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

Understand the Question To express a repeating decimal as a fraction or integer, we define it as a variable Since only one digit repeats, multiplying both sides by shifts the decimal point by one place. Subtracting the original equation from this new equation eliminates the infinite repeating decimal portion completely, allowing us to solve directly for . Step 1 · Define the Variable and Multiply by 10 Let Multiplying both sides by Step 2 · Subtract the Equations to Solve for Subtracting equation from equation Since and , it follows that . Answer > Common Mistakes Intuition vs. Mathematical Equivalence: Assuming is just "infinitely close to " rather than strictly equal. In the real number system, and represent the exact same quantity. Decimal Place Error: Incorrectly thinking that shifting by leaves fewer s behind the decimal point; because the sequence is infinite, the non-terminating tails are identical and cancel out to zero.

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

Understand the Question

To express a repeating decimal as a fraction or integer, we define it as a variable $x = 0.\overline{9} = 0.99999\dots$
Since only one digit repeats, multiplying both sides by $10$ shifts the decimal point by one place.
Subtracting the original equation from this new equation eliminates the infinite repeating decimal portion completely, allowing us to solve directly for $x$ .

Step 1 · Define the Variable and Multiply by 10

Let $x = 0.99999\dots \quad \dots (1)$

Multiplying both sides by $10$

\begin{aligned} 10 \times x &= 10 \times 0.99999\dots \\[0.6em] 10x &= 9.99999\dots \quad \dots (2) \end{aligned}

Step 2 · Subtract the Equations to Solve for $x$

Subtracting equation $(1)$ from equation $(2)$

\begin{aligned} 10x - x &= 9.99999\dots - 0.99999\dots \\[0.6em] 9x &= 9 \\[0.6em] x &= \dfrac{9}{9} \\[0.6em] x &= 1 \end{aligned}

Since $x = 0.\overline{9}$ and $x = 1$ , it follows that $0.\overline{9} = 1$ .

Answer

$0.\overline{9} = 1$

Common Mistakes

Intuition vs. Mathematical Equivalence: Assuming $0.\overline{9}$ is just "infinitely close to $1$ " rather than strictly equal. In the real number system, $0.\overline{9}$ and $1$ represent the exact same quantity.
Decimal Place Error: Incorrectly thinking that shifting by $10$ leaves fewer $9$ s behind the decimal point; because the sequence is infinite, the non-terminating tails are identical and cancel out to zero.

Question 4

More questions in Exercise 3.5