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0.33 recurring is the decimal equivalent of one third

Multiply both expressions by 3 and you have 0.99 recurring = 3/3 = 1

* * * * *

That is rather a circular argument, because it begs the question why 0.33 recurring equal to a third.

Instead, suppose f = 0.99....

multiply both sides by 10

10f = 9.99...

and since both numbers are recurring there is a nine in the same position after the decimal point in both equations.

Subtracting the first from the second gives: 9f = 9

Divide both sides by 9 and you have f = 1

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Q: How does 0.99 recurring equal 1?
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