Numbers in bases other than 10

Pre-College Topics High School Cafe
1

<p>I’m having a bit of trouble understanding rewriting base 10 numbers into other bases. For example:</p>

<p>Write the base 10 number 216 in base 4 (straight from AoPS).</p>

<p>The textbook begins by saying that 4^3 is the largest power we can start with (4^4 = 256 > 216). The biggest multiple of 64 (4^3) less than 216 is 192 (3 x 64), so the first digit must be 3. The explanation then says that we can’t just use 2 64’s because we would then need more than 3 16’s, but we are only allowed 3 nonzero digits to represent the number of 16’s. I don’t understand that. The answer btw, is base 4 number 3120.</p>

<p>Can’t base 10 number 216 also = base 4 number 2520? I’m using only 2 64’s (which is supposedly not allowed), but the numbers work out: 2<em>4^3 + 5</em>4^2 + 2<em>4^1 + 0</em>4^0 = 216.</p>

2

<p>No. The whole idea of base 4 is that you only use the digits 0, 1, 2, and 3. So using the 5 is illegal. It would be like trying to use 11 digits in base 10.</p>

<p>You have to use the largest multiple of the largest power without going over. You can’t arbitrarily choose 2 times 4^3, or 1 times 4^3. You have to use 3 times 4^3 in this case.</p>

<p>I love bases, they’re so much fun.</p>