Bases Explained Part 1 - Introduction

In this series, Bases Explained, I want to dive into the various aspects of bases. At the end of this series, you will be able to know what they are, how to convert between them and why they are awesome.

I suspect all of you have at least once touched upon the mathematical term "base". Things like binary and hexadecimal are probably familiar to most of you, but how good do you really know them? I mean, who can't use a hex color code, right? Well, some time ago I explored bases a bit more and I want to share my learnings with you.

Positional notation

At the heart of the concept of bases is the positional notation. That means, depending on the position of a digit within a number, its value is different. 1 from 100 is different than 1 from 1000. The more a digit is on the left side of a number, the higher its value.

This principle is the same throughout all bases. Base 3, 7, 4, 8, 342. They all share this common theme.

The term base in the context of numerical systems actually means "How many digits are used in this number system?" That is, base 2 has 2 digits, base 16 has 16 digits, base 10 has 10, and so on. The digits always range from 0 to one less than the base (e.g. 0-9 in base 10).

With these two things in mind - positional notation and number of digits - we have the building blocks to find out how bases work. As we'll see later, it is just mind blowing when you try to completely think in a different base system, or to imaging that you always thought in a certain base system. Would you see the world differently?

There are infinitely large amounts of bases out there, so it is definitely time to find out.

Bases are important

There are a number (ʕ•ᴥ•ʔ) of reasons why different bases are used for different jobs. It is very easy to think that bases are something abstract or only math related, but the reality is they are not only pretty damn useful, but they also bring a mental value once understood. In the end, we use numbers all the time, oftentimes without even realizing. So being aware of the power that this fundamental concept of our numerical system has can really step up your life-game.

Some applications of bases:

Electronics
Computers use, at their lowest level, base 2 systems to process data. This is due to a restriction in the hardware, e.g. logic gates are either on or off; nothing in-between. It is downright incredible that with just enough zeros and ones, you can have so much flexibility that all these crazy technologies that we use today can exist.

Storage
Storage devices leverage higher base systems, such as base 16, because it allows them to save much more data at less length. Compare 10000000₂ to 80₁₆. They represent the same data, the latter one is just much shorter.

It's fun
Simple and plain, it's worth playing around with. And getting your head to think in other base systems is also cool. Finding patterns in conversions to other base systems is even more fun! Take the number 100 for example. You can take this number and interpret it in different base systems. 100₂, 100₃, 100₄, etc.

Each of them translate to a different value (when viewed from base 10). 100₂ = 4₁₀, 100₃ = 9₁₀, 100₄ = 16₁₀. It is really fascinating to see just how relative our base ten system is.

Cultures
Different cultures have historically counted in different base systems. It is a mere coincident that our western culture uses the base 10 notation (some say we chose base 10 because we have ten fingers). Other cultures have used other systems as well such as base 12. That is most likely where words like "a dozen" (meaning 12) come from. But also, some languages such as english have even unique names for the 11^th and 12^th digit: eleven and twelve.

I want to dive into the math of bases a bit deeper in Part 2, e.g. how to convert between them.

Fun fact: you can count to 1 thousand using your two hands.

PS: Bonus points for everyone who can decipher the numbers in the header image.