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Objective 1.02
Understand Numbering Systems
COMPUTER PROGRAMMING I
Number Systems
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Number systems we will talk about:
Decimal (Base10)
Binary (Base2)
Hexadecimal (Base16)
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Decimal
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The number system we use in math and in life.
Base 10: ten one digit numbers:
0,1,2,3,4,5,6,7,8,9
After 9 comes 10
(the first two digit number) of course.
102= 100
Base
1 2
3
6
7
8
0
9
4 5
squared = 100
Computer Programming I
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Decimal
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What can we say about the decimal system? It is our day to day number
system…
The Decimal system has 10 digits; values are from 0 to 9.
0 10
1
11
2
12
3
13
4
14
5
15
6
16
7
17
8
18
9
19
20
21
22
23
24
25
26
27
28
29
30 40 50 60 70 80 90 100
31 41 51 61 71 81 91 101
32 42 52 62 72 82 92 102
33 43 53 63 73 83 93 103
34 44 54 64 74 84 94 104
35 45 55 65 75 85 95 105
36 46 56 66 76 86 96 106
37 47 57 67 77 87 97 107
38 48 58 68 78 88 98 108
39 49 59 69 79 89 99 109
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Binary
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Binary is Base 2
2 one digit numbers
0 and 1
For example: Base 10 of 4 = Binary 0100 or 100
102=100 – works in binary too! 2 squared = 4
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Machine Language
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Binary or Base2: Only contains 2 types of digits; 1 or 0.
The power of 2. Each digit from the right to the left is increased
by power of 2.
Each one (1) digit has a value representing on and each zero (0)
digit do not hold a value representing off.
OOOO OOOO
128 64 32 16
8
4
2
1
Ex: 0000 1001= The right most digit (1) = 1 (20)
The two middle digits are 0 therefore have no value.
The left most digit (1) = 8 (23 or 2x2x2). The other digits have
no value. The total value of all numbers would = 9. (8+0+0+1)
ex: 0000 1111 8+4+2+1 = 15 in decimal amount
1111 1111 128+64+32+16+8+4+2+1 = 255
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Why Binary?
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Computers operate on a series on electric impulses.
If the current is flowing the circuit is complete (1),
otherwise the current is off (0)
Write down the powers of 2 from 0-128.
27 26
128 64
25
32
24
16
23
8
22
4
21
2
20
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Powers of 2
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Remember from math the powers of 2:
1, 2, 4, 8, 16, 32, 64, 128 (first 8)
Remember any number to the zero power is 1 and
any number to the 1 power is that number.
So if Decimal 4= 100 in binary, what does decimal 5
equal in binary?
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Powers of 2
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Remember from math the powers of 2:
1, 2, 4, 8, 16, 32, 64, 128 (first 8)
Remember any number to the zero power is 1 and
any number of the 1 power is that number.
So if Decimal 4= 100 in binary, what does decimal 5
equal in binary?
A: 101
1
0
1
22
4
21
2
20
1
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The 1’s
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So let’s go beyond our basic example.
Remember the most right digit has the least
significant value and the most left digit has the most
significant value.
What is 1111 1111 in Decimal?
That would be 255. So… 1 0000 0000 would be 256,
right?
1
1
27 26
128 64
Computer Programming I
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1
1
1
1
1
25
32
24
16
23
8
22
4
21
2
20
1
1
0
0
0
0
0
0
0
0
28
27
26
25
24
23
22
21
256 128 64
32
16
8
4
2
20
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Let’s Try This…
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On your paper draw 8 columns
Above each column label a power of 2, starting at 128
in the first (left most) column. Finish with 1 in the
last (right most) column.
27 26
128 64
25
32
24
16
23
8
22
4
21
2
20
1
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Example Binary
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Figure out the following numbers in binary…
Decimal
56
100
198
64
18
84
231
Computer Programming I
27 26
128 64
25
32
24
16
23
8
22
4
21
2
20
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Example Binary Answers
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Figure out the following numbers in binary:
Dec Binary (Answer)
56 111000
100 1100100
198 1100110
64 1000000
18 10010
84 1010100
231 11100111
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Hexadecimal
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Hex is Base 16
There are fifteen one digit numbers:
0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
What comes after F?
Remember our rule: 102=100 (162=256 in decimal)
This works in Hex as it does for ANY number system.
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Hexadecimal
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Contains 16 digits starting with 0-9 & A-F containing the values from 0 – 15.
Hex Dec
Hex Dec
-Hex 20-2F and so on…
0 = 0
10 = 16
-Each digit carries a value of 16.
1 = 1
11 = 17
-Hex = 6 + Decimal = 10 (Hexadecimal =16)
2 = 2
12 = 18
-Hexadecimal is only 4 bits (binary value)
3 = 3
13 = 19
-ex: 1111 = 15 in decimal “F” in Hex value
4 = 4
14 = 20
-Another ex: 1001 1100 = 9C in Hex and
5 = 5
15 = 21
-156 in Decimal value.
6 = 6
16 = 22
7 = 7
17 = 23
8 = 8
18 = 24
9 = 9
19 = 25
Remember! Hex is only 4 bits long and its
A = 10
1A = 26
highest value is F in Hex or
B = 11
1B = 27
15 in decimal or 1111 in binary.
C = 12
1C = 28
D = 13
1D = 29
Mainframe computers use Hexadecimal to
E = 14
1E = 30
utilize less disk space.
F = 15
1F = 31
Hexadecimal Conversion
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Hex 9F to binary
9
F
1 0 0 1 1 1 1
23
8
22
4
21
2
20
1
23
8
22
4
Hex 9F to Decimal
9 F
161
16
160
16
21
2
1
20
1
9F Base 16 =
1001 1111 Base 2
9F Base 16 =
159 Base 10
(16 * 9) + (1 * 15) = 159 in Decimal Add the values…
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Hexadecimal
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Think if you had 3 hands. You would have 15 fingers
right?
That is what hex has!
So after 9 comes A (10), B (11), C (12), D (13), E (14)
and F (15)
Let try our example again in Hex.
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Another Conversion to Hexadecimal
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Figure out the following Decimal numbers to Hex:
Decimal
56
100
198
64
18
128
231
Computer Programming I
3
162
256
161
16
8
160
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1. Ask “How many of ‘256’ can come out of 56 (our
decimal number)?
0
2. Ask “How many of ‘16’ can come out of 56?
3 (3 * 16 = 48 with 8 left over)
Put the 3 in the 16’s spot
3. Ask “How many of ‘1’ can come out of 8 (the left
over)?
8 with 0 left over
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Example Hex Answers
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Figure out the following Decimal numbers to Hex:
Dec Hex
56 38
100 64
198 C6
64 40
18 12
128 80
256 100
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Conclusion
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In this lesson we learned about number systems used
in Programming.
Decimal
Binary
Hexadecimal
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For More Information
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http://software2i.com/viewthread.php?tid=56509&e
xtra=page%253D1&page=1
http://www.tpub.com/neets/book13/53e.htm
http://www.plcs.net/chapters/number23.htm
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