Show how square root of 5 can be represented on the number line.

Show how $\sqrt{5}$ can be represented on the number line.

To represent $\sqrt{5}$ on the number line, you can use a geometrical method based on the Pythagorean Theorem. Here's how it can be done:

1. Draw a number line and mark a point O on it, representing the origin (0).

2. From point O, draw a horizontal line segment OA of unit length (1 unit) to the right.

3. From point A, draw a perpendicular line segment AB upwards of 2 units in length.

4. Connect point O to point B with a diagonal line segment. This line segment OB will represent $\sqrt{5}$, because according to the Pythagorean theorem, in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Thus, for the triangle $OAB$, we have $OB^2 = OA^2 + AB^2$. Since $OA = 1$ and $AB = 2$, then $OB^2 = 1^2 + 2^2 = 5$, which gives $OB = \sqrt{5}$.

5. Extend the number line if necessary and measure the length of the segment OB using a compass. Then, mark this length from O on the number line to the right. The point where the arc intersects the number line is $\sqrt{5}$.

This geometrical construction accurately represents $\sqrt{5}$ on the number line through the use of the Pythagorean theorem.