MathSetuAsk a class whether 0.999999… (with the 9s going on for ever) is equal to 1, and most hands go up for "no". It looks a little less than 1. It feels a little less than 1. And yet it is exactly 1, not nearly, not almost. This is one of those small facts in Maths that make you stop and think, and that is why it is worth a few minutes today.
Way 1: the thirds
You already know that 1/3 = 0.333333… (the 3s never stop). Now multiply both sides by 3. On the left, 3 × 1/3 = 1. On the right, 3 × 0.333333… = 0.999999… So 1 = 0.999999… Nothing was added or lost on the way, so the two numbers are the same.
Way 2: the algebra trick
Let x = 0.999999… Multiply both sides by 10, so 10x = 9.999999… Now subtract the first line from the second:
- 10x − x = 9.999999… − 0.999999…
- 9x = 9
- x = 1
This is the same idea that Class 9 uses to turn a repeating decimal into a fraction, and you can use it for 0.666…, or for 0.272727…, in exactly the same way. Try them, and see which fractions you get.
Way 3: no gap is left
If two numbers are different, there is always a number between them (for example their average). So try to find a number between 0.999999… and 1. Suppose you pick a number smaller than 1, say 0.99999999, which is 0.00000001 short of 1. Then 0.999999… (with all its 9s) is already bigger than it, because it keeps adding 9s past that point. Whatever gap you choose, the 9s get past it. So nothing can sit between 0.999999… and 1, and then they are not two numbers. They are one number with two names, just as 1/2 and 0.5 are one number with two names.
Why it feels wrong
Our feeling says "0.9, then 0.99, then 0.999, always a tiny bit short of 1", and each of those numbers is indeed less than 1. But 0.999999… with the 9s going on for ever is not any of those numbers. It is what they get closer and closer to, and the "closer and closer" ends exactly at 1. Mathematicians call this idea a limit, and you will meet it again in Class 11 and 12, where the whole of calculus is built on it.
Try it yourself
- Write 0.444… as a fraction. (Hint: let x = 0.444… and multiply by 10.)
- Write 2.999… in its simplest form. What do you notice?
- Is 0.5 = 0.4999… ? Use any one of the three ways above to decide.
Maths is not about being fast. It is about being clear. When a result surprises you, do not argue with it and do not just memorise it. Ask why, and look for the idea that makes it obvious. That habit, more than any formula, is what turns a weak student into a strong one. If a topic in your own chapters feels like this, start from the notes for your class and take a short chapter test to see where you stand.