Binary / Hex
Binary and Hex
The number systems of
Computer Science
CMSC 104
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Main Memory
Capacitors on/off translates to values
1/0
 requires use of Binary number system
 Investigate Decimal number system first

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The Decimal Numbering System
The decimal numbering system is a
positional number system.
 Example:
5621
1 X 100
1000 100 10 1
2 X 101
6 X 102
5 X 103

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What is the base ?
The decimal numbering system is also
known as base 10. The values of the
positions are calculated by taking 10 to
some power.
 Why is the base 10 for decimal numbers
?

Because we use 10 digits. The digits 0
through 9.
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What is the base ?
The binary numbering system is called
binary because it uses base 2. The
values of the positions are calculated by
taking 2 to some power.
 Why is the base 2 for binary numbers ?

Because we use 2 digits. The digits 0
and 1.
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The Binary Numbering System
The Binary Numbering System is also a
positional numbering system.
 Instead of using ten digits, 0 - 9, the
binary system uses only two digits, the
0 and the 1.
 Example of a binary number & the
values of the positions.
1 0 0 0 0 0 1
CMSC 104
26 25 24 23 22 21 20

6
Computing the Decimal Values
of Binary Numbers
1 0 0 0 0 0 1
26 25 24 23 22 21 20
20 = 1
21 = 2
22 = 4
23 = 8
CMSC 104
24 = 16
25 = 32
26 = 64
1 X 20 = 1
0 X 21 = 0
0 X 22 = 0
0 X 23 = 0
0 X 24 = 0
0 X 25 = 0
1 X 26 = 64
65
7
Converting Decimal to Binary
First make a list of the values of 2 to the
powers of 0 to 8, then use the
subtraction method.
20 = 1, 21 = 2, 22 = 4, 23 = 8, 24 = 16,
25 = 32, 26 = 64, 27 = 128, 28 = 256
 Example:
42
42 10 2
- 32 - 8 - 2
1 0 1 0 1 0
5 24 23 22 21 20
2
CMSC 104

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Counting in Binary

CMSC 104
Binary
0
1
10
11
100
101
110
111

Decimal equivalent
0
1
2
3
4
5
6
7
9
Addition of Binary Numbers

Examples:
1001
+0110
1111
CMSC 104
0001
+1001
1010
1100
+0101
1 0001
10
Addition of Large Binary Numbers

Example showing larger numbers:
1010 0011 1011 0001
+ 0111 0100 0001 1001
1 0001 0111 1100 1010
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Working with large numbers
0101 0000 1001 0111
Humans can’t work well with binary
numbers. We will make errors.
 Shorthand for binary that’s easier for us
to work with - Hexadecimal

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Hexadecimal
Binary 0 1 0 1 0 0 0 0 1 0 0 1 0 1 1 1
CMSC 104
Hex
5
Written:
509716
0
9
7
13
What is Hexadecimal really ?
Binary 0 1 0 1 0 0 0 0 1 0 0 1 0 1 1 1
Hex
5
0
9
7
A number expressed in base 16. It’s
easy to convert binary to hex and hex to
binary because 16 is 24.
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Hexadecimal
Binary is base 2, because we use two
digits, 0 and 1
 Decimal is base 10, because we use
ten digits, 0 through 9.
 Hexadecimal is base 16. How many
digits do we need to express numbers
in hex ? 16 (0 through ?)
0123456789ABCDEF

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Counting in Hex
Binary
0000
0001
0010
0011
0100
0101
0110
CMSC 104 0 1 1 1
Hex
0
1
2
3
4
5
6
7
Binary
1000
1001
1010
1011
1100
1101
1110
1111
Hex
8
9
A
B
C
D
E
F
16
Another Binary to Hex Conversion
Binary 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1
Hex
7
C
3
F
7C3F16
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