Q) What is rational and irrational number?
Ans
Rational number:- Those numbers which can be written in the form of fractional value (a/b) or in decimal but the value after decimal will be repetitive number or can be a whole number are rational number.
For example 1/3, 4/9, 4, 3.171717 etc.
Irrational number:- Those numbers which is not rational or can't be written in the form of fractional value.
For example 2.34587...., π etc
Here the confusion comes on π which is an irrational number but we have seen it's value is written as 22/7 which is a rational number.
So the point here is the actual value of π is 3.14159..... which is not repetitive value and it consider as approx value in the form of ratio is 22/7.
Q) Prove √5 is a irrational number.
Ans
Let’s take it for granted that the number √5 is a rational number.
If √5 is rational, then it may be expressed as an equation of the type a/b, here integers are a and b and be should not be equal to 0.
√5/1 = a/b
√5b = a
Bringing both sides into balance
Now square both the side
5b² = a²
b²= a²/5 —- (i)
This indicates that a² is divided by 5.
Hence a also divided by 5.
If
a/5 = c
a = 5c
On squaring, we get
a² = 25c²
Replace a² in the equation with its value (1).
5b² = 25c²
b² = 5c²
b²/5 = c²
This indicates that b² may be divided by 5, and hence, b can likewise be divided by 5. Since this is the case, a and b share the factor 5 in common. However, this goes against the fact that a and b are in the coprime position.
we can not write √5 as in fractional form.
So our assumption of √5 is a rational number is wrong. As a result, we are forced to the conclusion that √5 is irrational
