Is π + 24 rational or irrational
The Correct Answer and Explanation is:
Answer:
The number π + 24 is irrational.
Explanation
To determine whether π + 24 is rational or irrational, we must understand the definitions and properties of rational and irrational numbers.
A rational number is any number that can be expressed as a fraction of two integers, where the denominator is not zero. For example, numbers like 1/2, 3, -4, and 0.75 are all rational because they can be written as fractions (1/2, 3/1, -4/1, 3/4).
An irrational number, on the other hand, is a number that cannot be written as a fraction of two integers. Its decimal expansion is non-terminating and non-repeating. Examples include √2, e (Euler’s number), and π (pi). The number π represents the ratio of a circle’s circumference to its diameter, and it is a well-known irrational number.
Now, let’s analyze the expression π + 24.
- π is irrational.
- 24 is rational (since 24 = 24/1).
A fundamental rule in mathematics is:
The sum of a rational number and an irrational number is always irrational.
So, when we add π (irrational) and 24 (rational), the result is irrational. This is because if π + 24 were rational, we could subtract 24 from both sides to get π = (π + 24) – 24, which would imply π is rational. But that contradicts the fact that π is irrational. Therefore, π + 24 must be irrational.
To conclude:
π + 24 is irrational because adding a rational number to an irrational number does not “cancel out” the irrationality. The irrational component (π) dominates the sum, and no combination of whole numbers or fractions can exactly represent π + 24.
