Is 0.999…= 1?

Many a times we have made 0.999….= 1. But we always thought it’s an approximation, they are not equal though.

It might be surprising for many of us to know that 0.999….. is actually EQUAL to the integer 1. It can be proved like this,

If x = 0.999…, then 10*x = 9.999… so by subtracting the first equation from the second, we get

9*x = 9.000…

Therefore, x=1.

Here’s another proof – The number 0.1111… = 1/9, so if we multiply both sides by 9, we obtain 0.9999…=1.

So by similar arguments, every rational number with a terminating decimal expansion has another expansion that ends in a never-ending string of 9’s. So, for instance, the rational 9/20 can be represented as 0.45 (the same as 0.35000…) or 0.44999…

General mathematical proof

Any rational number can be expressed in such a way that the digit in each place of a decimal expansion is associated with a positive or negative power of 10. The k-th place to the left of the decimal corresponds to the power 10^k. The k-th place to the right of the decimal corresponds to the power 10^(-k) or 1/10^k.

If the digits in each place are multiplied by their corresponding power of 10 and then added together, one obtains the real number that is represented by this decimal expansion.

So the decimal expansion 0.9999… actually represents the infinite sum

9/10 + 9/100 + 9/1000 + 9/10000 + …

Using the formula for finding the sum of infinite G.P. series i.e. {a/(1-r)}, we get,

(9/10)/(1 – 1/10) = 1

Hence 0.999… equals 1.

There can be many other proofs. Math enthusiasts are welcome to suggest more proofs. Alternatively if you can prove that they not equal, please post. You might find this post interesting, where I’ve proved, 2 + 2 =5. Obviously there is a fallacy, which you need to figure out.

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