Lesson RATIONAL AND IRRATIONAL NUMBERS

RATIONAL NUMBER
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Any number which be expressed in the form 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, , etc

Note: . 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 be expressed in the form 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. , , , etc

Note: Let us prove that is an irrational number.
Let us assume that is a rational number.
Then it can be expressed as where 'p' and 'q' are mutually prime integers and 'q' unequal to 1.
Squaring both sides
or ______(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 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 is a rational number must be wrong.
Hence, is an irrational number.


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