Numbers are simply numbers
It may sound strange, but there’s not much to learn in hexadecimal that’s different from the math you already know.
The mathematics of binaries, decimals, hexadecimals (which is the same as "base 16"), base 32, base 36, base 64, as well as any other basis, is the same. What changes is the amount of symbols to represent the digits.
"
Incidentally", as our common Arabic digits only range from 0 to 9, we have chosen to add the letters A to F to represent the extra digits, since hexadecimal is a 16-digit notation, instead of the 10 we are used to.
Instead of creating a new collection of drawings to represent the numbers after the 9, we used what was already ready, for convenience - the letters - but giving them a new meaning.
- In decimal, we use the Arabic digits
0 1 2 3 4 5 6 7 8 9;
- in hexadecimal, we use the digits
0 1 2 3 4 5 6 7 8 9 A B C D E F;
- on the basis of 36 we go further still, from
0 up to the letter Z!
- in binary, goes only from
0 to 1, but the logic is still the same.
Just like the last decimal number, which is 9, added to 1 turns 10 (i.e., goes back to zero and gains a new "house" next door), if we take the last "number" of Hexa, which is "F" and add "1", the result is 10 in hexadecimal. Similarly, in binary (which consists of 2 digits, "0" and "1" respectively), adding 1 + 1 we have 10 in binary.
DECIMAL HEXADECIMAL BASE36
9 + 1 = 10 F + 1 = 10 Z + 1 = 10
Summary: mathematics is the same.
In informatics
The Hexadecimal is very present in the computer by the ease that has to represent the numbers of another base: the binary.
Since both hexadecimal and binary are power numbers of two, you can reconcile some concepts of both bases interestingly. For this, let’s start from this tabelinha:
Decimal Hexa Binário
0 0 0
1 1 1
2 2 10
3 3 11
4 4 100
5 5 101
6 6 110
7 7 111
8 8 1000
9 9 1001
10 A 1010
11 B 1011
12 C 1100
13 D 1101
14 E 1110
15 F 1111 <-- Atenção a este caso!
16 10 10000
17 11 10001
18 12 10010
19 13 10011
...
Note that on reaching 15 decimal, both hexadecimal and binary arrived at the last house before "turning" another house. As this "turn" coincides, hexadecimal has become a very practical way of representing binary numbers.
For each of the 4-digit binaries (0000 to 1111), we have exactly one hexadecimal digit. Knowing or understanding how to count these 16 digits, we convert any hexadecimal to binary and vice versa.
So, looking at the above table, it becomes easy to convert a binary number:
111010110110110 -> original em binário
111 0101 1011 0110 -> dividido em grupos de nibbles
7 5 B 6 -> substituído pelos números da tabela.
75B6 -> reescrito em hexadecimal
Where the hexadecimal enters this history then?
The binary is the basis of the current technology, because in both Hds and memory, or in processors, the information is represented as "on" or "off" only (the so-called bits).
This doesn’t apply to quantum processors, but wait a little longer to have one of these on your desk.
For some time now, processors and memory in general have used spaces to store information organized in 4-bit packages (called Nibble), 8 bits (byte), 16 bits (word), 32 bits (dword), and so on.
- To represent a Nibble (0000 to 1111), just one hexadecimal digit (0 to F) is enough);
- to represent one byte, only two of them, from 00 to FF;
- and so we follow, for "measures" such as 0000 to FFFF, and larger.
Examples "Visual":
- It is much easier to write 5F than 01011111, no?
- A MAC Address Network card: 00-5F-FF-E0-AA-FF instead of 0-95-255-224-170-255
- Color #F00CC instead of rgb(255,0,204);
perhaps in this case the hexadecimal causes initially strangeness, but as the colors for the 216 web palette are multiples of 51 decimal (which is the same as 33 hexadecimal), we have the most common digits being 00, 33, 66, 99, cc and ff, which are abbreviated to one digit each (#33cc00 is the same as #3c0 for browsers).
- A hexdump would be very confusing if we represented bytes from 0 to 255 instead of 00 to FF
Example of hexdump:
00000000 255044462D312E34 0A312030206F626A %PDF-1.4.1 0 obj
00000010 0A3C3C0A2F426173 65466F6E74202F48 .<<./BaseFont /H
00000020 656C766574696361 0A2F456E636F6469 elvetica./Encodi
00000030 6E67202F57696E41 6E7369456E636F64 ng /WinAnsiEncod
...
Imagine if we had to use 3 houses, or separators to identify the numbers.
Thus, it has been agreed to use hexadecimal for many things in which it facilitates reading and context of information.
How to "read" the hexadecimal?
A: Without fear :)
In the same way that in decimal each "house" is multiplied by powers of ten:
279 = ( 2 * 10 * 10 ) + ( 7 * 10 ) + ( 9 )
In hexadecimal just multiply by powers of 16:
3AF = ( 3 * 16 * 16 ) + ( A * 16 ) + ( F )
é o mesmo que ( 3 * 16 * 16 ) + ( 10 * 16 ) + ( 15 ) em decimal.
It seems complex at first glance, but after you get used to the values from A to F, the rest becomes habit (in the same way we get used to the example of 0 to 9 in day-to-day life).
Just to complete the reasoning, see another example, this time in binary:
1101 = ( 1 * 2 * 2 * 2 ) + ( 1 * 2 * 2 ) + ( 0 * 2 ) + ( 1 )
Kept to proper proportions, it’s like that Cypher scene in the Matrix: "I don’t Even see the code. All I see now is Blonde, Brunette, Redhead." 1
1. "I don’t even see the code anymore. Now I just see blonde, brunette, redhead."
Very enlightening! I didn’t even know that hexadecimal notation made binary notation easier and more readable! Now it seems so obvious: 10 = 111111110000000. And, I didn’t understand your reference to the scene. I watch Matrix, but I don’t remember what that scene is. Does the speech in you actually refer to women?
– Sid
@Sid the Cypher is looking at the monitor, with those green symbols, and Neo asks, "Do you always see encoded?" and then Cypher replies (briefly): "The Matrix is too complex to decode in real time, but then you get used to it. I don’t even see the code anymore, I see blonde, brunette, redhead..."
– Bacco
By the way,
240 dec = F0 hex = 11110000 binand10 dec = 0A hex = 00001010 bin.1111111100000000 = FF00, that is to say,65280decimally. Separate to read:1111 1111 0000 0000 = F F 0 0, that is to say(15 * 16 * 16 * 16 ) + ( 15 * 16 * 16 ) + ( 0 * 16 ) + ( 0 ) = 65280– Bacco
Excellent reply, remember almost every case! + 1
– Jorge B.
I don’t Even see the code. All I see is 0xB00B5
– Luiz Vieira
Just fantastic this explanation
– RodrigoBorth
Perhaps it was worth remembering that on older architectures (such as the PDP-10 where the C was created), the "natural" unit was a group of 36 bits, and so on these architectures it was very common to use base octal 0-7, since a PDP-10 word is made up of six groups of octal two digits - hence the use of octals in Unix permissions and the support for this base in C.
– Wtrmute
@Wtrmute is a good observation. I don’t know if the question of the PDP-10 would have much to do with the original question to go into detail, but somehow, when it comes to tinkering with the answer more calmly, I think it pays at least for examples in octal as well, until pq is a basis that confuses the staff.
– Bacco
Great explanation. I like to think that the decimal is a box yesterday fit 10 items and to add more items would need another box. Hexadecimal is a box that fits 16 items and binary only two.
– Marcus Becker