# Avoiding Ambiguity

Reference > Mathematics > Number Theory > BasesIn 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,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: 482_{eight}. What is that number? It's 4 x 8^{2} + 8 x 8 + 2 = 322. But wait a minute...what is this number: 502_{eight}? It's 5 x 8^{2} + 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 *n *(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!

## Questions

_{seven}. Can you find the other base seven number that has the same value?