Using a number line, what whole number and rational number to the nearest half is sqrt 10 between?

Answer 1

To determine where $\sqrt{10}$ lies on a number line and find the whole number and rational number to the nearest half, we can start by calculating the approximate value of $\sqrt{10}$.

Since $3^2=9$ and $4^2=16$, we know that:

$$3<\sqrt{10}<4$$

Now, to find a more precise value, we can do some additional checks:

• ${3.1}^2=9.61$ which is less than 10
• ${3.2}^2=10.24$ which is greater than 10

Thus, we can narrow it down further:

$$3.1<\sqrt{10}<3.2$$

To estimate further, we can check:

• ${3.15}^2=9.9225$ which is less than 10
• ${3.16}^2=10.0576$ which is greater than 10

This gives us:

$$3.15<\sqrt{10}<3.16$$

To round $\sqrt{10}$ to the nearest half, we can look at the two nearest halves around $3.15$ and $3.16$:

• The nearest halves are $3.0$ and $3.5$.
• Since $3.15$ is closer to $3.0$ (specifically, $3.15-3.0=0.15$ and $3.5-3.15=0.35$), we round $\sqrt{10}$ down to $3.0$.

Thus, to the nearest half, we conclude that:

1. The whole number $\sqrt{10}$ is between is $3$.
2. The rational number to the nearest half is $3.0$.

Therefore, $\sqrt{10}$ is between $3$ and $3.0$ on the number line.