__Converting Bases__

First of all, you may be thinking, what's a base? Well, it's kind of hard to explain. To the best of my knowledge, a base is how many digits you use when writing numbers.

When a number x is written in base n, it is written x_{n}.

When we say that a base is how many digits you can use, we mean
that if a number is in base 3, then we cannot use the digit 3, or any higher
ones (because digits start at 0, not 1, so you can use 0, 1, and 2). We
use **base 10**, also called the **decimal system**.

So what digits can we use for base 2 **(also called binary: it
is what computers are written in)**?** **0 and 1

For base 4? 0, 1, 2, and 3

For base 6? 0, 1, 2, 3, 4, and 5

What about base 16? Well we obviously
don't have 16 digits. **Base 16** is also called **hexadecimal**.
It uses the digits 0=9, but for the 6 digits it uses that we don't have (10, 11,
12, 13, 14, and 15), it uses the letters A, B, C, D, E,
and F, respectively.

First I will show how to convert any base to base 10, and then
from base 10 to any base. So first let's take a number in a base...say
201_{3}.

Now stay with me on this. First we count each place
(digit) in the number from right to left starting at 0.

2 0 1

*2 1 0
*Then, since it is base 3, we multiply each digit by 3

1 * 3

0 * 3

2 * 3

Now we add the products up.

1 + 0 + 18 = 19

So 201

Try another: 110101_{2
}**1** => 0**,** **0** => 1**,** **1**
=> 2**,** **0** => 3**, 1** => 4**, 1** => 5

(1*2^{0}) + (0 * 2^{1}) + (1*2^{2}) + (0*2^{3})
+ (1*2^{4}) + (1*2^{5}) =

1 + 0 + 4 + 0 + 16 + 32 = 53

**110101 _{2} = 53_{10}**

If you want to try some more, the answers without explanations
(sorry) are on a separate page.

1. 456_{7
}2. 25_{6
}3. 100_{5
}4. 1A2_{16
}Answers

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