Is 6/23 minus square root 2 divided by 2 rational or irrational ?
The Correct Answer and Explanation is:
To determine whether the expression ( \frac{6}{23} – \frac{\sqrt{2}}{2} ) is rational or irrational, we need to analyze each component of the expression individually and then evaluate how they combine.
Step 1: Analyze the terms
- ( \frac{6}{23} ):
This is a simple fraction, where both the numerator (6) and denominator (23) are integers, and 23 is not equal to zero. Therefore, ( \frac{6}{23} ) is a rational number. By definition, a number is rational if it can be expressed as a ratio of two integers. - ( \frac{\sqrt{2}}{2} ):
The square root of 2, ( \sqrt{2} ), is a well-known irrational number. An irrational number cannot be expressed as a ratio of two integers. Since dividing an irrational number by a non-zero integer (in this case, 2) does not make it rational, ( \frac{\sqrt{2}}{2} ) is also irrational.
Step 2: Combine the terms
The expression we are evaluating is:
[
\frac{6}{23} – \frac{\sqrt{2}}{2}
]
Here, we are subtracting a rational number (( \frac{6}{23} )) from an irrational number (( \frac{\sqrt{2}}{2} )). The sum or difference of a rational number and an irrational number is always irrational. This is because the rational number cannot “cancel out” the irrationality of the other term. More formally:
- Let ( r ) represent a rational number and ( i ) represent an irrational number. Then, ( r – i ) results in an irrational number, since if ( r – i ) were rational, it would imply that ( i = r – (r – i) ), which would mean that the irrational number ( i ) is equal to a rational number, a contradiction.
Step 3: Conclusion
Since ( \frac{6}{23} ) is rational and ( \frac{\sqrt{2}}{2} ) is irrational, their difference, ( \frac{6}{23} – \frac{\sqrt{2}}{2} ), must be irrational.
Thus, the correct answer is irrational.