅km+n=

a+n⋅km+n

but a is rational so this number c is rational when d=n⋅km+n is rational and irrational when d is irrational.

So let us examine d

d is rational (irrational) when 1d is rational (irrational)

But

1d=m+nn⋅k=

mn⋅k+nn⋅k=

mn⋅k+1k

but again 1k is rational so if

mn⋅k

or

mn

is rational then c is rational otherwise c is irrational.

Conclusion.

In the formula

c=m⋅a+n⋅bm+n

plugging in numbers for m and n for which m/n is irrational you get an irrational number between a and b

Some examples

a<a⋅5–√+b⋅2–√5–√+2–√<b

or

a<a⋅e+b⋅πe+π<b

etc etc