How do you convert 8407 to base three?

Answer 1

8407 base ten = 1 0211 2101 base three

If you type "number base calculator" into whatever device you asked this question on, you will find a website that will immediately give you the answer we're about to find out the long way.

We're familiar with the place values in base ten. Ones, tens, hundreds, thousands and so on. They are ten raised to successive powers from zero to infinity. Other bases follow the same pattern. The place values in base three are ones, threes, nines, twenty-sevens, eighty-ones, two hundred forty-threes, seven hundred twenty-nines, two thousand one hundred eighty-sevens, six thousand five hundred sixty-ones and so on. They correspond to 3 to the zero power through 3 to the 9th power. Since 3 to the 10th power is more than 8407, we'll stop there.

The ninth place in base three is 6561. There is one of those in 8407. Write down a 1 and subtract 6561 from 8407. That leaves 1846. There aren't any 2187s in 1846. Write down a zero and move on. There are two 729s in 1846. Write the 2 down, subtract 1458 from 1846 and move on. That leaves 388. There is one 243 in 388. Write down the 1 and subtract 243 from 388. That leaves 145. There is one 81 in 145. Write down the 1 and subtract 81 from 145. That leaves 64. There are two 27s in 64, leaving 10. There is one 9 in ten, leaving 1. There are zero threes in one, but there is 1 one.

To sum up: (1 x 6561) + (2 x 729) + (1 x 243) + (1 x 81) + (2 x 27) + (1 x 9) + (1 x 1) = 8407