This is the sixth in a series of posts on the decadic numbers (previous posts: A curiosity, An invitation to a funny number system, What does “close to” mean?, The decadic metric, Infinite decadic numbers).
Last time I left you with some parting exercises; here are the answers, along with some additional commentary.
- What number is represented by ? How do you know?
It turns out that , and there are several ways to see this. The simplest is to note that adding it to itself nine times yields , which we already know is .
At this point we may also ask, if is , then what is ? It must be , since adding that to yields zero:
Hmm, so is , and is (since it is 8 times ). This also makes sense because :
It seems to make sense but it sure is weird! is negative but is positive. Can you figure out a relatively simple way to tell, just by looking at an infinite decadic number, whether it is positive or negative? (I don’t actually know the answer.)
- How about ?
If is then looks like since and .
Indeed, and here’s another way to see the same thing: if , then , so is
Hence , so . As a further check, , so
which ought to be : sure enough, adding to yields zero.
In general, we can see that the infinite repeating decadic number will be equal to where the number of ‘s in the denominator is the same as the number of digits in the numerator.
- Can you find an infinite decadic number which represents ? How about ? Or ?
is , and is ; by this point I hope you can see why.
However, is different: we would have to find some number which yields when multiplied by two. Obviously no such number exists: two times anything cannot possibly end in a . The same is true of any fraction with a denominator which is not relatively prime to (i.e. shares a common divisor with) .
However, we can extend our notion of decadic numbers to accommodate such fractions, by allowing digits after a decimal point. So for example will be , as usual, and then will be . It’s important to note that we can only allow finite strings of digits after the decimal point: a number like is meaningless because the sequence of numbers does not converge to anything; in fact, they are getting further and further apart! But any finite number of digits after the decimal point are OK.
In general, we call decadic numbers with no digits after the decimal point (including infinite ones) decadic integers. Obviously all the normal integers are also decadic integers; but so are fractions such as whose denominators are relatively prime to . Other fractions such as are not.
- Show that any fraction whose denominator has only and as factors is represented by a finite decimal number.
- Show that by allowing a finite number of digits after the decimal point, we can represent any fraction as a decadic number. (Hint: factor the denominator into one part consisting only of twos and fives and another part with everything else.)
You may recall that in a previous post I promised to show you a strange decadic number , which is not zero, but is equal to its own square. We’ve now finally seen enough for me to tell you what it is—which I will do in my next post. In the meantime you may want to try discovering it!
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Wait, for Question 1 from the last post, …11111 + …99999 = …00000, right? So wouldn’t that mean that since …99999 = -1, …11111 = 1?
Your explanation of why …11111 = -1/9 makes sense though…
No, …11111 + …99999 = …11110. After adding the 1 and 9 in the 1’s place you then have to carry a 1 to the next column, and so on.