The number (which means ) is a rational number. Using algebra (let , multiply by 10, and subtract), explain why is exactly equal to 1.

The World of Numbers | Exercise 3.5

Question 4

The number $0.\overline{9}$ (which means $0.99999\dots$ ) 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.

Solution

We will use algebra to show that $0.\overline{9}$ is equal to 1.

Step 1 — Assign a variable

Let's assign a variable to the given number.

We let x be the number $0.\overline{9}$ .

This is our first equation.

$x = 0.99999\dots$

$\boxed{x = 0.99999\dots \quad \text{(Equation 1)}}$

Step 2 — Multiply by 10

Now, we multiply both sides of Equation 1 by 10.

This shifts the decimal point one place to the right.

$10 \times x = 10 \times 0.99999\dots$

$10x = 9.99999\dots$

$\boxed{10x = 9.99999\dots \quad \text{(Equation 2)}}$

Step 3 — Subtract the equations

Next, we subtract Equation 1 from Equation 2.

This step eliminates the repeating decimal part.

$10x - x = 9.99999\dots - 0.99999\dots$

$9x = 9$

We then solve for x.

$x = \frac{9}{9}$

$x = 1$

$\boxed{x = 1}$

Answer

(i) We defined x as $0.\overline{9}$ . We found that x is 1. Therefore, $0.\overline{9}$ is exactly equal to 1.

Question 4

Answer

More questions in Exercise 3.5