To determine the nature of the decimal expansion of the number [tex]\(\sqrt{2}\)[/tex], let’s analyze the properties of irrational numbers, specifically [tex]\(\sqrt{2}\)[/tex].
1. Finite Decimal Expansion:
- A finite decimal expansion is one that stops after a finite number of digits. For example, the number 1.5 or 3.14.
- [tex]\(\sqrt{2}\)[/tex] is known to be an irrational number, which means its decimal representation cannot be finite.
2. The Decimal 1.41421:
- The value 1.41421 is an approximation of [tex]\(\sqrt{2}\)[/tex]. It only represents the square root of 2 up to five decimal places, but it is not the exact and complete value.
- Therefore, this is not the full decimal expansion of [tex]\(\sqrt{2}\)[/tex].
3. Non-Terminating Recurring Decimal:
- A non-terminating recurring decimal is one where digits continue infinitely with a repeating pattern. For example, [tex]\(\frac{1}{3} = 0.\overline{3}\)[/tex] is a non-terminating recurring decimal.
- Since [tex]\(\sqrt{2}\)[/tex] is irrational, its decimal cannot be repeating with a fixed pattern. Thus, it cannot be non-terminating recurring.
4. Non-Terminating Non-Recurring Decimal:
- A non-terminating non-recurring decimal is one where the digits continue infinitely without repeating in a pattern. This is characteristic of irrational numbers such as [tex]\(\pi\)[/tex], [tex]\(e\)[/tex], and indeed [tex]\(\sqrt{2}\)[/tex].
- Observations and computations confirm that the decimal expansion of [tex]\(\sqrt{2}\)[/tex] is non-terminating and does not have a repeating pattern.
Given these considerations, the correct description of the decimal expansion of the number [tex]\(\sqrt{2}\)[/tex] is:
d. non-terminating non-recurring.
So, the correct answer is:
```
4. non-terminating non-recurring.
```
