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Bases Larger than Ten

Reference > Mathematics > Number Theory > Bases

In 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 C4sixteen.

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

EXAMPLE
Find the base ten value of FFFsixteen

Since F represents 15, this is 15 x 162 + 15 x 16 + 15 = 4095

Incidentally, this is one less than 163; FFFsixteen is the largest base sixteen number you can write without using four digits. 4096 = 1000sixteen.

EXAMPLE
For this example, let's switch to base twelve. Find the base ten value of 3BAtwelve.

A and B still represent 10 and 11, so this is 3 x 122 + 11 x 12 + 10 = 574.

EXAMPLE
Find the base ten value of 4Htwenty-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

1.
The number 4G is written in base n. What is the smallest possible value of n, and why?
2.
What is the largest base we can have before we run out of letters of the alphabet?
3.
Find the value of 15sixteen as a base ten number.
4.
Find the value of EEsixteen as a base ten number.
5.
Find the value of EEeighteen as a base ten number.
6.
Find the value of 111eleven as a base ten number.
7.
What is the largest number that can be written in base twelve with three digits?
8.
What is the smallest number that can be written in base thirteen with three digits? What is the base ten value of that number?
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