*Published on:*

## Table of Contents

## Intro

Recently, I had a class which required conversion between a few numeric systems such as binary (base2), decimal (the "normal" one we use, base10), and base16.

(It was actually a couple more, but for this blogpost only those three are relevant.)

A classmate of mine asked the professor if it was alright for him, in a conversion between binary and base16, to convert to decimal as a middle step. This initially seems as a good practice. Since decimal is a system that we've all used since we learned to count, it makes sense to want to use it as a middleground between two unknown systems.

In a situation where time and success are critical, it's natural to want to stick to what you already know.

## The Problem with this Approach

In principle, I don't necessarily advise people to stray from their comfort
zone when performing a critical task; when dealing with little margin for error,
one should always exercise caution and stick to what works. However, in class,
it's **experimentation time**.

## Reasoning

For numerical systems that share a common base (base16 is, technically, base2^4),
conversion is actually **simpler** than it would be using a different system
with an incongruous base as a middleground. To prove this, let's do a little math...

### The Smart, But "Uncomfortable" Way

For a base16 number, each of its figures can be represented as 4 figures in binary.
Therefore, if the number were to be `52`

in base16, then the `5`

would be `0101`

and the `2`

would be `0010`

. Problem solved. Steps taken: 2. (3, if you wanna
count writing `01010010`

together as a step.)

### The "Comfortable", But Convoluted Way

Whereas converting it to decimal would entail multiplying `5`

by `16`

and then
adding `2`

to that, and **then** dividing `84`

(the result of the previous
conversion) by `2`

over and over again until the remainder is either `1`

or `0`

,
that's **six** divisions you have to carry out in order to convert a decimal
number of **two figures**.
Which then nets you `0101 0010`

, which is the exact same result we reached
earlier. Steps: 7 (if you consider conversion from base16 to decimal a single
step, and I know you don't.)

## Conclusion

This was merely an example of the myriad of situations in which stretching your limits a bit can save you tons of time. If you've read my previous post, then you know that, even though learning to use it may seem difficult at first, it'll definitely save you lots of headaches later on.