In previous sections, we learned how to convert numbers in other bases into base ten. But what if we have a number in base ten, and we want to convert it into another base?  That's actually a little more complicated, since we're used to working in base ten rather than other bases, so our minds automatically work in base ten.

Probably the best way to explore that question is with an example. Let's suppose you want to convert the number 245 into base twelve. What do you do?

Well, the first thing you should do is to make a list of the place values in base twelve. Remember that the place values are the powers of the base, like 12², 12³, and so on. We'll list our place values starting with the ones place, and keep going until we hit a value bigger than the number we're trying to convert.

12⁰ = 1
12¹ = 12
12² = 144
12³ = 1,728

Whoops! That last one is bigger than the number we're trying to convert (245), so we don't need that one.

Since 144 is less than our number, we ask "How many times does 144 go into 245?" The answer is: "1, with a remainder of 101". That tells us that our first digit is going to be a ONE.

Now take the remainder, and the previous place value, and ask, "How many times does 12 go into 101?" The answer is: "8, with a remainder of 5." Now we know that our next digit will be a EIGHT.

Now we take the remainder and the previous place value, and ask, "How many times does 1 go into 5?" Well, that's a silly question - FIVE, of course, with no remainder. So the last digit will be a FIVE.

Therefore, the answer is 185_twelve.

EXAMPLE
Convert 59 into base twenty.

First we make our list of powers of 20:

20⁰ = 1
20¹ = 20
20² = 400, which is too big.

How many times does 20 go into 59? Twice, with a remainder of 19.

How many times does 1 go into 19? 19, with no remainder.

Uh oh...we need the base twenty symbol for the number 19 now! Counting starting with A = 10 and B = 11, we find that J = 19.

Therefore, the answer is 2J_twenty.

EXAMPLE
Convert 1331 into base eleven.

First we make our list of powers of 11:

11⁰ = 1
11¹ = 11
11² = 121
11³ = 1331
11⁴ = 14641, which is too big.

How many times does 1331 go into 1331? Once, with zero remainder.

How many times does 121 go into 0? Zero times, with zero remainder.

How many times does 11 go into 0? Zero times, with zero remainder.

How many times does 1 go into 0? Zero times.

So the answer is 1000_eleven.

Once we obtained a zero remainder, we always had zeroes from that point onward, but we still needed to keep track of our place values, so we would know how many zeroes to put in our result!