Last week my daughter asked me about weird bases. “Do bases have to be integers?” “Do they have to be real?” (She’s heard about complex numbers such as i, the infamous square root of minus one.) Then she asked “How would you write 256 in base i+1?”
She swears that she made up the question on the spot, but the answer is suspiciously nice, as we can see by starting with i+1 and repeatedly squaring. If we multiply i+1 by itself we get (i+1)·(i+1) = i·i + i·1 + 1·i + 1·1, or –1 + i + i + 1; the –1 and the 1 cancel, so we get (i+1)2 = 2i. Square both sides: (i+1)4 = (2i)(2i) = (2)(2)(i)(i) = (4)(–1) = –4. Square again: (i+1)8 = (–4)(–4) = 16. Square one last time: (i+1)16 = (16)(16) = 256. So 256 equals 1 times (i+1)16, plus 0 times (i+1)15, plus 0 times (i+1)14, plus …, plus 0 times (i+1)1, plus 0 times (i+1)0, and we conclude that the base i+1 representation of 256 is 1 followed by sixteen 0’s: 10000000000000000. (What are the chances?)
I posted this on Twitter, and someone wrote “Impressive; how old is she?”. My daughter just turned 13, and 13 is (16)+(–4)+(1) which equals (i+1)8 + (i+1)4 + (i+1)0, so I wrote back “She just turned 100010001.”
But then I got to thinking: do I really know how to write every positive integer in base i+1? Or did I just get lucky?
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