Question

Change from base 10 to the give base

Answer 1

Writing numbers in base 10 means to write them as a combination of powers of 10, for example

$1925 = 1\cdot 10^3 + 9\cdot 10^2 + 2\cdot 10^1+5\cdot 10^0$

So, writing numbers in any base means to do the same thing: for example, in base 3, we have

$102_3 = 1\cdot 3^2 +0\cdot 3^1 +2\cdot 3^0 = 12_{10}$

So, in order to write 80 in base 3, we have to find out the largest power of 3 that fits: we have

$3^0=3,\ \ 3^1=3,\ \ 3^2=9,\ \ 3^3=27,\ \ 3^4=81$

So, since $3^4$ is too much, our number will have 4 digits:

$abcd_3 = a\cdot 3^3 +b\cdot 3^2 +c\cdot 3^1 + d\cdot 3^0$

In order to find the digits, again we see "how much it fits": we have

$0\cdot 3^3 = 0\ \ 1\cdot 3^3 = 27,\ \ 2\cdot 3^3 = 54$

So, we choose $a=2$. The remainder is $80-54=26$

Now we need to fix the coefficient for $3^2$: we have

$0\cdot 3^2 = 0\ \ 1\cdot 3^2 = 9,\ \ 2\cdot 3^2 = 18$

Again we choose $b=2$

Keep going like this and you'll find out that

$80_3 = 2222$

This was actually a special case, because our number is exactly one less than a power of 3: we have

$80=3^4-1$

and just like in base 10, when we subtract 1 from a power of 10 we get a number composed by 9 only:

$10000000-1 = 9999999$

In every base, when we subtract 1 from a power of the base we get a number composed by (base-1) only:

$(10000-1)_3 = 2222$