Proof that 0.9 recurring equals 1

“0.9 recurring” is often written , and is equal to (where there is an infinite number of nines after the decimal point):

         

This is a commonly misunderstood concept of maths. People often think that:

• is “almost” 1, but not quite equal to it. Or,
• is the largest number that is smaller than 1.

BUT NEITHER OF THESE STATEMENTS IS TRUE! is exactly equal to 1; they are the same thing:

         

And here is the proof. Let x be equal to ,

         

Multiply x by 10,

         

Then take x away from this,

         
         

Then, by dividing both sides by 9, we find that x is equal to 1,

         

So is equal to 1.