Q1(iv):

Classify the following numbers as rational or irrational: (iv) $\frac{1}{\sqrt{2}}$

Initial Setup & Theoretical Foundation

We are tasked with classifying the real number $\frac{1}{\sqrt{2}}$ as either rational or irrational. To do this rigorously, we must rely on the fundamental definitions and theorems governing the real number system.

• Rational Number: A number that can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$.
• Irrational Number: A real number that cannot be expressed as a simple fraction of two integers. Its decimal expansion is non-terminating and non-repeating.

Step 1: Analyzing the Components of the Expression

The given expression is a fraction where:

• The numerator is $1$, which is a non-zero rational number (since it can be written as $\frac{1}{1}$).
• The denominator is $\sqrt{2}$. [Per the fundamental theorem of arithmetic and the properties of square roots, the square root of any prime number is an irrational number. Since $2$ is a prime number, $\sqrt{2}$ is irrational.]

Step 2: Applying the Quotient Theorem of Real Numbers

We apply the established theorem regarding the arithmetic operations between rational and irrational numbers:

Theorem: The quotient of a non-zero rational number and an irrational number is always an irrational number.

Let $r = 1$ (a non-zero rational number) and $s = \sqrt{2}$ (an irrational number). The quotient $\frac{r}{s} = \frac{1}{\sqrt{2}}$ must, by definition, be irrational.

Step 3: Alternative Proof via Rationalization

To provide exhaustive proof, we can also manipulate the expression algebraically by rationalizing the denominator. This transforms the expression into a product, allowing us to apply the product theorem.

Multiply both the numerator and the denominator by $\sqrt{2}$:

$ \frac{1}{\sqrt{2}} = \frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2} $

This can be rewritten as a product:

$ \frac{\sqrt{2}}{2} = \frac{1}{2} \times \sqrt{2} $

Here, we have the product of $\frac{1}{2}$ (a non-zero rational number) and $\sqrt{2}$ (an irrational number). [Per the Product Theorem of Real Numbers: The product of a non-zero rational number and an irrational number is always irrational.] Therefore, the result is irrational.

Geometric Visualization of $\frac{1}{\sqrt{2}}$

To understand this number spatially, we can construct a right-angled isosceles triangle where the hypotenuse is exactly $1$ unit in length. By the Pythagorean theorem ($a^2 + b^2 = c^2$), the lengths of the two equal legs will be exactly $\frac{1}{\sqrt{2}}$. Because the hypotenuse is a rational integer ($1$), the legs represent an incommensurable (irrational) magnitude.

Final Conclusion

Whether analyzed through the quotient of a rational and irrational number, or by rationalizing the denominator to form a product, the mathematical logic strictly dictates that the resulting value cannot be expressed as a simple integer fraction.

Final Solution: The number $\frac{1}{\sqrt{2}}$ is an irrational number.