Question

find any 5 rational numbers between 1/3 and -1/4​

Answer 1

Step-by-step explanation:

Hi,

Given 2 rational numbers 1/3 and 1/4.

Let a = 1/4 and b = 1/3.

Now, for any 2 given rationals a and b, we know that (a + b)/2 is also rational

Hence , (1/3 + 1/4)/2 = 7/24 is also rational.

let c = 7/24.

Also, we know that ∀ a, b ∈ R, we have that

                        a ≤ (a + b)/2 ≤ b

Equality occurs if a =b.

Now, hence 1/4 < 7/24 < 1/3.

Hence, we have inserted one rational between 1/3 and 1/4.

Similarly consider pairs either 1/3 and 7/24 or 7/24 and 1/4,

say consider 1/3 and 7/24.

Now, (1/3 + 7/24)/2 = 15/48 lies between these 2 numbers

let this be d = 15/48

7/24 < 15/48 < 1/3

a = 1/4 <  7/24 < 15/48 < 1/3 = b

Hence , we have inserted 2 rationals 7/24 and 15/48 between 2 given

rational numbers.

Similarly, we can insert countable number of rationals.

These types of rational cuts were been proposed by 'Dedekind', hence

these cuts are known as 'Dedekind cuts'

Hope, it helped !

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