Transcript math_representation

Computer Math
CPS120: Data Representation
Representing Data
• The computer knows the type of data stored in a
particular location from the context in which the
data are being used;
– i.e. individual bytes, a word, a longword, etc
– 01100011 01100101 01000100 01000000
• Bytes: 99(10, 101 (10, 68 (10, 64(10
• Two byte words: 24,445 (10 and 17,472 (10
• Longword: 1,667,580,992 (10
Numbers
Natural Numbers
Zero and any number obtained by repeatedly adding
one to it.
Examples: 100, 0, 45645, 32
Negative Numbers
A value less than 0, with a – sign
Examples: -24, -1, -45645, -32
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Numbers (Cont’d)
Integers
A natural number, a negative number, zero
Examples: 249, 0, - 45645, - 32
Rational Numbers
An integer or the quotient of two integers
Examples: -249, -1, 0, ¼ , - ½
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Natural Numbers
How many ones are there in 642?
600 + 40 + 2 ?
Or is it
384 + 32 + 2 ? -- Octal
Or maybe…
1536 + 64 + 2 ? -- Hexadecimal
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Natural Numbers
642 is 600 + 40 + 2 in BASE 10
The base of a number determines the number
of digits and the value of digit positions
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Positional Notation
Continuing with our example…
642 in base 10 positional notation is:
6 x 10² = 6 x 100 = 600
+ 4 x 10¹ = 4 x 10 = 40
+ 2 x 10º = 2 x 1 = 2
= 642 in base
10
This number is in
base 10
The power indicates
the position of
the number
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Positional Notation
R is the base
of the number
As a formula:
dn * Rn-1 + dn-1 * Rn-2 + ... + d2 * R +
d1
n is the number of
digits in the number
642 is:
63 * 102 + 42 * 10 + 21
d is the digit in the
ith position
in the number
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Positional Notation
What if 642 has the base of 13?
+ 6 x 13² = 6 x 169 = 1014
+ 4 x 13¹ = 4 x 13 = 52
+ 2 x 13º = 2 x 1 = 2
= 1068 in base 10
642 in base 13 is equivalent to
1068 in base 10
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Representing Real Numbers
• Real numbers have a whole part and a fractional
part. For example 104.32, 0.999999, 357.0, and
3.14159 the digits represent values according to
their position, and those position values are
relative to the base.
• The positions to the right of the decimal point are
the tenths position (10-1 or one tenth), the
hundredths position (10-2 or one hundredth), etc.
Representing Real Numbers (Cont’d)
• In binary, the same rules apply but the base
value is 2. Since we are not working in base 10,
the decimal point is referred to as a radix point.
• The positions to the right of the radix point in
binary are the halves position (2-1 or one half),
the quarters position (2-2 or one quarter), etc.
Representing Real Numbers (Cont’d)
• A real value in base 10 can be defined by the
following formula:
• The representation is called floating point
because the number of digits is fixed but the
radix point floats.
Representing Real Numbers (Cont’d)
• Likewise, a binary floating –point value is defined by
the following formula:
sign * mantissa * 2exp
Representing Real Numbers (Cont’d)
• Scientific notation is a term with which you may
already be familiar, so we mention it here.
Scientific notation is a form of floating-point
representation in which the decimal point is kept
to the right of the leftmost digit.
• For example, 12001.32708 would be written as
1.200132708E+4 in scientific notation.
Representing Text
• To represent a text document in digital form, we simply need to
be able to represent every possible character that may appear.
• There are finite number of characters to represent. So the
general approach for representing characters is to list them all
and assign each a binary string.
• A character set is simply a list of characters and the codes
used to represent each one. By agreeing to use a particular
character set, computer manufacturers have made the
processing of text data easier.
Alphanumeric Codes
• American Standard Code for Information Interchange
(ASCII)
– 7-bit code
– Since the unit of storage is a bit, all ASCII codes are
represented by 8 bits, with a zero in the most significant digit
– H e l l o W o r l d
– 48 65 6C 6C 6F 20 57 6F 72 6C 64
• Extended Binary Coded Decimal Interchange Code
(EBCDIC)
The ASCII Character Set
• ASCII stands for American Standard Code for
Information Interchange. The ASCII character set
originally used seven bits to represent each
character, allowing for 128 unique characters.
• Later ASCII evolved so that all eight bits were
used which allows for 256 characters.
The ASCII Character Set (Cont’d)
The ASCII Character Set (Cont’d)
• Note that the first 32 characters in the ASCII
character chart do not have a simple character
representation that you could print to the
screen.
The Unicode Character Set
• The extended version of the ASCII character set is not
enough for international use.
• The Unicode character set uses 16 bits per character.
Therefore, the Unicode character set can represent
216, or over 65 thousand, characters.
• Unicode was designed to be a superset of ASCII. That
is, the first 256 characters in the Unicode character set
correspond exactly to the extended ASCII character set.
The Unicode Character Set (Cont’d)
A few characters in the Unicode character set