Continuing our introduction to binary, let’s think about how could we express numbers using an alphabet with only 2 elements:

The first two numbers are quite obvious:

But how can we go any further? Well, in decimal, when our alphabet is not enough to write a number, we increment the leftmost number and “reset” any number on its right side; so:

In binary we just do the same:

If it’s not clear enough, think about the numbers as a car’s odometer (click to learn more about it):

When a digit reaches 9 then it becomes 0 and the next digit is incremented. If you bring this to the binary world, when a digit reaches 1, it becomes 0 and the next digit is incremented.

So, we might say that:

Looking at those numbers, we can find a pattern: some of them have only one “1″ and some zeros: 1, 10, 100, 1000, 10000. If we look at their values, we’ll be able to notice that they are, respecively, 1, 2, 4, 8 and 16. All those numbers are powers of 2:

What if we replace the number 2 from the base of the multipliers for its binary notation?

To represent other numbers, we can work in an analogue fashion:

In other words: binary is just the same as decimal!

If we generalize the concept, we might say that:

Do I need to explain how to proceed for base 8 (octal) and base 16 (hexadecimal)?

Stay tuned and wait for An introduction to binary – Part 3 (I didn’t know how to sum).

* This post is a continuation for: An introduction to binary – Part 1

Before you start to understand binary numbers, you must first understand the decimal numbering system:

Let’s start from a very simple number:

What does that number mean? Well, we may say that:

or

Considering that (for those that are not familiar with logic, the “upside down V” means “AND”, and the arrow means “IT IMPLIES THAT”):

So, we can say that:

Now we can see a pattern here: considering that the rightmost digit in our number is called the less significant and the leftmost is the most significant, we might say that the most significant digit has the exponent 0 and as the digits become more significant, the exponent is incremented. From now on, we’ll call as n the position of a digit, starting from the less significant digit.

Digits x Alphabets:

Now the question is: why are all the digits multiplied by ?

Let’s go back to basics:

We can express any integer number in the decimal system using the “alphabet” defined on this set:

And, if you have half a brain, you’ll be able to count and notice that we have 10 (TEN) digits on this set. So, we say that the base of the decimal system is the number 10 (TEN).

If we dig a bit deeper, we can understand why the numeral “10″ represents the number TEN:

Using just one digit, we are able to represent the following numbers:

How do we represent the next number?

Well, consider that 9 can be represented as 09, what we do is just increment the leftmost number and “reset” everything on its right:

First thoughts about binary:

Now that we understand about the decimal system, just wonder how could we represent integer numbers using an alphabet with only two digits (BInary has the prefix BI, that means 2) defined as:

Just a tip:

Want to know more? See An introduction to binary – Part 2

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