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Hexadecimal has a
unique relationship to binary. Recall that all data types,
eventually, are reduced to bytes and bits.
These bites and bits
have only two settings, on and off. For this reason, all data must
be represented in base 2. Let's
look at some different bases:
“
A soft answer turns away wrath
” - Proverbs
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1510 = 15 in base 10. A
one in the tens place and a five in the ones place. 10+5=15
178 = 15 in base 8. A one in the eights place and a seven in the ones place. 8+7=15.
169 = 15 in base 9. A one in the nines place and 6 in the ones place.
9+6=15
217 = 15 in base 7.
A two in the sevens place and a one in the ones place. 7+7+1=15. |
To convert a standard base 10 number to another base, write out a factor table. Example for base 7:
|
Columns: |
4 |
3 |
2 |
1 |
|
Powers of 7: |
73 |
72 |
71 |
70 |
|
Decimal Value: |
343 |
49 |
7 |
1 |
|
Pass in: 200 |
0 |
4 |
0 |
4 |
Answer =
4047
To convert 200 from a decimal value, base 10, to base 7, look at the chart and choose which number to use first. The fourth column, 343, is
too large. It is greater than 200 so we know it is 0 and we need not worry about it. In the 49 column, 49 will go into 200 4 times, with 4 left over. Place a 4 under the 49's column. The 4 left over is for the 7's column. 0 sevens will go into 4, so place a 0 in that column.
4 is still left over, 4 ones will go into 4 four times. Place a 4 in the ones column.
So 20010 (in base 10) is 4047 (in
base 7).
Let's try converting 968 to base 6:
|
Columns: |
5 |
4 |
3 |
2 |
1 |
|
Powers of 6: |
64 |
63 |
62 |
61 |
60 |
|
Decimal Value: |
1296 |
216 |
36 |
6 |
1 |
|
Pass in: 968 |
0 |
4 |
2 |
5 |
2 |
Answer =
42526
There are no 1296's in 968, so column 5 is 0. There are 4 216's in 968, so column 4 is 4, with a remainder of 104 left over. There are 2 36's in 104, so column 3 is 2 with 32 left over. There are 5 6's in 32, so column 2 is 5, with 2 left over. There are 2 1's in 2, so column 1 is 2, with nothing left over.
So 96810 (in base 10) is 42526 (in
base 6).
To convert back from base 6 to base 10, we would multiply. Example for 42526:
4 * 216 =
864
2 * 36 =
72
5 * 6 =
30
2 * 1 =
2
968 |
Total = 968 |
Binary - Binary (base 2) is the ultimate base for computers, for 0 can be false and 1 true, 0 can be off and 1 on. You can represent the fundamental truth of every circuit - there is power or there isn't.
To convert 88 to base 2:
|
Columns: |
8 |
7 |
6 |
5 |
4 |
3 |
2 |
1 |
|
Powers of 2: |
27 |
26 |
25 |
24 |
23 |
22 |
21 |
20 |
|
Decimal Value: |
128 |
64 |
32 |
16 |
8 |
4 |
2 |
1 |
|
Pass in: 88 |
0 |
1 |
0 |
1 |
1 |
0 |
0 |
0 |
Answer =
10110002
There are no 88's in 128, so column 8 is 0. There is 1 64 in 88, so column 7 is 1, with 24 left over. There are 0 32's in 24, so column 6 is 0, with 24 left over. There is 1 16 in 24, so column 5 is 1, with 8 left over. There is one 8 in 8, so column 4 is 1, with nothing left over. The last 3 columns are 0 because nothing is left over.
To test the answer convert it back by multiplying:
1 * 64 =
64
0 * 32 =
0
1 * 16 =
16
1 * 8 =
8
0 * 4 =
0
0 * 2 =
0
0 * 1 =
0
88 |
Total = 88 |
Now let's look at some
physical memory sizes:
byte = 8 bits
nybble = 4 bits
kilobyte = 1024 bytes
megabyte = 1024 kilobytes
gigabyte = 1024 megabytes |
Notice that all of the
above can be reduced to base 2. With 8 binary digits you can represent 256 values, 255 if all 8 bits are set to 1 and 0 if all 8 bits are set to 0. In ASCII, every letter, punctuation, numeral and character is given a binary representation. The lowercase letter "a" is represented as "01100001" in binary. This translates to the number "97" in decimal (base 10). That is why the ASCII code for "a" is "97".
