To compare $-\sqrt{7}$ and $-3.12345\ldots$, we will calculate the approximate value of $\sqrt{7}$ and then compare the two numbers.

1. $\sqrt{7} \approx 2.64575$, so: $-\sqrt{7} \approx -2.64575$

2. The second number is $-3.12345\ldots$, which is already given.

Now let's compare:

• $-\sqrt{7} \approx -2.64575$
• $-3.12345\ldots$

Since $-3.12345\ldots$ is smaller (more negative) than $-2.64575$, we can conclude that: $-\sqrt{7} > -3.12345\ldots$

Justification:

The comparison is based on their decimal approximations. Since $-\sqrt{7}$ is closer to 0 than $-3.12345\ldots$, it is the larger of the two numbers.

Would you like further details on how the square root was calculated or any other questions?

5 Related Questions:

1. How do you approximate square roots without a calculator?
2. What is the exact value of $\sqrt{7}$?
3. Why is $-2.64575$ larger than $-3.12345$?
4. How can you compare square roots of non-perfect squares?
5. What other methods can be used to compare irrational and decimal numbers?

Tip:

When comparing negative numbers, the one closer to zero is always the larger number.