<PRE><B>Question 41656</B>
RATIONAL NUMBER
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Any number which {{{red(can)}}} be expressed in the form {{{p/q}}} where &#39;p&#39; and &#39;q&#39; (q not equal to 1) are integers mutually prime to each other (this means &#39;p&#39; and &#39;q&#39; have no common factors; in other words H.C.F. of &#39;p&#39; and &#39;q&#39; is 1) is called a rational number.
e.g. 56, -235.6, 5/7, {{{sqrt(16)}}}, etc


Note: {{{-235.6 = -2356/10 = -1178/5}}}. Thus -235.6 can be expressed as a ratio of two integers -1178 and 5 and -1178 and 5 have no factors common between them.



IRRATIONAL NUMBER
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Any number which {{{red(cannot)}}} be expressed in the form {{{p/q}}} where &#39;p&#39; and &#39;q&#39; (&#39;q&#39; not equal to 1) are integers mutually prime to each other (this means &#39;p&#39; and &#39;q&#39; have no common factors; in other words H.C.F. of &#39;p&#39; and &#39;q&#39; is 1) is called a rational number.
e.g. {{{sqrt(5)}}}, {{{pi}}}, {{{-sqrt(8)}}}, etc


Note: Let us prove that {{{sqrt(5)}}} is an irrational number.
Let us assume that {{{sqrt(5)}}} is a rational number.
Then it can be expressed as {{{sqrt(5) = p/q}}} where &#39;p&#39; and &#39;q&#39; are mutually prime integers and &#39;q&#39; unequal to 1.
Squaring both sides {{{5 = (p/q)^2}}}
or {{{5*q = p^2/q}}} ______(1)
Now, as &#39;q&#39; is an integer so &#39;5q&#39; is also an integer.
But as &#39;p&#39; and &#39;q&#39; has no common factors and &#39;q&#39; is not equal to 1, so {{{p^2/q}}} cannot be an integer.
So, there is a contradiction!
Left side of eqn.(1) is an integer but the right side is not.
This cannot be true.
So our very assumption that {{{sqrt(5)}}} must be wrong.
Hence, {{{sqrt(5)}}} is an irrational number.</PRE>