Hexadecimal numbers

Anyone who spends enough time with a computer or other digital device will likely come across mysterious records consisting of numbers and Latin letters, such as $10FEF$. For people not familiar with it, they may seem like some cipher. What is behind these symbols? It turns out that these are simply numbers in HEX (Hexadecimal Number System). HEX is an alternative way to represent numbers.

HEX can also be written in lowercase (hex, an example of usage: Wikipedia article), but in this topic, we, for consistency, will use only uppercase HEX or, simply, hexadecimal.

Why HEX-a-decimal?

For years before the invention of computers, people used the decimal (base-$10$) system for counting – because it's convenient when you have $10$ fingers and $10$ toes.

In computers, however, there are only two options: on and off, which created the idea of a binary digit or bit, for short.

For convenience, computer engineers tended to group bits. In the 1960s, they could group $3$ bits at a time: a group of $3$ bits is the base $2\cdot2\cdot2=$ a base-$8$ number.

As computers got more powerful, people learned to cluster bits by $4$ instead of $3$. A group of $4$ bits, written as one symbol, can have $16$ values instead of $8$ – in other words, two times more than a group of $3$ bits. So, this system is more compact than decimal, octal (base-$8$), and binary systems!

Hexadecimal digits

The hexadecimal number system uses numbers $0,1\dots9$ and the first six Latin letters: $A, B, C, D, E, F$ (corresponding to decimals $10,11\dots15$), as hexadecimal digits.

Yes, in some number systems, letters can be digits, too!

The table below compares how the numbers $0,1\dots15$ are written in the binary, the decimal, and the HEX number systems:

So, as you see, we start using letters when we run out of digits in HEX. How is the decimal number $16$ represented in HEX? The answer is $10$!

Binary to HEX and vice versa

HEX is widely used in computer science because we can easily translate binary numbers into HEX. Moreover, HEX is a simplified way to represent binary numbers.

Remember: $16$ is $2^4$. That means we can split (starting from the right or, in other words, from the end) any binary number into a sequence of $4$-digit binary numbers. These numbers are then mapped to HEX digits using the table above.

Let's try converting the number $1110011001110001_2$ to HEX:

$1110011001110001_2 = 1110\ 0110\ 0111\ 0001_2 = E671_{16}$

Of course, everything is not as simple as it seems. If the length of our binary number is not divisible by $4$ without remainder, then the leftmost number of our sequence will contain less than $4$ digits:

$101100_2 = 10\ 1100_2 =\ ???_{16}$

How can it help us? Remember that $10_2=0010_2$, and, therefore:

$101100_2 = 10\ 1100_2 =2C_{16}$

So, if the leftmost number in our split-sequence of $4$-digit binary numbers contain less than $4$ digits, we add leading zeroes to it until it is a $4$-digit binary number and then match it, along with the rest in the sequence, to the corresponding HEX digit using the table. The resulting sequence is ready to be mapped to HEX digits using the table above.

Converting a hexadecimal number into a binary is even easier. Each HEX digit corresponds to a $4$-digit binary sequence. All you have to do is to map digits to sequences with the table.

$BE_{16} = 1011\ 1110_2 = 10111110_2$

If you need to convert a number from decimal to hexadecimal, you must divide it by $16$ multiple times during the calculation. The binary to hexadecimal conversion method is computationally simple and dividing by $2$ is easier than dividing by $16$. Therefore, it is reasonable to convert in two steps using the conversion to binary number. Thus, to convert a number from decimal to hexadecimal, convert the decimal number to binary and then the resulting binary number to hexadecimal using the method above. For example, we convert the number $44$ to binary and get $101100_2$, and then convert $101100_2$ to hexadecimal and get $2C_{16}$. To convert back from hexadecimal to decimal, use the same idea. To achieve that, convert the hexadecimal number to binary and the resulting number to decimal.

Applications of HEX

We use the hexadecimal system to store error codes during the work of various software products. For example, in some operating system errors are encoded in this way. If you decode your error code, you'll find out the exact error that occurred. Besides, in URLs, character codes are written as hexadecimal pairs prefixed with %. You can see for yourself by googling a symbol @. The link in the address bar of your browser would look like this:

https://www.google.com/search?q=%40

HEX numbers are also used for writing programs in low-level languages and in some encodings. For example, Unicode (a computer standard for symbol encoding) represents every symbol as a hexadecimal number. HEX numbers encode even color schemes. Thus, in RGB encoding, every color can be defined as $3$ pairs of hexadecimal numbers, each for the Red, Green, or Blue color components, respectively.

Conclusion

As you can see, there are many ways to use HEX. In this topic, we have covered that:

• A HEX number is basically a $4$-bit group with $16$ values that can include numbers from $0$ to $9$ and $A, B, C, D, E, F$ as hex-digits;

• Converting a HEX number into a binary requires splitting into a sequence of $4$-digit binary numbers;

• In the real world, the application of HEX numbers is vast — from error codes to symbol encoding.

We hope that this information will be helpful to you in the future. But for now, let's turn to practice!