| $0F
(Pronounced "OH EFF" or "ZERO EFF")
Note the
use of the $ to show hex. You'll also see a hex number like this:
0x0F
In fact,
the '0x' in front of a hex number is more current than the '$'. The '$'
is sort of old school. Whether you use a '$' or a '0x', it tells you that
you're working with a hex number, not a decimal one.Why use a special
symbol to denote hex? Isn't it obvious that '0F' isn't an ordinary decimal
number? Sure, but what if you had this number:
5396
Hmmm. How
do you know whether that is a decimal 5396 or a hex 5396? See, not all
hex numbers will have an 'A' thru 'F' in them. Many do, but not all. But
as soon as we put a '$' or a '0x' in front of that number
$5396
or 0x5396
We know it's
a hexadecimal number. Look, I'm already tired of typing it both ways,
so I'm going to keep using the '$' to denote hex instead of showing you
both methods from now on. If you feel more comfortable with the '0x',
feel free to substitute that for the rest of the tutorial.
Here's another:
$A0FF
(Pronounced
"A OH EFF EFF" or "A ZERO EFF EFF")
And here's
some others (yes, they're legitimate hex numbers):
(Numbers in
parentheses are the decimal versions)
| $BEAD
(48,813) |
$DEAD
(57,005) |
$FEED
(65,261) |
$DEAF
(57,007) |
$FADE
(64,222) |
$BABE
(47,806) |
| $ABBA
(43,962) |
$DEED
(57,069) |
$FACE
(64,206) |
$BEEF
(48,879) |
$CAD
(3,245) |
$ADD
(2,781) |
| $CAFE
(51,966) |
$CAB
(3,243) |
$BAD
(2,989) |
$ACE
(2,766) |
$FAD
(4,013) |
$BED
(3,053) |
Now, you
may be wondering, "Hey, it's all well and good that you're showing
us hex examples, but how do we go about translating those hex numbers
into decimal?"
Well, I'll
tell you... (and to keep things simple, we'll stick to smaller numbers
for now):
Let's look
at a typical small decimal number.
235
Now, in decimal,
the above number is equivalent to:
2
x 100 + 3 x 10 + 5 or 200 + 30 + 5
Easy enough.
Now look at a slightly larger decimal number:
1,236
This is equivalent
to:
1
x 1000 + 2 x 100 + 3 x 10 + 6 or 1,000 + 200 + 30 + 6
Well, hex
numbers follow a similar pattern where each digit is in a "place".
But instead of 10's place or 100's place or 1000's place, hex uses the
following progression (from right to left):
|
4096's
Place
|
256's
Place
|
16's
Place
|
1's
Place
|
Then, to
translate a hex number such as $A0FF, you set up a chart
like this:
Remember
that hex 'A' actually is equal to decimal 10, and hex 'F' is actually
decimal 15 (look at the chart above if you just felt your brain drop into
your underwear and are totally lost. I don't know where that image came
from...). Here's our revised chart:
Now multiply
10 times 4096, then 0 times 256, then 15 times 16, then 15 times 1. Then
add the results (10 times 4096=40960 + 0 times 256=0 + 15 times 16=240
+ 15 times 1=15 which comes out to decimal 41215.
Still with
me? No? Here's a smaller example for the 'larger example impaired':
2 Digit
Hex Number: $B9
Remember that hex 'B' = 11 and hex '9' is still just 9:
|
16's
Place
|
1's
Place
|
|
11
|
9
|
Or
Since 'B'
= 11, you multiply 11 times 16 which equals 176, then add a 9 times 1.
The final
translation of $B9 to decimal would thus be 11 times 16 + 9 times 1=decimal
185.
Another 4
Digit Hex Number: $FEBC
Remember
that hex 'F' = 15, hex 'E' = 14, hex 'B'=11, and hex 'C' = 12:
|
4096
|
256
|
16
|
1
|
|
15
|
14
|
11
|
12
|
So multiply
15 times 4096, 14 times 256, 11 times 16, and 12 times 1 and add the results-
15 x 4096
=61440 + 14 x 256=3584 + 11 x 16=176 + 12 x 1 = 65212.
| WHAT
ABOUT REALLY LARGE HEX NUMBERS?!?!? |
To translate
(or convert as the big boys say) large hex numbers, you need to know more
powers of 16. We've already seen the first four powers of 16: 4096 (16^3),
256 (16^2), 16 (16^1), and 1 (16^0). Whipping out my calculator (you think
I studied multiplications tables in school larger than 12?!?!?), I find
that after 4096 (16^3), the powers of 16 are:
|
16^7
|
16^6
|
16^5
|
16^4
|
16^3
|
16^2
|
16^1
|
16^0
|
|
268,435,456
|
16,777,216
|
1,048,576
|
65536
|
4096
|
256
|
16
|
1
|
Whew! Big
numbers!!! So if we had a hex number like:
$FE973BDC
We could
convert it by setting up the table like this:
|
268,435,456
|
16,777,216
|
1,048,576
|
65536
|
4096
|
256
|
16
|
1
|
|
F
|
E
|
9
|
7
|
3
|
B
|
D
|
C
|
This is going
to be a big number. You'll definitely want your calculator for this one.
