# Bases Larger than Ten

Reference > Mathematics > Number Theory > BasesIn base eight, and base five, and all the other bases we used in the previous sections, we used a subset of the digit symbols we use in base ten. Another way of saying that is, we use some of the symbols from base ten, and no extra symbols.

But that's because eight and five are less than ten. What happens if we use a base larger than ten? What if, for example, we use base sixteen? Well, now we need symbols for all the digits from 0 to...15, right? Those symbols are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9...

Uh oh...we've run out of symbols!

Never fear; whenever we run out of symbols, we always go visit our friends the English teachers and ask if we can use some of theirs. Okay, actually, we don't ask; we just steal them. We're going to use some *letters* to represent digits. So let's try again. The symbols we use for base sixteen are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F. (Double check that for me - are there sixteen symbols?)

Notice that we're using upper case letters; we're more likely to use lower case letters as variables, so it makes sense to avoid ambiguity as much as possible.

The only problem we have now is, we need to be able to look at those digits A, B, C, D, E, and F, and quickly recognize what they mean.

A = ten

B = eleven

C = twelve

D= thirteen

E = forteen

F = fifteen

And that works out exactly right; we have a digit for every value less than sixteen. Let's see if we can do some conversions now.

**EXAMPLE**

Find the base ten value of C4_{sixteen}.

Since C represents twelve, 12 x 16 + 4 = 196

**EXAMPLE**

Find the base ten value of FFF_{sixteen}

Since F represents 15, this is 15 x 16^{2} + 15 x 16 + 15 = 4095

Incidentally, this is one less than 16^{3}; FFF_{sixteen} is the largest base sixteen number you can write without using four digits. 4096 = 1000_{sixteen}.

**EXAMPLE**

For this example, let's switch to base twelve. Find the base ten value of 3BA_{twelve.}

A and B still represent 10 and 11, so this is 3 x 12^{2} + 11 x 12 + 10 = 574.

**EXAMPLE**

Find the base ten value of 4H_{twenty-four}.

Hopefully you noticed that we've got an H in there, and we haven't yet used an H; we've only gone up to F. But we can quickly figure out what H must represent; if F is fifteen, G is sixteen, and H is seventeen. Thus, the value is 4 x 24 + 17 = 113.

## Questions

_{sixteen}as a base ten number.

_{sixteen}as a base ten number.

_{eighteen}as a base ten number.

_{eleven}as a base ten number.