0.9 repeating as a fraction

How it gets there

1. Call the number x, so x = 0.(9). It cannot go over a power of ten the way a terminating decimal can, because the digits never stop.
2. Multiply by 10, so that it and the original have the same endless tail: 10x = 9.9… against x = 0.9…
3. Subtract. The tails are identical, so everything after the point cancels exactly: 9x = 9, which is 9/9.
4. Divide both by 9: 1.

Is it really exactly 1?

Yes, and not as an approximation or a convention. The subtraction above is an equality at every step, so 0.(9) and 1 are two spellings of one number, the way 2/4 and 1/2 are.

The usual objection is that there must be some gap between them. Name it: whatever positive number you pick, enough 9s will fit inside it, so the gap is smaller than every positive number and therefore zero. Two numbers with nothing between them are the same number. The 9s do not approach 1 — the notation, all of it at once, is 1.

It is a fact about decimal notation rather than about arithmetic: base ten simply gives some numbers two names. In base three, a third is 0.1 exactly and this particular argument never comes up.

Questions people actually ask

What is half of 0.(9)?

1/2, or 0.5 as a decimal. Halving a fraction doubles the denominator rather than halving the top, unless the numerator is even and cancels. Doubled instead, 1 is 2 or 2.

What is 0.(9) as a fraction?

1, in lowest terms. 9/9 is the same value before reducing, and dividing both by 9 gets there.

Why can't 0.(9) be written over a power of ten?

Because the digits never stop, and a power of ten only clears a decimal that terminates. The algebra above multiplies by two different powers of ten so the endless tails line up, then subtracts them away — which is the standard proof that every repeating decimal is a rational number.

Is 0.(9) a rational number?

Yes. A number is rational exactly when it can be written as one whole number over another, and 1 is that. Every repeating decimal is rational and every rational number's decimal either terminates or repeats — the two statements are the same fact from opposite ends.

Is 1 in simplest form?

Yes. The numerator and denominator share no factor above 1, which is what simplest form means. 9/9 is the same number written before that reduction.

What is 1 as a decimal?

0.(9). This page runs the conversion in the direction people ask for it, and dividing 1 by 1 runs it back the other way.

The same number, other ways

Fraction, lowest terms

1

Before reducing

9/9

As a percentage

100%

Nearest sixteenth on a tape

1 (exact)

Work it out for any decimal

The two calculators this page is built from take any numbers you like: reduce a fraction shows the divisor it used, and fraction to decimal goes the other way, to the nearest 16th, 32nd and 64th.

Decimals either side

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APA

Rule Calculator. (2026). 0.9 repeating as a fraction. Rule Calculator. https://rulecalculators.com/fraction/0-9-repeating-as-a-fraction

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Rule Calculator. "0.9 repeating as a fraction." Rule Calculator. Last reviewed October 1, 2026. https://rulecalculators.com/fraction/0-9-repeating-as-a-fraction.

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