Question

Find and plot √2.

1. Approximate between two whole numbers.

     12 = 1, 22 = 4

2. Approximate further by squaring the number halfway between 1 and 2.

     1.52 = 2.25

3. Plot the square root on the number line.

The square root lies between
and 1.5 on the number line.

Answer 1

To find and plot \( \sqrt{2} \), we'll go through the steps you've outlined:

### Step 1: Approximate between two whole numbers.
We need to find two whole numbers whose squares are close to 2:

- \( 1^2 = 1 \)
- \( 2^2 = 4 \)

Thus, \( \sqrt{2} \) lies between 1 and 2.

### Step 2: Approximate further by squaring the number halfway between 1 and 2.
The number halfway between 1 and 2 is 1.5. Now we square 1.5:

\[
1.5^2 = 2.25
\]

Since \( 2.25 \) is greater than 2, we conclude \( \sqrt{2} \) is less than 1.5.

### Further Approximation
Next, let's test a value between 1 and 1.5. Let’s use 1.4:

\[
1.4^2 = 1.96
\]

Since \( 1.96 < 2 \), we know that \( \sqrt{2} \) is greater than 1.4.

Now, let's also test 1.41:

\[
1.41^2 \approx 1.9881
\]

Since \( 1.9881 < 2 \), we know \( \sqrt{2} \) is greater than 1.41.

We can now test 1.42:

\[
1.42^2 \approx 2.0164
\]

Since \( 2.0164 > 2 \), we know \( \sqrt{2} \) is less than 1.42.

### Conclusion
Now we have:
- \( 1.41 < \sqrt{2} < 1.42 \)

### Step 3: Plot the square root on the number line.
Based on our approximations:
- \( \sqrt{2} \) lies between 1.41 and 1.42 on the number line.

Here's how you can visualize it on a number line:

```
1.0 1.4 1.5 2.0
|------------------------|------------------------|-----------------|
[-------]
1.41 sqrt(2) 1.42
```

In summary, \( \sqrt{2} \) is approximately \( 1.41 < \sqrt{2} < 1.42 \) and can be plotted accordingly on the number line.