Question

Describe a procedure for converting decimal (base 10) expansions of integers into hexadecimal expansions.

Short answer

We can determine the hexadecimal expansion of by consecutively dividing the integer and the quotients by until we obtain a quotient of .

The successive remainders of each divisions then represents the hexadecimal expansions from right to left, while we let $\mathrm{A}=10,\mathrm{B}=11,\mathrm{C}=12,\mathrm{D}=13,\mathrm{E}=14,\mathrm{F}=15$.

Step-by-step solution

Step 1

DEFINITIONS

If $(\mathrm{base}\mathrm{b}\mathrm{represention}\mathrm{of}\mathrm{n}\}\mathrm{is}{\mathrm{a}}_{\mathrm{k}}....{\mathrm{a}}_2{\mathrm{a}}_1{\mathrm{a}}_0,$, then

$\mathrm{n}={\mathrm{a}}_{\mathrm{k}}{\mathrm{b}}^{\mathrm{k}}+{\mathrm{a}}_{\mathrm{k}-1}{\mathrm{b}}^{\mathrm{k}-1}+...+{\mathrm{a}}_1\mathrm{b}+{\mathrm{a}}_0$

In hexadecimal notation:

$$\begin{gathered} \mathrm{A}=10 \\ \mathrm{B}=11 \\ \mathrm{C}=12 \\ \mathrm{D}=13 \\ \mathrm{E}=14 \\ \mathrm{F}=15 \end{gathered}$$

Step 2

SOLUTION

Let nbe an integer.

We can determine the hexadecimal expansion of n by consecutively dividing the integer and the quotients by until we obtain a quotient of .

The successive remainders of each divisions then represents the hexadecimal expansions from right to left, while we let $\mathrm{A}=10,\mathrm{B}=11,\mathrm{C}=12,\mathrm{D}=13,\mathrm{E}=14,\mathrm{F}=15$.

For example, let us determine the hexadecimal notation of 231:

231=16.14+7

14=16.0+14

The successive remainders of each division represents the hexadecimal expansion from right to left.

role="math" localid="1668498041287" $({\mathrm{E}}_7{)}_{16}$.

Thus $({\mathrm{E}}_7{)}_{16}$is the hexadecimal notation of 231.