Binary / Hex
Binary and Hex
The number systems of
Computer Science
CMSC 104, Lecture 05
1
Number Systems in Main Memory



The on and off states of the capacitors in
RAM can be thought of as the values 1 and 0,
respectively.
Therefore, thinking about how information is
stored in RAM requires knowledge of the
binary (base 2) number system.
Let’s review the decimal (base 10) number
system first.
CMSC 104, Lecture 05
2
The Decimal Numbering System
The decimal numbering system is a
positional number system.
 Example:
5 6 2 1
1 X 100 =
1
103 102 101 100
2 X 101 =
20
6 X 102 = 600
5 X 103 = 5000

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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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4
The Binary Number System
The binary numbering system is also
known as 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.
CMSC 104, Lecture 05
5
The Binary Number System (con’t)



The Binary Number System is also a
positional numbering system.
Instead of using ten digits, 0 - 9, the binary
system uses only two digits, 0 and 1.
Example of a binary number and the values
of the positions:
1 0 0 0 0 0 1
26 25 24 23 22 21 20
CMSC 104, Lecture 05
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Converting from Binary to Decimal
1 0 0 1 1 0 1
26 25 24 23 22 21 20
20 = 1
21 = 2
22 = 4
23 = 8
CMSC 104, Lecture 05
24 = 16
25 = 32
26 = 64
1 X 20 = 1
0 X 21 = 0
1 X 22 = 4
1 X 23 = 8
0 X 24 = 0
0 X 25 = 0
1 X 26 = 64
77
7
Converting from Binary to Decimal (con’t)
Practice conversions:
Binary
Decimal
11101
1010101
100111
CMSC 104, Lecture 05
8
Converting Decimal to Binary
•
First make a list of the values of 2 to the powers of 0 to the
number being converted - e.g:
20 = 1, 22 = 4, 23 = 8, 24 = 16, 25 = 32, 26 = 64,…
• Perform successive divisions by 2, placing the remainder of 0 or 1
in each of the positions from right to left. Continue until the
quotient is zero; e.g.: 4210
25 24 23 22 21 20
32 16 8 4 2 1
1
0 1 0 1 0
• alternative method: “subtract maximum powers and repeat”. E.g.:
42 - 32 = 10 - 8 = 2 - 2 = 0
25 24 23 22 21 2
1 0 1 0 1 0
CMSC 104, Lecture 05
9
Converting From Decimal to Binary (con’t)
Practice conversions:
Decimal
Binary
59
82
175
CMSC 104, Lecture 05
10
Counting in Binary

Binary
0
1
10
11
100
101
110
111
CMSC 104, Lecture 05

Decimal equivalent
0
1
2
3
4
5
6
7
11
Addition of Binary Numbers

Examples:
1001
+0110
1111
CMSC 104, Lecture 05
0001
+1001
1010
1100
+0101
1 0001
12
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 there are too many digits to deal with. We will
make errors.
Memory addresses and other data can be
quite large. Therefore, we sometimes use the
hexadecimal number system (shorthand for
binary that’s easier for us to work with)
CMSC 104, Lecture 05
14
The Hexadecimal Number System
The hexadecimal number system is also
known as base 16. The values of the
positions are calculated by taking 16 to
some power.
 Why is the base 16 for hexadecimal
numbers ?
o Because we use 16 symbols, the digits
0 and 1 and the letters A through F.

CMSC 104, Lecture 05
15
The Hexadecimal Number System (con’t)
Binary
0
1
10
11
100
101
110
111
1000
1001
Decimal
Hexadecimal
Binary
Decimal
Hexadecimal
0
0
1010
10
A
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1011
1100
1101
1110
1111
11
12
13
14
15
B
C
D
E
F
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The Hexadecimal Number System (con’t)

Example of a hexadecimal number and
the values of the positions:
3 C 8 B 0 5 1
166 165 164 163 162 161 160
CMSC 104, Lecture 05
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Binary to Hexadecimal Conversion
Binary 0 1 0 1 0 0 0 0 1 0 0 1 0 1 1 1
Hex
5
Written:
509716
CMSC 104, Lecture 05
0
9
7
18
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.
CMSC 104, Lecture 05
19
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
CMSC 104, Lecture 05
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Summary of Number Systems



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 F: 0 1 2 3 4 5 6 7 8 9 A B C D E F)
CMSC 104, Lecture 05
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Example of Equivalent Numbers
Binary: 1 0 1 0 0 0 0 1 0 1 0 0 1 1 12
Decimal: 2064710
Hexadecimal: 50A716
Notice how the number of digits gets
smaller as the base increases.
CMSC 104, Lecture 05
22