Question

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.

Accepted Answer

Let x = 0.99999... This means the number has an infinitely repeating block of 9s. Next, we multiply by 10 to shift the decimal point one place to the right. This gives 10x = 9.99999... Now, we subtract the first equation from the second: 10x - x = 9.99999... - 0.99999... 9x = 9 Dividing both sides by 9, we get x = 1. Therefore, 0.99999... is exactly equal to 1. This demonstrates the non-uniqueness of decimal representations, where a terminating decimal like 1.000... has an alternative form with repeating 9s. — The question asks to prove 0.999... = 1 algebraically. By setting x = 0.999..., multiplying by 10 gives 10x = 9.999.... Subtracting the original equation yields 9x = 9, so x = 1. The context also mentions that rational numbers can have two decimal forms, showing non-uniqueness where any terminating decimal has an alternative with repeating 9s.

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