Let's carefully examine each of the given numbers to determine which one is irrational.
1. Number 0:
- The number 0 is considered a rational number because it can be expressed as the ratio of two integers, [tex]\( \frac{0}{1} \)[/tex].
2. Number [tex]\( -\frac{1}{2} \)[/tex]:
- The number [tex]\( -\frac{1}{2} \)[/tex] is rational because it is also a ratio of two integers, [tex]\( -1 \)[/tex] and [tex]\( 2 \)[/tex].
3. Number [tex]\( 0 . \overline{1257} \)[/tex]:
- This number represents a repeating decimal, specifically [tex]\( 0.125712571257... \)[/tex]. Any repeating or terminating decimal can be expressed as a fraction. Therefore, this is a rational number.
4. Number [tex]\( \sqrt{29} \)[/tex]:
- To determine if [tex]\( \sqrt{29} \)[/tex] is irrational, we note that 29 is not a perfect square (no integer squared equals 29). The square root of a non-perfect square is always an irrational number because it cannot be expressed as a ratio of two integers. The decimal representation of [tex]\( \sqrt{29} \)[/tex] is non-terminating and non-repeating.
Based on these observations:
- [tex]\( 0 \)[/tex] is rational.
- [tex]\( -\frac{1}{2} \)[/tex] is rational.
- [tex]\( 0 . \overline{1257} \)[/tex] is rational.
- [tex]\( \sqrt{29} \)[/tex] is irrational.
Therefore, the number that is irrational among the given options is [tex]\( \sqrt{29} \)[/tex].
Numerically, the value of [tex]\( \sqrt{29} \)[/tex] approximately equals [tex]\( 5.385164807134504 \)[/tex] which confirms the irrational nature of this number.
