The number 0.9 (which means 0.99999 … ) is a rational number. Using algebra (let x = 0.9, multiply by 10, and subtract), explain why 0.9 is exactly equal to 1.

Step-by-step solution

Idea: The infinite tail of 9s is identical in x and in 10x, so it cancels exactly when we subtract, leaving a whole number.

1. Let x = 0.9 = 0.999…½ mark
2. Multiply by 10: 10x = 9.9 = 9.999…½ mark
3. Subtract: 10x − x = 9.999… − 0.999… = 9, so 9x = 9.½ mark
4. x = 1. So 0.9 = 1 exactly. (Another way: 13 = 0.333…, and 3 × 13 = 1 while 3 × 0.333… = 0.999….)½ mark

Check: 19 = 0.111…, and 9 × 19 = 1 while 9 × 0.111… = 0.999… ✓.

Answer to write in the exam

Common mistakes that cost marks

• Thinking 0.999… is ‘just a tiny bit less than 1’. There is no gap: the difference 1 − 0.999… is smaller than every positive number, so it is 0.
• Writing 10x − x = 9.9, forgetting that the tails of 9s cancel completely.
• Treating 0.999… as 0.9 or 0.99, which are terminating and smaller than 1.

How this can come in the exam

Try one yourself

Write 3.9 as a whole number using algebra.

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