## Ask Professor Puzzler

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Louis from Uganda asks, "What is the place value of 3 in 131 base three?"

Well, Louis, funny thing is, I can't answer that question. Because 131 is not a base three number. You see, if you are working in base n, there is no symbol for the number n. In base ten, for example, we have symbols for 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, but there is no symbol for the number 10. We require *two *symbols to represent that number (the symbol 1 and the symbol 0).

In base three, there is no symbol for the number three. "3" as a symbol does not exist. So your number doesn't actually exist.

It would be like asking me, "What is the place value of ¥ in 75¥23 base ten?" The answer is, that's not a base ten number, because there is no symbol "¥" in the base ten system.

In base three, you have three symbols to work with. They are 0, 1, and 2. Thus, you could have the number 101, 111, or 121, but 131 doesn't exist.

So let's suppose you meant to write "What is the place value of 2 in 121 base three?" In that case, I can answer you fairly easy; that is the "threes place". Whatever number is multiplied by three to find its actual value.

I could give you a more detailed answer, but we actually already have a fairly detailed unit on bases here: Number Theory Unit on Bases, so you might want to take a look at that for more information! If you're wondering why you can't use the symbol "3" in base three, take a look at this page, which elaborates a bit more: Avoiding Ambiguity in Notation.

Thanks for asking, Louis!