You may have heard of the binary numeral system before, but what exactly is it? You may have come across it in some classes, and the binary system is used extensively in computers. You may also have heard that this system is related to our normal numeral system. What is all the commotion about? I will help demystify this system in this post.

A very short answer would be that our normal system is based on ten, while the binary numeral system is based on two. I will explain at length what that means and what it entails.

Our counting system consists of two parts. The first part is a set of digits ranging from 0 to 9. They say this corresponds to the fact that we have ten fingers on our hands, which are also called digits.

The second part is a system for positioning these digits to describe numbers. When we want to write numbers larger than nine, we position the digits further to the left. When we want to write numbers smaller than one, we shift them to the right.

To transition to the binary numeral system, we first have to reduce the total number of available digits from ten to two. We do this by only considering the digits 0 and 1. With these two symbols, we can write 0 to mean zero and 1 to mean one as in our normal system.

Before taking the next step, we should make an observation about our normal system. The ones place is for multiples of one, the tens place is for multiples of ten, the hundreds place is for ten times ten, the thousands place is for ten times ten times ten, and so on. In addition, the tenths place is for multiples of one tenth, the hundredths place is for one tenth of one tenth, and so on.

To transition to the binary numeral system, we should replace all the tens (multiply by ten) and tenths (multiply by one tenth) described above with “twos” and “halves”, respectively.

The ones place should be as before, but the tens place should be replaced with a “twos place”, which corresponds to two. For example, 10 in the binary numeral system represents two because the digit 1 here is in the “twos place”. We can go further. Since the hundreds place corresponds to ten times ten, what replaces the hundreds place should correspond to two times two. So, the hundreds place should be replaced with a “fours place” as four is two times two. For the thousands place, we note that it corresponds to ten times ten times ten. So what replaces that should correspond to two times two times two. That is, the hundreds place should be replaced with an “eights place” as eight is two times two times two.

Recall that the tenths place corresponds to one tenth. So as we want to replace ten with two, the appropriate replacement here would be one half. That is, we should replace the tenths place with a “halves place”.  For instance, 0.1 in the binary numeral system represents one half because the digit 1 here is in the “halves place”. What about the hundredths place? Since the hundredths place corresponds to one tenth or one tenths, we should replace that with one half of one half. So as one half of one half is one quarter, we should replace the hundreds place with a “quarters place”.

Lastly, to calculate what a number in the binary numeral system is, we add the various “places” occupied by 1 together. If 101 were written in the binary numeral system, what number would that equal to? Note that the “fours place” and the “ones place” are occupied by 1. So to calculate 101, we would need to add four and one, which is five. In summary, 101 is five.

If 1.1 were written in the binary numeral system, we can read it like the following. Note that the “ones place” and the “halves place” are occupied by 1. So to calculate 1.1, we would need to add one and one half, which is one and a half. What if 0.11 were written in the binary numeral system? Note that the “halves place” and the “quarters place” are occupied by 1. So to calculate 0.11, we would need to add one half and one quarter, which is three quarters. In summary, 1.1 is one and a half and 0.11 is three quarters.

This could make for good fun. For instance, someone who is sixty years old could write 111,100 as their age. Why is that? Let us consider the places. We discussed the “ones place”, the “twos place”, and the “fours place”. Going further to the left, we can speak of the “eights place”, the “sixteens place”, and the “thirty-twos place”. How come? Because eight is two to the power of three, sixteen is two to the power of four, and thirty-two is two to the power of five.

Looking at 111,100, we notice that the “thirty-twos place”, the “sixteens place”, the “eights place”, and the “fours place” are all occupied by 1. So to calculate 111,100, we add thirty-two, sixteen, eight, and four together to obtain sixty.

So if someone writes 111,100 as their age on their birthday cake, don’t be fooled. They are not actually over a hundred thousand years old; they are much younger!

The binary numeral system presents an alternative that is much like our system but, at the same time, is quite different. Hopefully, this post can further clarify what this interesting numeral system is all about. You may wonder about how we obtained 111,100 above. I may describe how to do this in more detail in another post.