What is the rationalising factor of a ?

We know, The expression that is multiplied with an irrational expression to obtain a rational number is called the rationalising factor. If the product of two surds is a rational number, then each surd is a rationalising factor to other. For example, if 2 is multiplied with 2 , it will be 2 , which is rational number, so 2 is rationalising factor of 2 . Given, a Therefore, rationalising factor of a is a . ⇒ a × a = a , which is rational. Hence, the required answer is a .

What is the rationalising factor of a ?

We know, The expression that is multiplied with an irrational expression to obtain a rational number is called the rationalising factor. If the product of two surds is a rational number, then each surd is a rationalising factor to other. For example, if 2 is multiplied with 2 , it will be 2 , which is rational number, so 2 is rationalising factor of 2 . Given, a Therefore, rationalising factor of a is a . ⇒ a × a = a , which is rational. Hence, the required answer is a .

Represent 9.3 on the number line.

To represent 9.3 on the number line, we first need to mark a line segment O B = 9.3 on number line. Now, take B C of 1 unit. Find the mid-point D of O C and draw a semi-circle on O C while taking D as its Centre. Draw a perpendicular to line O C passing through point B . Let it intersect the semi-circle at E . Taking B as Centre and B E as radius, draw an arc intersecting number line at F . B F is 9.3 . Now, B E = B F = 9 . 3 . Hence, 9 . 3 is located on the number line.

Recall, π is defined as the ratio of the circumference (say c ) of a circle to its diameter (say d ). That is, π = c d . This seems to contradict the fact that π is irrational. How will you resolve this contradiction?

There is no contradiction at all. When we measure a length with scale or any other instrument, we only obtain an approximate rational value like 22 7 which is rational. We never obtain, the exact value. For this reason, we may not realize, that either c or d is irrational. Therefore, the fraction c d is irrational. Hence, π is irrational.

Visualize 4 . 26 ¯ on the number line, up to 4 decimal places.

To visualize 4 . 26 - on the number line, we must follow these steps: (i) First, we see that 4.2 lies between 4 and 5 (ii) Now, divide this portion into 10 equal parts. (iii) Next, we locate 4.26 . We observe that this lies between 4.2 and 4.3 (iv) To get a more accurate visualization, we further divide this portion into 10 parts and locate it. (v) Further, we visualize 4.262 and observe that it lies between 4.26 and 4.27 (vi) To locate this, we divide the portion between 4.26 and 4.27 into 10 equal parts and locate 4.262 (vii) We observe that, 4.2626 lies between 4.262 and 4.263 . (viii) To find this, we divide the portion further into 10 parts and hence, locate 4.2626 .

Show how 5 can be represented on the number line.

To represent 5 on the number line, take O A = 2 units and make a perpendicular at A , so that A B = 1 unit , as shown in the figure. Then, ∆ O A B is a right-angled triangle So, by applying Pythagoras theorem, we get, O B 2 = O A 2 + A B 2 . ⇒ O B 2 = 2 2 + 1 2 ⇒ O B 2 = 5 ⇒ O B = 5 units . Now, taking O as Centre and O B as radius, draw an arc intersecting the number line at P . Hence, O P is the required distance that represents 5 .

Show that 5 + 7 is an irrational number.

Let us assume that 5 + 7 is a rational number. So, 5 + 7 = a b , where b ≠ 0 and a , b are integers ⇒ 7 = a b - 5 ⇒ 7 = a - 5 b b ∵ a , b are integers ∴ a – 5 b are also integers. So a - 5 b b is a rational number. But 7 is an irrational number. Since it is contradicting our assumption, therefore the assumption was wrong. Hence, 5 + 7 is an irrational number.

Represent 7 . 28 geometrically on the number line.

Draw a line segment A B = 7 . 28 units. Extend this line from B to C such that B C = 1 unit. Find the mid-point M of A C . With M as centre and M A as radius, draw a semicircle. Now, draw B D ⊥ A C , intersecting the semicircle at D . Then, B D = 7 . 28 units. With B as centre and B D as radius, draw an arc, meeting A C produced at E . Then, B E = B D = 7 . 28 units. Thus, B E represents 7 . 28 on the number line. Hence, the required diagram is shown below,