Real Numbers Test - 13

√9 is an irrational number but because √9 = √3² = 3 and 3 is a rational number.

√p is an irrational number because the square root of every prime number is an irrational number. (for example √3 is an irrational number)

The value of π = 3.141592653589……….

∴ Value of π is not-repeating decimal number

Therefore, π is an irrational number.

Rational Numbers say  4/9, p/q, √4 , fraction, whole numbers, terminating decimal, repeating decimal, perfect square,  can be expressed as a ratio of two integers provided the denominator is not equal to zero

Irrational Numbers √2 ,√5 ,√7 , π not a fraction, decimal does not repeat, decimal does not end, non-perfect square,  we cannot express as a ratio but both can be expressed as decimal numbers

The difference between a rational and an irrational number is always an irrational number.

e.g. rational - irrational = irrational   say 2-√2  = irrational

The difference of two distinct irrational numbers can be either a rational number or an irrational number.

e.g difference between pi and  (pi - 3) is equal to 3 which is rational

√2 and √2 + 1 both are irrational but their difference is 1 which is rational

similarly √2  and √3 are irrational and their difference ( √3 - √2 ) is also irrational

The product of a rational number and an irrational number can be either a rational number or an irrational number.

e.g √5 × √2 = √10 which irrational

but  √8 × √2 = √16  = 4 which is a rational number

Thus, product of two irrational numbers can be either rational or irrational 

similarly, product of rational and irrational numbers can be either rational or irrational 

5 × √2 = 5√2  which is irrational.

but 4 × √4 = 4  2× = 8 which is rational.

The sum of two irrational numbers can be either a rational number or an irrational number.

e.g 5√3 + 3√2 = 5√3 + 3√2 sum is irrational

(2+ 6√7) + ( - 6√7) = 2 sum is rational

Hence sum can be either rational or irrational

Here, 3 is rational and 2√5  is irrational.

We know that the sum of a rational and an irrational is an irrational number, therefore, 3 + 2√5 is irrational.

(1 + √2) + (1 - √2) = 1 + √2 + 1 - √2 = 1 + 1 = 2 And 2 is a rational number.

Let aa be rational and √b is irrational.

If possible let a+√b be rational.

Then a + √b is rational and aa is rational.

⇒ [(a + √b)−a] is rational [Difference of two rationals is rational]

⇒ √b  is rational.

This contradicts the fact that √b is irrational.

The contradiction arises by assuming that a + √b is rational.

Therefore, a + √b is irrational.