√2, √3, √4, √5, √6, ... - এই সংখ্যাগুলো কি মূলদ সংখ্যা?

To represent $\sqrt{n}$ on a number line, we use the Pythagorean theorem ($H^2 = P^2 + B^2$). We construct a right-angled triangle such that its hypotenuse ($H$) has a length of $\sqrt{n}$. This is done by choosing appropriate lengths for the perpendicular ($P$) and base ($B$).

• Draw a number line and mark the origin as point O (representing 0).
• From O, mark point A at 1 unit distance to the right (representing 1). So, $OA = 1$.
• At point A, draw a line segment AB perpendicular to the number line, with length 1 unit. So, $AB = 1$.
• Connect O and B. Triangle OAB is a right-angled triangle.
• Using the Pythagorean theorem: $OB^2 = OA^2 + AB^2 = 1^2 + 1^2 = 1 + 1 = 2$. Therefore, $OB = \sqrt{2}$.
• With O as the center and OB as the radius, draw an arc that intersects the number line at point P.
• The point P represents $\sqrt{2}$ on the number line.

• First, construct $\sqrt{2}$ as described in Step 2. Let P be the point representing $\sqrt{2}$ on the number line.
• At point P, draw a line segment PQ perpendicular to the number line, with length 1 unit. So, $PQ = 1$.
• Connect O and Q. Triangle OPQ is a right-angled triangle.
• Using the Pythagorean theorem: $OQ^2 = OP^2 + PQ^2 = (\sqrt{2})^2 + 1^2 = 2 + 1 = 3$. Therefore, $OQ = \sqrt{3}$.
• With O as the center and OQ as the radius, draw an arc that intersects the number line at point R.
• The point R represents $\sqrt{3}$ on the number line.

• Since $\sqrt{4} = 2$, this is a rational number.
• Simply mark the point 2 on the number line.

• Draw a number line and mark the origin as point O (representing 0).
• From O, mark point C at 2 units distance to the right (representing 2). So, $OC = 2$.
• At point C, draw a line segment CD perpendicular to the number line, with length 1 unit. So, $CD = 1$.
• Connect O and D. Triangle OCD is a right-angled triangle.
• Using the Pythagorean theorem: $OD^2 = OC^2 + CD^2 = 2^2 + 1^2 = 4 + 1 = 5$. Therefore, $OD = \sqrt{5}$.
• With O as the center and OD as the radius, draw an arc that intersects the number line at point S.
• The point S represents $\sqrt{5}$ on the number line.

• First, construct $\sqrt{5}$ as described in Step 5. Let S be the point representing $\sqrt{5}$ on the number line.
• At point S, draw a line segment ST perpendicular to the number line, with length 1 unit. So, $ST = 1$.
• Connect O and T. Triangle OST is a right-angled triangle.
• Using the Pythagorean theorem: $OT^2 = OS^2 + ST^2 = (\sqrt{5})^2 + 1^2 = 5 + 1 = 6$. Therefore, $OT = \sqrt{6}$.
• With O as the center and OT as the radius, draw an arc that intersects the number line at point U.
• The point U represents $\sqrt{6}$ on the number line.

For question 11, the representation of $\sqrt{2}$ is covered in Step 2.

The final answer is the description of the construction steps for each number.