The number 0.9 (which means 0.99999 ...) is a rational number. Using algebra let x = 0.9, multiply by 10 Class 9

The number 0.9 (which means 0.99999 ...) is a rational number. Using algebra let x = 0.9, multiply by 10 Class 9

Question 1.

The number $0 . \overline{9}$ (which means 0.99999 ...) is a rational number. Using algebra (let x = $0 . \overline{9}$, multiply by 10, and subtract), explain why $0 . \overline{9}$ is exactly equal to 1. Class 9

Solution:

Let x = $0 . \overline{9}$ ...(i)

Multiplying both sides by 10, we get

10x = $9 . \overline{9}$ ...(ii)

Subtracting (i) from (ii), we get

10x - x = $9 . \overline{9}$ - $0 . \overline{9}$

∴ 9x = 9

∴ x = 1

$0 . \overline{9}$ = 1

Question 2.

We have seen that the repeating block of $\frac{1}{7}$ is a cyclic number. Try to find more numbers (n) whose reciprocals $\left(\frac{1}{n}\right)$ produce decimals with repeating blocks that are cyclic. Class 9

Solution:

Beyond n = 7, some other numbers whose reciprocals show cyclic repeating blocks are:

1. 17: $\frac{1}{17}$ has a repeating block of 16 digits: $0 . \overline{0588235294117647}$.
2. 19: $\frac{1}{19}$ has a repeating block of 18 digits: $0 . \overline{052631578947368421}$.
3. 23: $\frac{1}{23}$ has a repeating block of 22 digits.
4. 29: $\frac{1}{29}$ has a repeating block of 28 digits.
5. 47: $\frac{1}{47}$ has a repeating block of 46 digits.