Why Does 0.999... Equal 1?

The ten-times argument

This is the proof that appears in every textbook, and it takes five lines.

The step the whole thing turns on is the subtraction. After the decimal point, 9.999... and 0.999... have identical digits - nines all the way down, infinitely many of them - so when one is taken from the other, everything after the point cancels completely and 9 is what remains. With a decimal that stopped, the tails would not match and the cancellation would leave something behind.

It is worth being honest about what this argument assumes. It takes for granted that 0.999... names a definite number which can be multiplied and subtracted like any other. It does, and making that precise is the work of a first-year university course on limits. The school proof is not a trick; it is a correct argument resting on a foundation laid elsewhere.

• x = 0.999...
• 10x = 9.999...
• 10x - x = 9.999... - 0.999...
• 9x = 9
• x = 1

The argument from one third

Divide 1 by 3 by hand and the threes never stop: 1/3 = 0.333.... Anyone who has done that division knows the remainder of 1 comes back at every stage, and that is why it recurs.

Now multiply both sides by 3. On the left, 3 × 1/3 = 1. On the right, 3 × 0.333... = 0.999..., because tripling each digit gives a 9 and nothing carries. So 1 = 0.999..., and every step was one the student already accepted before the question was asked.

This is the argument that convinces the people who resist the first one, because nothing surprising happens in it. Anyone who wants to reject the conclusion has to reject 1/3 = 0.333..., and almost nobody does - it is the first recurring decimal anyone meets.

Why there must be a tiny bit left over fails

The objection is always the same, and it deserves a real answer rather than a repetition of the proof. It says that 0.999... falls short of 1 by 0.000...1: a point, then infinitely many zeros, then a 1.

That string does not name a number. Every decimal place is a finite number of steps to the right of the point - the tenths, the hundredths, the thousandths - and after infinitely many zeros is not one of those places. There is nowhere for the final 1 to sit.

Put it as arithmetic instead. Suppose 1 - 0.999... = d and that d is bigger than zero. Then d is smaller than 0.1, smaller than 0.01, smaller than 0.001, and smaller than one over ten to the power of anything you care to name. No positive number has that property, so d must be zero, so the two are equal.

There is a third way in, and it is the shortest. Between any two different numbers there is always another one - their midpoint, for a start. Name a number between 0.999... and 1. There is not one, and two numbers with nothing between them are not two numbers.

This is a feature of decimals, not a quirk of 0.999

Every decimal that terminates has a second name ending in recurring nines. 0.5 is also 0.4999..., 2 is also 1.999..., and 7.25 is also 7.24999.... Decimal notation is simply not a one-to-one naming system, and this is the price paid for a system in which addition and multiplication are easy.

The GCSE and IGCSE skill living next door to this uses the same ten-times move, and it is worth practising because it appears on Higher and Extended papers as a two or three mark question. To convert 0.4747... to a fraction, multiply by 100 rather than 10, because the repeating block is two digits long: 100x - x = 47, so 99x = 47 and x = 47/99.

Where the recurring part does not start immediately, shift twice. For 0.1666..., 10x = 1.666... and 100x = 16.666..., so 90x = 15 and x = 15/90, which simplifies to 1/6. The rule underneath is always the same: multiply by whatever power of ten makes the tails identical, then subtract.