Converting Base 16 digits to base 2 and 10 - Analysis Question

Question

With base $16$, the digits are denoted as $0, 1,\ldots, 9, A,\ldots, F$. Let $n = AB3$. Rewrite $n$ with bases $10$, $2$.

I have no clue what this question means and how I should attempt to do this.

Answer 1

Hint: a number $n$ in base 10 can be written as follow: $$n=a_i \cdot 10^{i} + a_{i-1} \cdot 10^{i-1} \cdot \dots \cdot a_1 \cdot 10 + a_0 \cdot 10^0 $$ where $a_i \in \{0, 1, \dots, 9\}$. Similar to this case...

Answer 2

In base 16, each number is composed of a series of the digits 0-9 and A-F, where, A represents 10, B 11, C 12 and so on up to the maximum for any digit which is F, representing 15.

As such, AB$3$ represents $10$ in the $16^2=256$s column, $11$ in the $16^1=16$s column and 3 in the units column, so in decimal (base 10) that's $10(256)+11(16)+3=2739$.

In general, for a natural number $n$, given a number in base $n$, the digit at the far right tells you how many units, the digit to the left of that how many $n$s, the next digit to the left how many $n^2$s etc. None of the digits in a number in base $n$ is allowed to exceed $n-1$.

For how to convert the number to base 2, see the algorithm presented in Adi Dani's answer.

Answer 3

$$n=(AB3)_{16}=A\cdot16^2+B\cdot 16^1+3\cdot 16^0=10\cdot256+11\cdot16+3\cdot1=2739$$ $$2739:2=1369+1$$ $$1369:2=2\cdot684+1$$ $$684:2=2\cdot342+0$$ $$342:2=2\cdot171+0$$ $$171:2=2\cdot85+1$$ $$85:2=2\cdot42+1$$ $$42:2=2\cdot21+0$$ $$21:2=2\cdot10+1$$ $$10:2=2\cdot5+0$$ $$5:2=2\cdot2+1$$ $$2=2\cdot1+0$$

$$2739=(101010110011)_2$$