To solve the problem [tex]\(\sqrt{2} = 0\)[/tex], let's understand and analyze it step-by-step.
1. Understanding the notation: The symbol [tex]\(\sqrt{2}\)[/tex] represents the principal (positive) square root of 2. The square root of a number [tex]\(x\)[/tex] is a value that, when multiplied by itself, gives the number [tex]\(x\)[/tex].
2. Mathematical properties: For any positive real number [tex]\(x\)[/tex], its square root [tex]\(\sqrt{x}\)[/tex] is also a positive real number. Specifically, the square root of 2 is a positive real number since 2 is positive.
3. Checking the value of [tex]\(\sqrt{2}\)[/tex]:
- The value of [tex]\(\sqrt{2}\)[/tex] is an irrational number, meaning it cannot be exactly represented as a fraction of two integers.
- [tex]\(\sqrt{2}\)[/tex] is approximately equal to 1.4142135623730951, a non-zero value.
4. Comparison: Compare [tex]\(\sqrt{2}\)[/tex] to 0. Since [tex]\(\sqrt{2} \approx 1.4142135623730951\)[/tex], which is clearly a positive value and not equal to 0, the statement [tex]\(\sqrt{2} = 0\)[/tex] is false.
5. Conclusion: Therefore, the correct solution is that [tex]\(\sqrt{2}\)[/tex] is approximately 1.4142135623730951 and certainly not equal to 0. As such, the assertion [tex]\(\sqrt{2} = 0\)[/tex] is false. The correct value of [tex]\(\sqrt{2}\)[/tex] is a positive number and specifically around 1.4142135623730951.
So, the step-by-step analysis concludes that:
- The statement [tex]\(\sqrt{2} = 0\)[/tex] is false.
- The actual approximate value of [tex]\(\sqrt{2}\)[/tex] is 1.4142135623730951.
