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- State whether the following statements are true or false. Justify your answers.

(i) Every irrational number is a real number.

(ii) Every point on the number line is of the form √m where m is a natural number.

(iii) Every real number is an irrational number.

Answer:

(i) Every irrational number is a real number.

True

Irrational Numbers - A number is said to be irrational if it cannot be written in the p/q, where p and q are

integers and q ≠ 0.

i.e., Irrational numbers = 0, 19/30, 2, 9/-3, -12/7, √2, √5, , 0.102….

Real numbers - The collection of both rational and irrational numbers are known as real numbers.

i.e., Real numbers = √2, √5, , 0.102…

Therefore, Every irrational number is a real number, however, every real number is not irrational numbers.

(ii) Every point on the number line is of the form √m where m is a natural number.

False

The statement is false since as per the rule, a negative number cannot be expressed as square roots.

E.g., √9 =3 is a natural number.

But √2 = 1.414 is not a natural number.

Similarly, we know that there are negative numbers on the number line but when we take the root of a

negative number it becomes a complex number and not a natural number.

E.g., √-7 = 7i, where i = √-1

∴ The statement that every point on the number line is of the form √m, where m is a natural number is false.

(iii) Every real number is an irrational number.

False

The statement is false, the real numbers include both irrational and rational numbers. Therefore, every real

the number cannot be an irrational number.

Real numbers - The collection of both rational and irrational numbers are known as real numbers.

i.e., Real numbers = √2, √5, , 0.102…

Irrational Numbers - A number is said to be irrational if it cannot be written in the p/q, where p and q are

integers and q ≠ 0.

i.e., Irrational numbers = 0, 19/30, 2, 9/-3, -12/7, √2, √5, , 0.102….

∴ Every irrational number is a real number, however, every real number is not irrational.

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