Numbering
systems
(This chapter is provided to help you understand certain parts of chapter
19, Memory.)
Normally, when we use a number such as 110, we understand it to mean "one
hundred and ten," but in this chapter you will see how this is not
always the case.
Hexadecimal numbers
We generally use
the base 10 (decimal) numbering
system, where each digit must be between 0-9; but the "hexadecimal"
system (base 16) can also have digits
A, B, C, D, E, and F (16
total digits).
(The hexadecimal numbers in this tutorial are red.)
0 = Zero
1 = One
2 = Two
3 = Three
4 = Four
5 = Five
6 = Six
7 = Seven
8 = Eight
9 = Nine
A = Ten
B = Eleven
C = Twelve
D = Thirteen
E = Fourteen
F = Fifteen
In the base 10 system, you add another digit when you get past the number
9; but with base 16, it isn't added
until after F
(or fifteen).
10 = Sixteen
11 = Seventeen
12 = Eighteen
13 = Nineteen
14 = Twenty
15 = Twenty one
16 = Twenty two
17 = Twenty three
18 = Twenty four
19 = Twenty five
1A = Twenty six
1B = Twenty seven
1C = Twenty eight
1D = Twenty nine
1E = Thirty
1F = Thirty one
20 = Thirty two
21 = Thirty three
22 = Thirty four
23 = Thirty five 24 = Thirty six
.
.
.
In the decimal system (base 10), we multiply ten for each time a digit goes to the left.
10 = 10
100 = 10 * 10
1000 = 10 * 10 * 10
10000 = 10 * 10 * 10 * 10
.
.
.
But in the hexadecimal (base 16) system, we multiply sixteen, instead.
10 = 16 (16)
100 = 16 * 16
(256)
1000 = 16 * 16 * 16
(4096)
10000 = 16 * 16 * 16 * 16
(65536)
.
.
.
Therefore, since
10 is 16 and 100
is 256, the number 110 is two hundred and seventy two (272).
110 = (100 + 10) = (256 + 16)
= 272
TIP: To enter a hexadecimal number in QBasic, use &H.
&H110
Binary numbers
The
"binary" system (base 2)
can only have two digits, 0 and 1. Therefore, no binary number has a
digit between 2 and 9.
(Binary numbers are shown in dark blue.)
0 = Zero
1 = One
10 = Two
11 = Three
100 = Four
101 = Five
110 = Six
111 = Seven
1000 = Eight
1001 = Nine
1010 = Ten
1011 = Eleven
1100 = Twelve
1101 = Thirteen
1110 = Fourteen
1111 = Fifteen
10000 = Sixteen
10001 = Seventeen
10010 = Eighteen
10011 = Nineteen 10100 = Twenty
.
.
.
Notice how
binary numbers can be found by excluding numbers that have a 2, 3, 4, 5, 6, 7,
8, or 9.
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22 .
.
.
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115 .
.
.
In base 10, as explained above, we multiply ten for each time a digit goes to the left.
10 = 10
100 = 10 * 10
1000 = 10 * 10 * 10
10000 = 10 * 10 * 10 * 10
.
.
.
But in binary, we multiply by two.
10 = 2 (2)
100 = 2 * 2
(4)
1000 = 2 * 2 * 2
(8)
10000 = 2 * 2 * 2 * 2
(16)
.
.
.
So, since 10 is 2 and 100 is 4, the number 110
is six.
110 = (10 + 100) = (2 + 4) =
6
TIP: Binary (and
hexadecimal) numbers are often written with leading 0's.
0000 (same
as 0)
0001 (same
as 1)
0010 (same
as 10)
0011 (same
as 11)
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