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CSE
1400
Applied Discrete Mathematics
Conversions Between Number Systems
Department of Computer Sciences
College of Engineering
Florida Tech
Fall
2011
Conversion Algorithms: Decimal to Another Base
1
Conversion of Natural Numbers
1
Conversion of Integers
3
Biased Notation for Integers
4
Conversion of Rational Numbers
5
Conversion of Floating Point Numbers
7
Problems on Converting Decimal to Another Base
7
Conversion Algorithms: Another Base to Decimal
8
Conversion of Natural Numbers
9
Conversion of Integers
10
Conversion of Rational Numbers
11
Conversion of Floating Point Numbers
11
Floating Point Arithmetic
12
Machine Epsilon
13
Problems on Converting Another Base to Decimal
14
Abstract
Decimal notation using Arabic numerals is the standard system for
writing numbers in everyday life. However, in many computing ap
plications, other notations are used. Chieﬂy: binary and hexadecimal.
Learning algorithms to translate names between these numerical sys
tems forms a foundation for developing translations between more
complex languages.
Conversion Algorithms: Decimal to Another Base
Conversion of Natural Numbers
The
unsigned whole numbers, the natural numbers, written in deci
mal 0, 1, 2, 3, 4, 5, .
. . can be converted to binary by repeated division
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View Full Document cse 1400 applied discrete mathematics
conversions between number systems 2
by 2. Dividing a number by 2 produces one of two remainders:
r
=
0
or
r
=
1. The natural numbers can be converted to hexadecimal by
repeated division by 16. Dividing a number by 16 produces one of 16
remainder:
r
=
0 through
r
=
15
=
F
.
• Convert
q
=
14 to binary.
Repeatedly divide 14 and following quotients by 2 pushing the
remainder bits onto a
stack.
R
epeated
R
emaindering
M
od
2
Q
uotients
14
7
3
1
R
emainders
0
1
1
1
∴
(
14
)
10
= (
1110
)
2
• Convert
q
=
14 to hexadecimal.
Repeatedly divide 14 and following quotients by 16 pushing the
remainder hexdigits onto a
stack.
R
epeated
R
emaindering
M
od
16
Q
uotients
14
R
emainders
E
∴
(
14
)
10
= (
E
)
16
• Convert
q
=
144 to binary.
R
epeated
R
emaindering
M
od
2
Q
uotients
144
72
36
18
9
4
2
1
R
emainders
0
0
0
0
1
0
0
1
∴
(
144
)
10
= (
1001 0000
)
2
• Convert
q
=
144 to hexadecimal.
R
epeated
R
emaindering
M
od
16
Q
uotients
144
9
R
emainders
0
9
∴
(
144
)
10
= (
90
)
16
cse 1400 applied discrete mathematics
conversions between number systems 3
• Convert
q
=
143 to binary.
R
epeated
R
emaindering
M
od
2
Q
uotients
143
71
35
17
8
4
2
1
R
emainders
1
1
1
1
0
0
0
1
∴
(
143
)
10
= (
1000 1111
)
2
• Convert
q
=
143 to hexadecimal.
R
epeated
R
emaindering
M
od
16
Q
uotients
143
8
R
emainders
F
8
∴
(
143
)
10
= (
8
F
)
16
• Convert
q
=
255 to binary.
R
epeated
R
emaindering
M
od
2
Q
uotients
255
127
63
31
15
7
3
1
R
emainders
1
1
1
1
1
1
1
1
∴
(
255
)
10
= (
1111 1111
)
2
• Convert 161 to binary.
R
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This note was uploaded on 02/11/2012 for the course MTH 2051 taught by Professor Shoaff during the Fall '11 term at FIT.
 Fall '11
 Shoaff
 Math, Integers, Natural Numbers

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