A Conversation for Number Systems

Uniqueness of representation

Post 1

Max C

Many people hearing about different number systems for the first time wonder whether there is more than one way to write a particular number in bases other than decimal. Is 14 really written only as 1110 (eight+four+two) or is there another way to make a total of 14?

The answer is that there is more than one way to write many numbers, even in decimal! Here's a proof in plain English:

One third in decimal is 0.333...
Three times one third is clearly one.
But 3 times 0.333... is 0.999...
So the two decimal representations 0.999... and 1.000... both represent precisely the same number: one. When you think about the "number" 0.999999..., the number you are considering is the number one, also written 1.0.

If you stick to representations without decimal points in them then there is only one way to write each number. But the proof of that is less interesting.

Incidentally, I think the main purpose of base64 in http form submissions is to ensure that characters that affect the syntax of the URL do not appear as a result of being entered on the form.

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Uniqueness of representation

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