Lesson RATIONAL AND IRRATIONAL NUMBERS

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RATIONAL NUMBER
__________________
Any number which red%28can%29

be expressed in the form p%2Fq

where 'p' and 'q' (q not equal to 1) are integers mutually prime to each other (this means 'p' and 'q' have no common factors; in other words H.C.F. of 'p' and 'q' is 1) is called a rational number.
e.g. 56, -235.6, 5/7, sqrt%2816%29

, etc

Note: -235.6+=+-2356%2F10+=+-1178%2F5

. 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
____________________
Any number which red%28cannot%29

be expressed in the form p%2Fq

where 'p' and 'q' ('q' not equal to 1) are integers mutually prime to each other (this means 'p' and 'q' have no common factors; in other words H.C.F. of 'p' and 'q' is 1) is called an irrational number.
e.g. sqrt%285%29

, etc

Note: Let us prove that sqrt%285%29

is an irrational number.
Let us assume that sqrt%285%29

is a rational number.
Then it can be expressed as sqrt%285%29+=+p%2Fq

where 'p' and 'q' are mutually prime integers and 'q' unequal to 1.
Squaring both sides 5+=+%28p%2Fq%29%5E2

______(1)
Now, as 'q' is an integer so '5q' is also an integer.
But as 'p' and 'q' has no common factors and 'q' is not equal to 1, so p%5E2%2Fq

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%285%29

is a rational number must be wrong.
Hence, sqrt%285%29

is an irrational number.

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