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Avoiding Ambiguity

Reference > Mathematics > Number Theory > Bases

In the previous section we looked at base eight, and you might have been thinking "Why in the world can't we just go ahead and use the symbols 8 and 9 in base eight? Why do we have to stop at the symbol 7?"

Well, it's all about ambiguity. Can you imagine how confusing it would be if you could write the same integer in multiple ways? That happens in fractions - for example,
is the same as
, even though they look different. The only reason this doesn't get overly messy and confusing is that we require everyone to simplify their fractions, so no one will write
as a final answer; they'll all write
.  The same is true with square roots:
is the same as
, but because we're required to rationalize the denominator, no one will write 
as their solution, and that keeps things from getting too confusing.

Well, the same thing is true in bases. If we were in base eight, and we allowed you to use the symbol 8, you might come up with a number like this: 482eight. What is that number? It's 4 x 82 + 8 x 8 + 2 = 322. But wait a minute...what is this number: 502eight? It's 5 x 82 + 2 = 322!

Oh, that's ugly! If we let you use an 8 when you're using base eight, we can write the number 322 two different ways! And that's confusing! If we let you use an 8, then you would also have to simplify your answer, just like fractions and radicals have to be simplified. That's not something you want, right?

Okay, we're agreed - if you're working in base n (where n is some integer) than you don't want to have a symbol for the number (or higher values)!

And that also clarifies something we mentioned at the beginning of this unit - we always spell out the word eight if we're working in base eight. Why? Because if you're working in base eight, the symbol 8 doesn't even exist!


If you could use the symbol 7 in base seven, you could write the number 17seven. Can you find the other base seven number that has the same value?
146 is a number in base n (with n<10). What are the possible values of n, and why?
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