Question

Insert a rational between $5\,and\,6$.
$\left( 1 \right)\,\,\,5.5$
$\left( 2 \right)\,\,\,4$
$\left( {3\,} \right)\,\,4.5$
$\left( 4 \right)\,\,6$

Answer 1

Hint: From the given options it is very easy to check which number is our required number. A rational number is any number that can be expressed as a fraction, thus producing a ratio between two numbers. Rational numbers may be positive or negative, and they can be written as fractions, mixed numbers, decimals, or integers.

Complete answer:
In the given questions, z
We know that,
$5 = \sqrt {25} $
$6 = \sqrt {36} $
All $\dfrac{n}{m}$(with positive integers n and m) for which two inequalities hold:
$n\, \geqslant 5m\,\,and\,\,n \leqslant 6m$.
There are infinitely many such rational numbers. Ordered by increasing numerator, and of course just mentioning the “smallest” representation of equal numbers, the first few are:
$\dfrac{5}{1},\,\dfrac{6}{{1\,}},\,\dfrac{{11}}{2},\,\dfrac{{16}}{3},\,\dfrac{{17}}{3},\,\dfrac{{21}}{4},\,\dfrac{{23}}{4},\,\dfrac{{26}}{5},\,\dfrac{{27}}{5}\,,\,\dfrac{{28}}{5}\,,\,\dfrac{{29}}{5},\,\dfrac{{31}}{6},\,\dfrac{{35}}{6},\,\dfrac{{36}}{7},\,\dfrac{{37}}{7},\dfrac{{38}}{7},\,\dfrac{{39}}{7},\,\dfrac{{40}}{7},\dfrac{{41}}{7},\dfrac{{41}}{8},..........$
Therefore, a rational number between $5$ and $6$ from the given options can be $5.5$
Hence, the correct option is $1.$

Additional information: A real number that is not rational is called irrational. Irrational numbers include $\sqrt 2 ,\pi ,\varphi \,\,and\,{e^{}}$. The decimal expansion of an irrational number continues without repeating. Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational. Rational numbers together with addition and multiplication form a field which contains the integers, and is contained in any field containing the integers. In other words, the field of rational numbers is a prime field, and a field has characteristic zero if and only if it contains the rational numbers as a subfield.

Note:
A rational number is a number that can be expressed as the quotient or fraction $\dfrac{p}{q}$ of two integers, a numerator p and a non-zero denominator q. The decimal expansion of a rational number either terminates after a finite number of digits or eventually begins to repeat the same finite sequences of digits over and over.