Question

State whether the following statements are true or false. Justify your answers.
i. Every rational number is a real number.
ii. Every point on the number line is of the form \[\sqrt m \], where \[m\] is a natural number.
iii. Every real number is an irrational number.

Answer 1

Hint: Using the definition of number system, we can find whether the given statements are true or false.
We know that the real number system is the combination of rational and irrational numbers.
Again, we know that the roots of the natural numbers are always positive.

Complete step-by-step solution:
We have to find out whether the given statements are true or false.
i. Every rational number is a real number.
We know that real numbers are simply the combination of rational and irrational numbers, in the number system. So, every rational number is a real number.
Hence, the statement is true.
ii. Every point on the number line is of the form \[\sqrt m \], where \[m\] is a natural number.
The number line may have negative or positive numbers. Since, no negative can be the square root of a natural number, thus every point the number line cannot be in the form \[\sqrt m \], where, \[m\] is a natural number.
Hence, the statement is false.
iii. Every real number is an irrational number.
All numbers are real numbers and non-terminating numbers are irrational numbers. For example, 1, 2, 3…, 10 these are real numbers but they are rational numbers.
Hence, the statement is false.

Note: Real numbers are simply the combination of rational and irrational numbers, in the number system.
Rational numbers are represented in \[\dfrac{p}{q}\] form where \[q\] is not equal to zero. It is also a type of real number. Any fraction with non-zero denominators is a rational number.
Irrational numbers are the real numbers that cannot be represented as a simple fraction. It cannot be expressed in the form of a ratio, such as \[\dfrac{p}{q}\], where p and q are integers, \[q \ne 0\]