Here goes:
Convert the
hex digits to decimal digits using the chart at the top:
|
268,435,456
|
16,777,216
|
1,048,576
|
65536
|
4096
|
256
|
16
|
1
|
|
15
|
14
|
9
|
7
|
3
|
11
|
13
|
12
|
Multiply
the decimal value in the bottom box by the powers of 16 values in the
top box:
15 x 268,435,456
+ 14 x 16,777,216 + 9 x 1,048,576 + 7 x 65,536 + 3 x 4,096 + 11 x 256
+ 13 x 16 + 12 x 1 =
4,026,531,840
+ 234,881,024 + 9,437,184 + 458,752 + 12,288 + 2,816 + 208 + 12 =4,271,324,124.
Ow! That
hurts my brain. I'm resting for awhile...
| HEXADECIMAL
AND BINARY'S RELATIONSHIP |
Alright,
I'm back.
There is
a wonderful (if you're a geek like me) relationship between hexadecimal
and binary that might not be readily apparent. In fact, they're so closely
tied together, that many programmers learn both equally well (especially
assembler programmers). Let's take a look.
Here's a
binary number (again, review my Binary
Tutorial if you're a bit rusty. Get it? A 'bit' rusty. I kill me):
#10111101
Notice
the pound sign to signify that it is a binary number so we don't confuse
decimal 10,111,101 with binary 10111101.
So, based
on our tutorial, we set up something like this first:
|
128
|
64
|
32
|
16
|
8
|
4
|
2
|
1
|
|
1
|
0
|
1
|
1
|
1
|
1
|
0
|
1
|
Now, add
together all the numbers in the top boxes that have a 1 below them and
ignore the numbers with zeroes below them.
128+32+16+8+4+1=189
in decimal.
Now if you
split that little 8-bit binary number into 2 sets of 4 bits, let's call
them the leftmost 4 bits and the rightmost 4 bits, we get
|
8
|
4
|
2
|
1
|
|
8
|
4
|
2
|
1
|
|
1
|
0
|
1
|
1
|
|
1
|
1
|
0
|
1
|
What the?!? Notice,
we dumped the 128, 64, 32 and 16, because we're now working with 2 sets
of 4 bit numbers instead of 1 big 8-bit number.
Wow that makes it
so much easier...
To convert our binary
number to hex, figure out what the leftmost 4 bits is equal to, and the
rightmost 4 bits.
|
LEFTMOST
|
1011
|
=8+2+1
|
=11
|
=$B
|
|
RIGHTMOST
|
1101
|
=8+4+1
|
=13
|
=$D
|
And since
11 decimal = $B hex and 13 decimal = $D hex, the final hexidecimal conversion
of binary #10111101 = $BD.
For larger
binary numbers, it still works. Take this 16 bit number:
#1011010010100111
Break it
up into four 4-bit sets:
|
8
|
4
|
2
|
1
|
|
8
|
4
|
2
|
1
|
|
8
|
4
|
2
|
1
|
|
8
|
4
|
2
|
1
|
|
1
|
0
|
1
|
1
|
|
0
|
1
|
0
|
0
|
|
1
|
0
|
1
|
0
|
|
0
|
1
|
1
|
1
|
And add up
the numbers which have a 1 below them, but be sure to keep it in 4 different
sets:
|
8+0+2+1
|
0+4+0+0
|
8+0+2+0
|
0+4+2+1
|
|
11
|
4
|
10
|
7
|
|
B
|
4
|
A
|
7
|
So #1011010010100111
is equal to $B4A7 hex and 46247 decimal.
Here's a
32-bit binary number to convert to hex:
#11010100010100111111011100010101
Break it
up into eight 4-bit chunks...
|
8
|
4
|
2
|
1
|
|
8
|
4
|
2
|
1
|
|
8
|
4
|
2
|
1
|
|
8
|
4
|
2
|
1
|
|
8
|
4
|
2
|
1
|
|
8
|
4
|
2
|
1
|
|
8
|
4
|
2
|
1
|
|
8
|
4
|
2
|
1
|
|
1
|
1
|
0
|
1
|
|
0
|
1
|
0
|
0
|
|
0
|
1
|
0
|
1
|
|
0
|
0
|
1
|
1
|
|
1
|
1
|
1
|
1
|
|
0
|
1
|
1
|
1
|
|
0
|
0
|
0
|
1
|
|
0
|
1
|
0
|
1
|
Add up all
numbers that have a 1 below them:
|
8+4+0+1
|
0+4+0+0
|
0+4+0+1
|
0+0+2+1
|
8+4+2+1
|
0+4+2+1
|
0+0+0+1
|
0+4+0+1
|
|
13
|
4
|
5
|
3
|
15
|
7
|
1
|
5
|
|
D
|
4
|
5
|
3
|
F
|
7
|
1
|
5
|
So binary
11010100010100111111011100010101 is equal to $D453F715 hex. Cool.
Click [HERE]
to see a chart of decimal, hexadecimal and binary numbers.
|
