# Base Conversions: An Essential Skill in Math and Computer Science

Numbers, digits, operations. These are the basic building blocks of mathematics, floating near the shore of an endless ocean. Every day, whether you like it or not, we encounter math, either in the forms of calculating costs, sorting quantities for a recipe, or even checking the progress bar in a YouTube video. So, what if I were to tell you, that even after studying linear equations, imaginary numbers, algebra, statistical relationships, logarithms, geometric properties, and just about any other complex math topic for years at school, you still don’t know how to count. To elaborate further, you most likely know how to count when ten digits, from 0 to 9 are presented to you. But do you know how to count when only zeros and ones, or sixteen different digits are presented? If your answer to the above question is “no”, then continue reading to learn about math concepts that will change the way you think about arithmetic as a whole.

Now, before I talk about the whole universe of base systems that most humans just blatantly ignore, let’s converse about the composition of the number system that is closer to home: the base 10, or decimal system. Most people define the word “base 10” as synonymous to numbers, and decimals as numbers that contain digits that come after the dot. Well, these definitions are, mathematically, incorrect. First off, base 10 does not mean all the numbers in the world; it is a system in which numbers are constructed using digits from 0 to 9. And, the word decimal (note the Latin prefix deci-) literally means “of ten”. Right now, you might be rolling your eyes thinking, “both definitions of base 10 mean the same thing”. On the contrary, this assumption is invalid, for there are way more numbers written in base systems besides base 10. Next, let’s take a look at the math behind decimal digits by referring to the chart below:

The above table denotes the number 95,826 as a decimal. The bottom row seems pretty self-explanatory, but for those of you who haven’t studied base systems, the 10x row might perplex you. In reality, though, this concept is pretty simple: the base of the exponent represents the base system in which the number should be written in, and the exponent represents the place value of the digit. In this case, the number has a base of 10, and the digit 6 is in the one’s place, for 100 is one. Furthermore, a digit multiplied by its respective exponent yields the digit’s numerical value; e.g. 2 * 101 is 20. Two main rules when dealing with base systems is that the bases of all the digits have to be uniform throughout the number (you can’t have part of a number written in one base and part of a number written in another base) and the digits have to be integers less than the base’s value (the maximum value for a base 10 digit is 9). To solidify your knowledge about the decimal system, try representing the base 10 numbers 604, -8259, and 90.5 in a chart like the one above. Hint: you can use negative exponents to represent the fractional parts of the number.

# The Universe of Base Systems:

Now that you’ve learned about how the base 10 system works, let’s try applying those principles to understand how other base systems work. Let’s start with the base 2, or the binary number system: