What is the value of 22 7 ?

22 7 is a rational number, as its numerator and denominator are integers. The number can be expressed in mixed fraction as 3 1 7 In decimal form, 22 7 may be expressed as 3 . 142857142857 . . . or 3 . 142857 . The decimal digits under the bar do repeat without end. We can describe the decimal number as non-terminating, recurring decimal numbers. We use 22 7 as the value of π while calculating area, and perimeter of the circular regions, surface area, and volume involving curved surfaces. However, π ≈ 22 7 and let us note that π ≠ 22 7 , also π ≠ 3 . 14 . These two values are only approximations of the value of π . In normal calculations where high precision is not required, we use 3 . 14 as the value of π or 22 7 to for ease of calculation. The decimal digits of π is non-terminating and non-recurring. Therefore π cannot be expressed in rational form, and hence π is an irrational number.

What is the value of 22 7 ?

22 7 is a rational number, as its numerator and denominator are integers. The number can be expressed in mixed fraction as 3 1 7 In decimal form, 22 7 may be expressed as 3 . 142857142857 . . . or 3 . 142857 . The decimal digits under the bar do repeat without end. We can describe the decimal number as non-terminating, recurring decimal numbers. We use 22 7 as the value of π while calculating area, and perimeter of the circular regions, surface area, and volume involving curved surfaces. However, π ≈ 22 7 and let us note that π ≠ 22 7 , also π ≠ 3 . 14 . These two values are only approximations of the value of π . In normal calculations where high precision is not required, we use 3 . 14 as the value of π or 22 7 to for ease of calculation. The decimal digits of π is non-terminating and non-recurring. Therefore π cannot be expressed in rational form, and hence π is an irrational number.

Show that 4 √ 2 is an irrational number.

Let us assume that 4 √ 2 is a rational number. ⇒ 4 √ 2 = a b where, b ≠ 0 and a , b are integers ⇒ √ 2 = a 4 b Here, a , b are integers. So, 4 b is also integer. ⇒ a 4 b is rational number which can't be possible. ⇒ a 4 b = √ 2 , which is an irrational number. It is contradicting our assumptions. Hence, the assumption was wrong. So we can say that, 4 2 is an irrational number.

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 .

You know that 1 7 = 0 . 142857 ¯ . Can you predict what the decimal expansions of 2 7 , 3 7 , 4 7 , 5 7 , 6 7 are, without actually doing the long division? If so, how?

Yes, we can predict the expansions without actually doing the long division. Given, 1 7 = 0 . 142857 ¯ To get the decimal expansions of 2 7 , 3 7 , 4 7 , 5 7 , 6 7 , we need to multiply the numerator of each rational number by 0 . 142857 . Then, we get 2 7 = 2 × 1 7 = 2 × 0. 142857 ¯ = 0. 285714 ¯ . 3 7 = 3 × 1 7 = 3 × 0. 142857 ¯ = 0. 428571 ¯ . 4 7 = 4 × 1 7 = 4 × 0. 142857 ¯ = 0. 571428 ¯ . 5 7 = 5 × 1 7 = 5 × 0. 142857 ¯ = 0. 714285 ¯ . 6 7 = 6 × 1 7 = 6 × 0. 142857 ¯ = 0. 857142 ¯ .