SOLUTION: What is the difference between a rational number and an irrational number?

Question 41656This question is from textbook Elementary and Intermediate Algebra
: What is the difference between a rational number and an irrational number? This question is from textbook Elementary and Intermediate Algebra

Answer by psbhowmick(878) (Show Source): You can put this solution on YOUR website!
RATIONAL NUMBER
__________________
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
____________________
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. , , , 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 must be wrong.
Hence, is an irrational number.
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