Question

Represent `sqrt(6)` on the number line.

Answer 1

Given: Represent `sqrt(6)` on the number line.

Step wise calculation:

1. We know that `sqrt(6)` is between two perfect squares `sqrt(4) = 2` and `sqrt(9) = 3`.

So, `sqrt(6)` lies between 2 and 3 on the number line.

2. To locate `sqrt(6)` more accurately, find a number whose square is close to 6 and lies between 4 and 9.

3. For example, consider 2.4 since 2.4² = 5.76 which is less than 6 and 2.5 since 2.5² = 6.25 which is more than 6.

4. So, `sqrt(6)` is between 2.4 and 2.5 on the number line.

5. To precisely represent `sqrt(6)`, use the Pythagorean theorem or geometric constructions:

Draw a line segment of length `sqrt(2)`.

From one endpoint, draw a perpendicular segment of length `sqrt(4) = 2`.

The hypotenuse of this right triangle will be

`sqrt(2^2 + (sqrt(2))^2)`

= `sqrt(4 + 2)`

= `sqrt(6)`

6. Place this length on the number line by transferring this segment from zero.

`sqrt(6)` lies between 2.4 and 2.5 on the number line. It can be represented using geometric construction of right triangle with legs 2 and `sqrt(2)` and the hypotenuse of this right triangle gives `sqrt(6)` on the number line.