binary numbers douglas wilhelm harder department of electrical and computer engineering university...
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Binary Numbers
Douglas Wilhelm Harder
Department of Electrical and Computer Engineering
University of Waterloo
Copyright © 2007 by Douglas Wilhelm Harder. All rights reserved.
ECE 204 Numerical Methods for Computer Engineers
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
k
kkb
0
2
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 1
10 2
11 3
100 4
101 5
110 6
111 7
1000 8
1001 9
1010 10
1011 11
1100 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
k
kkb 2
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
mk
kkb 2
Binary Numbers
• For example, 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, 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
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:
– for example, add
1011.1 = 1.0111 23
and
1.1101 = 1.0111 20
yielding 1.1010101 23 = 1101.0101
1010101.1
0011101.
0111.11
Binary Numbers
• Multiplication of numbers in scientific notation is similar, multiply the mantissa and add the exponents– for example, multiply
101100 = 1.011 25 and
0.1101 = 1.101 2–1
yielding 10.001111 24
=1.0001111 25 001111.10
011000.1
101100.0
000000.0
001011.0
101.1
011.1
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