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ECSE2610
Computer Components & Operations
(CoCO)
Fall 2006
Part 2:
Number Systems
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View Full Document Decimal Representation:
10 digits = {0,1,2,.
..,9}
Positional Number Notation:
Weight of each digit determined by its position.
Positional Number Notation:
Positional Number Notation:
Weight of each digit determined by its position.
Example:
246 = 200
+ 40
+ 6
= 2 x 10
2
+ 4 x 10
1
+ 6 x 10
0
In general, for the Base 10 representation:
N
=
Σ
N
i
x 10
i
where each
N
i
∈
{0, 1, 2, . . . , 9} is the weight on the
base raised to the exponent
i
.
Note:
Decimal fractions occur when
i
is negative:
0.35 = 3 x 10
1
+ 5 x 10
2
In general, for the Base 10 representation:
N
=
Σ
N
i
x 10
i
where each
N
i
∈
{0, 1, 2, . . . , 9} is the weight on the
base raised to the exponent
i
.
Note:
Decimal fractions occur when
i
is negative:
0.35 = 3 x 10
1
+ 5 x 10
2
Binary Representation:
2 digits = {0, 1}
Positional Number Notation for the Binary Representation
Positional Number Notation for the Binary Representation
Positional Number Notation for the Binary Representation
N
=
Σ
N
i
x 2
i
with each
N
i
∈
{0,1}
Example (Converting from binary to decimal)
111001
2
= 100000
2
+ 10000
2
+ 1000
2
+ 000
2
+ 00
2
+ 1
2
= 1 x 2
5
+ 1 x 2
4
+ 1 x 2
3
+ 0 x 2
2
+ 0 x 2
1
+ 1 x 2
0
= 1 x 32
+ 1 x 16
+ 1 x 8
+ 0 x 4
+ 0 x 2
+ 1 x 1
= 57
10
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View Full Document Quote of the Day
There are 10 kinds of people in this world:
Those who understand binary …
and those who don’t.
Hexadecimal (Hex) Representation:
16 digits = {0,1,2,.
..,9,A,B,C,D,E,F}
Positional Number Notation for the Hexadecimal Representation
Positional Number Notation for the Hexadecimal Representation
Positional Number Notation for the Hexadecimal Representation
Note
:
A  F represent the decimal values 10  15, respectively.
Note
:
A  F represent the decimal values 10  15, respectively.
N
=
Σ
N
i
x 16
i
with each
N
i
∈
{0,1,2,. . .,9,A,B,C,D,E,F}
Example (Converting from Hex to Decimal representation)
8A9B
16
= 8000
16
+ A00
16
+ 90
16
+ B
16
= 8 x 16
3
+ A x 16
2
+ 9 x 16
1
+ B x 16
0
= 8 x 4096 + 10 x 256 + 9 x 16 + 11 x 1
= 35483
10
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View Full Document Converting Decimal to Binary:
Successive Division
Successive Division
Idea
:
Use remainders from dividing the decimal number by powers of 2.
Idea
:
Use remainders from dividing the decimal number by powers of 2.
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This homework help was uploaded on 04/22/2008 for the course ECSE 2610 taught by Professor Ji during the Spring '08 term at Rensselaer Polytechnic Institute.
 Spring '08
 Ji

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