0.414

103/90

1/999

\(\frac{1}{7}=0.142857142857 \ldots=0 . \overline{142857}\) and \(\frac{2}{7}=0.28571428571428 \ldots=0 . \overline{285714}\)
Here, the two decimal expansions are non-terminating recurring.
Hence, 1/7​ and 2/7​ are two rational numbers.
We know, between any two rational numbers, there are infinitely many irrational numbers.
An irrational number has non-terminating non-recurring decimal expansions.
Then an irrational number between \(\frac{1}{7} \text { and } \frac{2}{7}\) is 0.15015001500015.
Similarly, 0.21020020002... is another irrational number between \(\frac{1}{7} \text { and } \frac{2}{7}\)

101/144, 47/72

If possible let √2 be rational and let its simplest form be \(\frac{a}{b}\) where a and b are integers having no common factor other than 1 and b≠0
Now, √2 = \(\frac{a}{b}\)⇒2 = \(\frac{a^{2}}{b^{2}}\) [on squaring both sides]
⇒ 2b² = a² ...(1)
⇒ 2 divides a² 
⇒ 2 divides a
Let a = 2c for some integer c.
Putting a = 2c in (1), we get
2b² = 4c² ⇒ b² = 2c²
⇒ 2 divides b²
⇒ 2 divides b
Thus 2 is a common factor of a and b
But this contradicts the fact that a and b have no common factor other than 1 the contradiction arise by assuming that √2 is rational.
Hence √2 is irrational.

\(\frac { 2\sqrt { 3 } +3-\sqrt { 21 } }{ 3 } \)

0

a = 16, b = 0

\(\frac { 3\sqrt { 6 } }{ 4 } \)