Friday, August 12, 2011

Is 0.999... really equal to 1.000...?

What does (1/3) mean? It means you break the number 1 into 3 equal parts.

When you sum up these 3 parts, or perform the multiplication operation (1/3) x 3, you get back unity or 1. That makes perfect sense. But under decimal theory, 1/3=0.333...,  where the “3..” represents an endless recurrence of 3s. And then 0.333....x3 = 0.99999...

This equality implies that 0.999....=1.000...

Yes, I realize that Mathematics speaks for itself, but how can we understand this strange equality and how do we know that this equality is true?

In order to answer these questions, we firstly need to understand the concept of real numbers. So the question I ask you is this: how can you quantitatively define 2 distinct real numbers? Think about this for a second before moving on.

Here’s the crunch!

Any two distinct real numbers A and B will always have an infinite number of numbers lying in between them. Under mathematical formalism, one would say that the set of real numbers between A and B has an infinite cardinality. Cardinality refers to the number of elements in a set. But anyways, we don’t have to delve into strict mathematical terminology to understand the ideas here.   
Now I’m going to prove this idea to you. Let’s just take any two arbitrary real numbers for analysis. Let’s take the numbers 1.000... and 1.0100... So how many numbers are there in between these two numbers? Well, there is 1.0090.., 1.00990.., 1.009990..., 1.0099990... and so on. This already shows you that there can be an infinite number of numbers in between the two numbers we are studying. Of course, there could be even more combinations and permutations of numbers, leading to even more possible numbers (but still infinite). There could be 1.0080..., 1.00912330.., 1.0091241240.... and many many more. Don’t bother listing them, you’ll go on forever... literally.

Try this out for other distinct real numbers. You’ll realize the same thing. There are always an infinite number of numbers in between 2 distinct real numbers. The reason for this is the ability of real numbers to expand, or to be more precise, contract in decimal places forever and ever. 

So now I ask you, how many numbers separate A and B, when A and B are exactly the same? Quite intuitively, the answer is none. If they are the same, no numbers should separate them.

Now, let’s look at our strange equality 0.999...= 1.0000...

So I ask you now, how many numbers separate 0.999... and 1.000...?

To answer this question, let’s just try to generate a possible number that could separate them. Let’s try the number 0.999990... Does it separate them? Nope. 0.9999999... is greater than 0.999990... and hence it can’t separate the two numbers.

Let’s try something smaller. Perhaps 0.990...?  Nope, 0.999... exceeds 0.990... and hence 0.990... cannot be in between the numbers.

This same pattern of analysis can be repeated and one will realize that for any real number you try to define in between 0.999... and 1.00..., 0.999.... will always be greater than it or 1.000... will always be smaller than it.

We can only conclude that there must be no real number separating 0.999... and 1.000... And according to what we deduced from the earlier part, since no real numbers separate 0.999... and 1.000..., they must be equal.


One final question you may ask is this: even after this proof, why does it still intuitively seem that 0.999... cannot be equal to 1.000...? Well this is because when we were taught basic arithmetic in school, we were not exactly taught about real numbers and their analysis, but rather simpler forms of numbers like whole numbers. 

Thus, when we see "0.99..", we automatically tend to reject the idea that it could be equal to "1.00.." , just like we would reject the idea of a whole number like 99 being equal to 100. The key idea here is to unlearn our previous  ideas of numbers as whole numbers or simple numbers and to replace our understanding with a more complex one, such as that of the real numbers. Only after that will this strange equality start to make more sense intuitively. 

If you liked this mini-proof that I gave and want to learn more, do check out the following links:


  1. A great read, though I needed more explanation to understand it fully.

  2. Well written piece. Can I share this article with your juniors from Mayflower?

  3. Hey Mr Chng! Thanks so much sir! Sure, no problem. Are you still teaching the sec 4s?

  4. Yup. Alternating between sec 3 and sec 4 it seems... (:

  5. Intriguing... you have a very interesting blog!