Question

How do you prove that the square root of 14 is irrational?

Answer 1

Hint: Here the number is given we have to apply the square root to the number. When we apply the square root to the number sometimes, we obtain the different kinds of numbers. Here we have to determine what kind of number we will obtain when we apply the square root to the number.

Complete step-by-step solution:
In mathematics we have different form of numbers.
Natural numbers - Contain all counting numbers which start from 1.
Example: All numbers such as 1, 2, 3, 4, 5, 6, …
Whole Numbers - Collection of zero and natural number.
Example: All numbers including 0 such as 0, 1, 2, 3, 4, 5, 6, …
Integers- The collective result of whole numbers and negative of all-natural numbers.
Example: \[ - \infty , \cdot \cdot \cdot 0,1,2,3, \cdot \cdot \cdot + \infty \]
Rational Numbers- Numbers that can be written in the form of \[\dfrac{p}{q}\] where \[q \ne 0\]
Example: 3, -7, -100, \[\dfrac{1}{2}\], \[\dfrac{5}{3}\], 0.16, 0.4666 etc
Irrational Numbers- All the numbers which are not rational and cannot be written in the form of \[\dfrac{p}{q}\]
Example: \[\sqrt 2 \], \[\pi \], \[\sqrt 3 \], \[2\sqrt 2 \] and \[ - \sqrt {45} \] etc
Real numbers: Real numbers can be defined as the union of both the rational and irrational numbers. They can be both positive or negative and are denoted by the symbol “R”.
Now consider the given number 14
When we apply the square root to the number 14, we get
\[\sqrt {14} = 3.74165738677\]
We obtain the decimal number. The decimal expansions that neither terminate nor become periodic. This cannot be expressed in the form of fraction.
Therefore, we can’t write in the form \[\dfrac{p}{q}\]. Therefore, the number 14 is an irrational number.

Note: As per definition, the rational numbers include all the integers, fractions and repeating
decimals, so while deciding keep the definitions in mind. Also, remember that the decimal expansion
for rational numbers that execute finite or recurring decimals.