On the previous page we looked at what it means that we use "base ten." In this section, we'll explore what math would look like if we were using another base.

We use base ten because we have ten fingers. So what if human beings all had eight fingers instead of ten? Well, we probably would have created a very different system of counting, and it's what we call "base eight."

If you only had eight fingers, you would only need eight symbols (remember that in base ten, we had ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9). In base eight you don't need symbols for eight and nine.  Let's see how this works. Consider the number 14 (written in base eight). What does this number mean? Well, it means ONE person's fingers, plus 4 more fingers. But how many fingers does one person have? EIGHT! So 14 (base eight) means 8 + 4, which (written in base ten) is 12. It seems a little strange at first, but after awhile you get used to it. Let's try another example.

EXAMPLE
What would 27 (base eight) look like if you wrote it in base ten?

Well, the two means TWO people's fingers, and seven means SEVEN more fingers. Two people have a total of 16 fingers (2 x 8 = 16), so 27 (base eight) is 2 x 8 + 7 = 23 (base ten).

By the way, we do have a short-hand for writing numbers in different bases. Instead of writing 27 (base eight), we can write 27_eight. 

So the answer to the previous problem could be written: 27_eight = 23_ten. 

We could shorten the answer even more, because if we don't write a base number, we assume it's in base ten. Thus, 27_eight = 23.

EXAMPLE
Find the base ten value of 246_eight.

Here we need to remember what we learned on the last page about interpreting base ten. If this was a base ten number, we would think of it as:

2 x 10² + 4 x 10 + 6.

But since it's base eight, we need to think of it as:

2x 8² + 4 x 8 + 6 = 166 (in base ten - notice that by not writing a base as a subscript, it's automatically assumed to be base ten.)

From here, you can easily see how to work in other bases, like base five.

EXAMPLE
Find the value in base ten of 304_five.

This is the same as 3 x 5² + 0 x 5 + 4 = 79.

1.

What digits (symbols) can you use in base nine?

2.

What digits can you use in base four?

3.

Although base one (called unary) is occasionally used, it can't be written the way we normally write numbers in bases. Why not?

4.

Which base uses five symbols?

5.

What is the value of 32_eight?

6.

What is the value of 45_seven?

7.

What is the value of 102_three?