Transcript d - Electrical and Computer Engineering

ECE 204 Numerical Methods for Computer Engineers
Decimal Numbers
Douglas Wilhelm Harder
Department of Electrical and Computer Engineering
University of Waterloo
Copyright © 2007 by Douglas Wilhelm Harder. All rights reserved.
Decimal Numbers
• This topic is a quick review of decimal
numbers, both integers and real numbers
– standard notation
– scientific notation
– addition
– multiplication
Decimal Numbers
• We represent an integer as a sequence of
decimal digits:
dn dn – 1 dn – 2 · · · d2 d1 d0
where dn  0
• This represents the number
n
d
k 0
k
 10
k
Decimal Numbers
• For example, 214932 represents
200000 + 10000 + 4000 + 900 + 30 + 2
or
2·105 + 1·104 + 4·103 + 9·102 + 3·101 + 2·100
Decimal Numbers
• The sequence of integers is
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ..., 17, 18, 19, 20, 21, ...
where we increment the next decimal digit
to the left whenever we add one to a 9
• Note that there is nothing special about
“ten”, the base of our number system
• Our decimal system uses ten digits:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Decimal Numbers
• We represent a real number as a
sequence of decimal digits:
dn dn – 1 · · · d1 d0 . d–1 d–2 d–3 · · ·
where dn  0, and n is any integer
• This represents the number
n
d
k  
k
 10
k
Decimal Numbers
• For example, p = 3.14159265··· represents
3 + 0.1 + 0.04 + 0.001 + 0.0005 + 0.00009 + ···
or
3·100 + 1·10–1 + 4·10–2 + 1·10–3 + 4·10–4 + 9·10–5 + ···
Decimal Numbers
• We cannot store an infinite number of
decimal digits, and therefore, we
approximate real numbers using a finite
number of decimal digits:
dn dn – 1 · · · d0 . d–1 · · · dm
where dn  0, n is any integer and m  n
• This represents the number
n

k m
d k  10 k
Decimal Numbers
• For example, 214.932 represents
200 + 10 + 4 + 0.9 + 0.03 + 0.002
or
2·102 + 1·101 + 4·100 + 9·10–1 + 3·10–1 + 2·10–
2
• Also, 3.14 represents
3 + 0.1 + 0.04 = 3·100 + 1·10–1 + 4·10–2
Decimal Numbers
• These formats can be inconvenient if n
and m are either both very large or both
very small:
52320000000000.
0.00000000000005232
and thus it is common to use scientific
notation to represent real numbers or their
approximations
Decimal Numbers
• Any real number can be written as
d0.d-1 d-2 d-3 · · · d–m  10n
where d0  0
• This is often more conveniently written as
d0.d-1d-2d-3···d–men
where
n is the exponent,
e denotes exponent, and
the digits d0.d-1d-2d-3···d are the mantissa
Decimal Numbers
• For example:
– 214.932 may be written as
2.14932  102 or 2.14932e2
– 3.14 may be written as
3.14  100 or 3.14e0
– 52320000000000. may be written as
5.232  1013 or 5.232e13
– 0.00000000000005232 may be written as
5.232  10–14 or 5.232e-14
Decimal Numbers
• To add two decimal numbers:
– line up the decimal points, add the columns,
carrying 1 to next column if the sum is greater
than 9:
– for example, add
1
3259.3 = 3.2593e3
3.2593
and
 .0053549
5.3549 = 5.3549e0
yielding 3.2646549e3
3.2646549
Decimal Numbers
• Multiplication of numbers in scientific
notation is similar, multiply the mantissa
and add the exponents
– for example, multiply
27000 = 2.7e4
and
0.32 = 3.2e-1
yielding 8.64e3 where 3 = 4 + –1
2 .7
 3 .2
.54
8.10
8.64
Decimal Numbers
• In both addition and multiplication, an
adjustment may have to be made to keep
the numbers in scientific notation
• For example,
9.25e0 + 5.23e0 = 14.48e0 = 1.448e1
Decimal Numbers
• The justification for these apparently
obvious examples is that we will next be
working with binary numbers
– computers can only store binary numbers
– the representation is slightly different, but the
operations are the same
Usage Notes
• These slides are made publicly available on the web for
anyone to use
• If you choose to use them, or a part thereof, for a course
at another institution, I ask only three things:
– that you inform me that you are using the slides,
– that you acknowledge my work, and
– that you alert me of any mistakes which I made or changes
which you make, and allow me the option of incorporating such
changes (with an acknowledgment) in my set of slides
Sincerely,
Douglas Wilhelm Harder, MMath
[email protected]