Question

8. Prove that vp is not a rational number, if
is not a perfect square​

Answer 1

Answer:

(a) All numbers that are not rational are considered irrational. An irrational number can be written as a decimal, but not as a fraction. An irrational number has endless non-repeating digits to the right of the decimal point. Here are some irrational numbers:

π=3.141592…

2

=1.414213…

Therefore,

2

is an irrational number.

(b) Let us take a rational number a=

1

2

and an irrational number b=

2

, then their product can be determined as:

a×b=2×

2

=2

2

which is also an irrational number.

Therefore, if a is a rational number and

b

is an irrational number than a

b

is an irrational number.

(c) By definition, a surd is a irrational root of a rational number. So we know that surds are always irrational and they are always roots.

For eg,

2

is a surd since 2 is rational and

2

is irrational.

Surds are numbers left in root form

to express its exact value. It has an infinite number of non-recurring decimals.

Therefore, every surd is an irrational number.

(d) Let us take a positive integer 4, now square root of 4 will be:

4

=2 which is not an irrational number

Hence, the square root of every positive integer is not always irrational.