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Frequently Asked Questions

0.999... = 1


This is perhaps one of the most common objections met by students of mathematics.

First of all, let us consider one of the standard demonstrations.

Let x = 0.999...
10x = 9.999...
Algebraically, 10xx = 9x
Numerically, 9.999... − 0.999... = 9

So it follows that 9x = 9 ⇒ x = 1

The argument against it usually goes, "I hear what you're saying, but surely it's a bit less than one?"

Suppose that 0.999... is less than 1. Can there be any number greater than 0.999... but less than 1? Clearly not.

Given two numbers, x and y, ½(x + y) is midway between the two numbers and is called the arithmetic mean...

You guessed it. If 0.999... is the 'last' number before 1, what is ½(0.999...+1) equal to? This can only be resolved if 0.999... = 1.