Question

Which of the following is an irrational number: −1 5/29 , 0.343434..., √2/3 , or 87.9821?

Answer 1

To determine which of the given numbers is irrational, we need to examine each of them:

1. **−1 5/29**: This is a mixed number that can be converted to a simple fraction. The whole number part is -1, and the fractional part is \(\frac{5}{29}\). Combined, this gives:
\[
-1 5/29 = -\frac{29}{29} - \frac{5}{29} = -\frac{34}{29}
\]
Since -34 and 29 are both integers, this is a rational number.

2. **0.343434...**: This is a repeating decimal, which can be represented as a fraction. It can be expressed as:
\[
0.343434... = \frac{34}{99}
\]
Therefore, this is also a rational number.

3. **√2/3**: The square root of 2, \(\sqrt{2}\), is known to be an irrational number. When you divide an irrational number by a rational number (in this case, 3), the result is still irrational. Therefore, \(\frac{\sqrt{2}}{3}\) is irrational.

4. **87.9821**: This is a finite decimal, which can also be expressed as a fraction. Thus, it is a rational number.

Based on this analysis, the only irrational number among the options provided is:

**√2/3**.