Transcript 1 b - Electrical and Computer Engineering

ECE 204 Numerical Methods for Computer Engineers
Binary Numbers
Douglas Wilhelm Harder
Department of Electrical and Computer Engineering
University of Waterloo
Copyright © 2007 by Douglas Wilhelm Harder. All rights reserved.
Binary Numbers
• This topic introduces binary numbers
– standard notation
– scientific notation
– addition
– multiplication
Binary Numbers
• Decimal numbers are convenient for
humans because we have 10 fingers (also
called “digits”)
• Otherwise, they are exceptionally useless
as 10 has only two divisors:
2 and 5
• Even our day is more conveniently broken
up into base 12 which has four divisors:
2, 3, 4, and 6
Binary Numbers
• Representing ten digits is exceptionally
inconvenient for computers
• It is easiest to represent just two digits with
closed and open circuits (5 V and 0 V)
• We use 0 and 1 to represent these two
binary digits, or bits
Binary Numbers
• We represent a binary integer as a
sequence of bits
bn bn – 1 · · · b1 b0
which equals
n
b
k
k 0
2
k
Binary Numbers
• For example, 101010 represents
1·25 + 0·24 + 1·23 + 0·22 + 1·21 + 0·20
= 32 + 16 + 2 = 42
• As another example, 1110001 represents
1·26 + 1·25 + 1·24 + 0·23 + 0·22 + 0·21 + 1·20
= 64 + 32 + 16 + 1 = 113
Binary Numbers
• In counting, we always increment the next
digit whenever we increment a “1”:
1
10
11
100
101
110
111
1000
1001
1010
1011
1100
Binary Numbers
• We use these numbers to represent the
integers in the natural order:
1
10
11
100
101
110
111
1000
1001
1010
1011
1100
1
2
3
4
5
6
7
8
9
10
11
12
Binary Numbers
• Binary numbers are very convenient for
computers, but very inconvenient for
humans
– for example, how much smaller is 110010
than 111110100?
– these are 50 and 500, respectively
Binary Numbers
• Just like decimal real numbers, we can
represent a real number as a sequence of
bits:
bn bn – 1 · · · b1 b0 . b–1 b–2 b–3 · · ·
where bn  0 (i.e., bn = 1), and n is any
integer
• This represents the number
n
b
k
k  
2
k
Binary Numbers
• We cannot call the point in
bn bn – 1 · · · b1 b0 . b–1 b–2 b–3 · · ·
a “decimal point”, as this is not a decimal
number
• The generic name for the point is the radix
point
Binary Numbers
• As with decimal digits, we are only
concerned with a finite number of bits:
bn bn – 1 · · · b0 . b–1 · · · bm
where bn = 1, n is any integer and m  n
• This represents the number
n

k m
bk  2 k
Binary Numbers
• For example, 101.101 represents
1·22 + 0·21 + 1·20 + 1·2–1 + 0·2–2 + 1·2–3
= 4
+ 1 + 0.5
+ 0.125
= 5.625
• As another example, 1.01101 represents
1·20 + 0·2–1 + 1·2–2 + 1·2–3 + 0·2–4 + 1·2–5
= 1
+ 0.25 + 0.125
+ 0.03125
= 1.40625
Binary Numbers
• Notice that 101.101 is 1.01101 differ only by
shifting the radix point by two locations
• This is equivalent to multiply by 2–2 or
dividing by 4
• We also see that 5.625/4  1.40625
Binary Numbers
• As with decimal numbers, we can
represent binary numbers using scientific
notation:
• Any real number can be written as
1.b-1 b-2 b-3 · · · b–m  2n
which often written as
1.b-1b-2b-3···b–men
Binary Numbers
• To differentiate between binary numbers
and decimal numbers it is common to add
a subscript 2 or subscript 10:
1010102 = 4210
0.101012 = 0.6562510
11111010002 = 100010
100000000002 = 102410
• Note that 42/64 = 0.65625
Binary Numbers
• To add two binary numbers:
– line up the radix points, add the columns,
carrying 1 to next column if the sum is greater
than 1:
1
– for example, add
1.0111
1011.1 = 1.0111  23
 .0011101
and
1.1010101
1.1101 = 1.0111  20
yielding 1.1010101  23 = 1101.0101
Binary Numbers
• Multiplication of numbers in scientific
notation is similar, multiply the mantissa
and add the exponents
1.011
– for example, multiply
101100 = 1.011  25
and
0.1101 = 1.101  2–1
yielding 10.001111  24
=1.0001111  25
 1.101
0.001011
0.000000
0.101100
1.011000
10.001111
Binary Numbers
• We can verify this by converting to decimal
numbers:
1011002 = 4410
0.11012 = 0.812510
where 44  0.8125 = 35.75
and 1.0001111  25 = 100011.11
= 32 + 2 + 1 + 0.5 + 0.25
= 35.75
Binary Numbers
• Thus, we have seen that the mechanics of
binary numbers are identical to the
mechanics of decimal numbers
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]