Negative Digits

Number Theory Level 3

Digits are usually numbers from 0 to some $n$ . But they may also be negative. Below are numbers in base 3 with digits $+1$ , represented by the symbol $+$ , zero, and $-1$ , represented by a $-$ .

Write the number 2018 in this system.

Which of the following templates, where a dot may stand for any digit, fits your representation of 2018?

A B C D E F

2 solutions

Marta Reece
Jul 9, 2018

$2018=+\text{ } 0 - +\text{ } 0 - + -$

You can obtain this from usual representation of 2018 in base $3$ , which is $2202202$ .

Start from the right and replace $2$ with its representation in the " $+-$ " system, that is $+-$ .

That will give you $-$ at that location and $+1$ , that is $+$ , to carry over.

Every time you get $2$ after carrying over, replace it again with $+-$ in the same way.

Every time you get $3$ , replace it with its representation in the system, that is $+\text{ }0$ .

Stephen Mellor
Jul 25, 2018

First we get the placeholder powers of 3 to show the value of each symbol in each location which are $2187,729,243,81,27,9,3,1$ .

It is clear that we need the 2187 to be positive or 2018 is unobtainable using the remaining numbers.

We now need -169. A 0 for the 729 is the only way to keep us close enough using the remaining numbers.

Continuing on this way, we get 2018 = +0-+0-+-, which matches with D

Calculate 2018+2187+729+243+81+27+27+9+3+1.
Write it in base 3 (21201202)
21201202-(2187+729+243+81+27+27+9+3+1)=+0+-0+-+

EDIT: Now i see that it isn't correct (but still matches with D), any thought why?

Piotr Szewczyk - 2 years, 10 months ago

You need to invert all of the digits except the first (e.g. + goes to -, 0 goes to 0, - goes to +) although I'm not sure why

Stephen Mellor - 2 years, 10 months ago

Negative Digits

2 solutions

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