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Archive for June 26th, 2006

Comprehending 1=0.99(9)

Posted by Tanas Gjorgoski on June 26, 2006

On first look it is weird when they tell you that 1 equals 0.999(9) where 9 is repeating.
Or, which is the same thing – that 3 * 0.333(3) , equals both 0.999(9) and 1.

I know I searched once on web for the explanation of why is it so – an explanation I could easily grasp/comprehend,  as intuitively it seems there is something wrong – our intuition tells us – “As much nines are repeating there, they won’t add-up to 1”. All I found is proofs that it is so, but from the philosophical point (well from common-sense point even) we can’t be happy (me at least) just with symbolic proofs, we need to comprehend the equality. So here goes the easy explanation…

When we have integer, let’s say four, we can say that we have a quantum of four things, a quantum which can be divided to four ones. Say you have four apples. You can divide them to four people – each person will get one apple.
Decimal numbers can be comprehended in similar way, just that if we add decimal fractions, we need to imagine that even the ones are further dividable – that they are made from parts. We need to imagine each one as made from ten parts, and each of those ten parts as made from even smaller ten parts, and so on…

In such way 1 would mean the item, 0.1 would mean one of the ten parts of the item (1), 0.01 would mean one of the ten part of the 0.1 part, and so on…

Now, let’s try to divide one item to three people. Because the item consist of ten parts, we will give three parts to each person, and we will be left with one of those ten parts that we need to divide:

1/3= 0.3 + 0.1/3

This 0.1 is again consisting of ten parts and we will likewise divide it between those three people – three parts to each, but we will be left again with one (now ten times smaller) part to divide further…

1/3= 0.3 + 0.03 + 0.01/3 , and dividing further…
1/3 = 0.3 + 0.03 + 0.003 + 0.001/3 , and so on..

Or if we sum the parts which are already divided we get

1/3 = 0.3 + 0.1/3
1/3 = 0.33 + 0.01/3
1/3 = 0.333 + 0.001/3 etc…

We see that this adding of cipher 3 to infinity, is not consequence of the part which is already divided, but of the part (left-over) which is left to be divided. As many threes are generated there will be still a part to be divided which would generate new threes. We can now say the following… the infinity in the never-ending repeating cipher 3, is not about the infinity of how much ciphers 3 are generated, but about the impossibility to finish the dividing process.

So, it is the wrong comprehension of what 0.333(3) means, that is causing the intuition against 1 being equal to 0.333(3) * 3. If we imagine 0.33(3) vaguely as some never ending string of cipher 3 – a decimal number in decimal notation that just has infinity of ciphers, our intuition tells you that however far you multiply with 3, you will only get 0.999(9), and however far you go, it won’t be 1.
But comprehending 0.33(3) in that way, is ignoring the reason of that infinity, for example in the given process – ignoring the left over which is still to be divided that causes us to add new and new ciphers 3, and which will never end. So, we need to see repeating cipher as a new notation added to the decimal system, which tell us about that impossibility to finish the description in the normal decimal notation, and not a shortcut for normal decimal notation, in which we could write the cipher 3 infinite number of times.

By the way we can divide 1 in similar way to get 0.99(9) directly…
We divide 10 of the parts (0.1) to 10 people, so that we divide 9 or those parts to 10 people, and leave one more to divide; and that we repeat that:
1/10=0.9/10+0.1/10=0.9/10 + 0.09/10 + 0.01/10, etc…
We will get 1 = 0.99(9), again this repeating 9 signifying the impossibility to finish the process.

* It isn’t really necessary for each part consist of 10 parts. What we need is just that each part can be divided to 10 parts, and if it can, we can treat it as consisting of 10 parts. For example if we don’t have quantum, but a continuous magnitude, it can be divided in any number of parts we want, hence to 10 also.

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