Hexadecimal - Base 16 is called hexadecimal. Translating from base 2(binary), to base 16(hexadecimal) is much more efficient than translating from base 2 to base 10.
This is why it is so useful in programming. In base 16, there are 16 numerals:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
After 0-9, hexadecimal continues up the scale with A=10, B=11, C=12, D=13, E=14, F=15.
To convert 3980 to hexadecimal:
|
Columns: |
9 |
8 |
7 |
6 |
5 |
4 |
3 |
2 |
1 |
|
Powers of 16: |
168 |
167 |
166 |
165 |
164 |
163 |
162 |
161 |
160 |
|
Decimal Value: |
4,294,967,296 |
268,435,456 |
16,777,216 |
1,048,576 |
65,536 |
4,096 |
256 |
16 |
1 |
|
Pass in: 3980 |
0 |
0 |
0 |
0 |
0 |
0 |
15 |
8 |
12 |
Answer = F8C16 |
Remember: A=10, B=11, C=12, D=13, E=14,
F=15,
so 15 and
8 and
12 =
F and
8 and
C
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There are 0 4096's in 3980, so column 4 is 0. There are 15(or F) 256's in 3980, so column 3 is F, with 140 left over. There are 8 16's in 140, so column 2 is 8, with 12 left over. There are 12(or C) ones in 12, so column 1 is C with nothing left over.
To translate F8C from hexadecimal to decimal, multiply:
F * 256 =
3840
8 * 16 = 128
C * 1 =
12
3980 |
15 * 256 =
3840
8 * 16 = 128
12 * 1
= 12
3980 |
To translate FC to binary, first translate it to base 10, and then to binary:
|
Columns: |
2 |
1 |
|
Powers of 16: |
161 |
160 |
|
Hex
value: |
F |
C |
|
Translates to: |
15 |
12 |
|
Value: |
15 x
16 |
12 x
1 |
|
Now add the
results:
F(15) * 16 = 240
C(12) * 1 = 12
Total =
25210 |
Now that we know that FC in hexadecimal (base 16) is 252 in
base 10, let's translate it to binary (base 2):
|
Columns: |
9 |
8 |
7 |
6 |
5 |
4 |
3 |
2 |
1 |
|
Powers of 2: |
28 |
27 |
26 |
25 |
24 |
23 |
22 |
21 |
20 |
|
Decimal Value: |
256 |
128 |
64 |
32 |
16 |
8 |
4 |
2 |
1 |
|
Pass in: 252 |
0 |
1 |
1 |
1 |
1 |
1 |
1 |
0 |
0 |
Answer = 111111002
There are no 256's in 252, so column 9 is 0. There is 1 128 in 252, so column 8 is 1, with 124 left over. There is 1 64 in 124, so column 7 is 1, with 60 left over. There is 1 32 in 60, so column 6 is 1, with 28 left over. There is 1 16 in 28, so column 5 is 1, with 12 left over. There is 1 8 in 12, so column 4 is 1, with 4 left over. There is one 4 in 4, so column 3 is 1, with nothing left over. The remaining columns are 0 because nothing is left.
Shortcut: If you split this 8-digit binary
number into 2 sets of 4 digits each, you can use a shortcut to
convert them to hexadecimal:
1111
1100
The left set, 1111, is 15 in decimal:
1 * 1 =
1
1 * 2 =
2
1 * 4 =
4
1 * 8 =
8
Total = 15
15 =
F in
hexadecimal. |
The right set, 1100, is 12 in decimal:
0 * 1 =
0
0 * 2 =
0
1 * 4 =
4
1 * 8 =
8
Total = 12
12 =
C in
hexadecimal. |
Therefore, FC is the
hexadecimal value of 1111 1100.
This shortcut always works. Take any binary number, split it into sets of 4 digits, translate each set of 4 into hexadecimal, and put the hexadecimal results together to get the entire number in hexadecimal.
This shortcut could definitely save us some time.
©2004 C. Germany
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