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2-1
Recall . . .Recall . . .
• The definition of a computer• Analog/Digital• Electric/Mechanical• General Purpose/Special Purpose• The General Purpose Electronic Computer• General Purpose Binary• Binary?
2-2
2-3
Binary CircuitryBinary Circuitry
• Binary circuitry: – cheap
– reliable
– able to be extended to very complicated logic
• built on only two states
» ON (1)
» OFF (0)
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Computers work in BinaryComputers work in Binary
• Computers are not only powered by electricity they “compute” with electricity– they shift voltage pulses around internally
– circuits allow for electricity to flow or to be blocked depending on the type of circuit
Closedcircuit
Opencircuit
ON or 1 OFF or 0
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Representation of DataRepresentation of Data
• So, our binary computer can represent:– 0s and 1s. . .
– We need to represent considerably more than that:• Numbers
• Characters
• Visual Data
• Audio Data
• Instructions
… and we need to do it with only 0’s and 1’s
2-6
Representation of NumbersRepresentation of Numbers
• Representing numbers is considerably more than something that looks like the symbol “1” or “2” or “430”
– We’re trying to represent numbers; which have conceptual meaning
9 = 3+3+3 = 4+5 = 10-1 = 3*3 = 3^3 = 9
2-7
Representation of NumbersRepresentation of Numbers
• Decimal numeration system: (aka base 10)
– Uses 10 symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.– The place values of each position are increasing powers of ten.
• A number such as 1428 literally means:– Eight Ones– Two Tens– Four Hundreds– One (A single) Thousand
= (1 x 1000) + (4 x 100) + (2 x 10) + (8 x 1)
1010101010 01234
110100100010000
2-8
CombinationsCombinations
• Imagine we have three light-bulbs in a row, and each bulb can be on (1) or off (0).
• How many unique combinations of lights can we have?– (Hint, start with all lights off)
2-9
CombinationsCombinations
• The number of unique combinations we can have of one light with two states per lights is two:
• The number of unique combinations we can have of two lights with two states per light is four:
• The number of unique combinations we can have of three lights with two states per lights is eight: 000
001
010
011
100
101
110
111
0
1
00
01
10
11
2-10
Representation of NumbersRepresentation of Numbers
• Our three-light system– has eight possible combinations of
on and off.
• With eight unique combinations, we could represent the numbers
0, 1, 2, 3, 4, 5, 6, 7
0 = 000 4 = 100
1 = 001 5 = 101
2 = 010 6 = 110
3 = 011 7 = 111
2-11
Representation of NumbersRepresentation of Numbers
• Binary numeration system (aka base 2):– Will use 2 symbols: 0, and 1.
(Each is called a bit for binary digit)– The place values of each position are powers of two.
– A binary number such as 10110two will be expanded as:
• Zero Ones• One Two• One Four• Zero Eights• One Sixteen
= (1 x 16) + (0 x 8) + (1 x 4) + (1 x 2) + (0 x 1) = 22 in decimal
22222 01234
124816
0 1 1 01
2-12
Binary-to-Decimal ConversionBinary-to-Decimal Conversion
• Convert the following binary number (base two) in decimal (base ten)
1 0 0 0 0 0 1 1
2-13
Binary ConversionBinary Conversion
1 0 0 0 0 0 1 1
• Step 1: Make a table with the same number of columns as places in the binary string and copy the string into the table
1 0 0 0 0 0 1 1
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Binary ConversionBinary Conversion
• Step 2: Write out the powers of two corresponding to each position in the binary number:
27 26 25 24 23 22 21 20
1 0 0 0 0 0 1 1
2-15
Binary ConversionBinary Conversion
• Step 3: Write out the powers of two corresponding to each position in the binary number in decimal:
27 26 25 24 23 22 21 20
128 64 32 16 8 4 2 1
1 0 0 0 0 0 1 1
2-16
Binary ConversionBinary Conversion
• Step 4: multiply the second and third rows and put the result in the fourth row:
27 26 25 24 23 22 21 20
128 64 32 16 8 4 2 1
1 0 0 0 0 0 1 1
128 0 0 0 0 0 2 1
2-17
Binary ConversionBinary Conversion
• Step 5: (final step) – Add up all the numbers in the fourth row
27 26 25 24 23 22 21 20
128 64 32 16 8 4 2 1
1 0 0 0 0 0 1 1
128 0 0 0 0 0 2 1
128+0+0+0+0+0+2+1 = 131
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Decimal to BinaryDecimal to Binary
2(number of places) = number of unique combinations we can achieve with some number of places in binary; but we have to use one of the mappings for zero so. . .
2(number of places) - 1 = largest binary number we can store with that many places
1 21 – 1 = 1 with 1 place we can store numbers from 0 to 1
5 25 – 1 = 31 with 5 places we can store numbers from 0 to 31
7 27 – 1 = 127 with 7 places we can store numbers from 0 to 127
8 28 – 1 = 255 with 8 places we can store numbers between 0 and 255
2-20
Binary ConversionBinary Conversion
Step 2: Write out a binary conversion table with n many places, and fill in the values of the first two rows:
27 26 25 24 23 22 21 20
128 64 32 16 8 4 2 1
2-21
Binary ConversionBinary Conversion• Step 3: Subtract out the decimal powers of two from left to
right. – If you can subtract that amount and the result is non-negative, write a 1 in
the binary string and continue with the result– If the subtraction results in a negative number, write a 0 and continue
with the last positive number
27 26 25 24 23 22 21 20
128 64 32 16 8 4 2 1
245-128 = 117
117 - 64 = 53
53 - 32 = 21
21 - 16 =
5
5 - 8 = -3 error
5 - 4 = 1
1 - 2 = -1 error
1 - 1 = 0
1 1 1 1 0 1 0 1
245
2-22
Addition/SubtractionAddition/Subtraction
• Addition in any number system. . . – We add in the places from right to left– If the sum of the two numbers exceeds the symbols we have at our
disposal we “carry” some amount. . .
4 + 8 = “twelve” ; in base ten, we have no single numeric symbol that equals twelve
• We don’t need to express “twelve” in a single symbol, because we can do so by breaking the number down and incrementing the base
• Instead we write 12 which is: One Ten and Two Ones.
• Logically, we subtract 10 (the base amount) from our sum, and write the result in that place, and we carry a one into the next place, which represents One additional ten (our base number)
2-23
Binary AdditionBinary Addition
• Addition of binary numbers:0 + 0 = 00 + 1 = 11 + 0 = 11 + 1 = 2ten How do we write this?
So we’ll carry the symbol 1 into the next place (which represents two in decimal), and we’ll write down the sum minus our base amount (2)
1 + 1 = 10two
Example: adding two binary numbers
2-24
Binary SubtractionBinary Subtraction
• Subtraction of binary numbers:
0 - 0 = 00 - 1 = -1 problem! We need to borrow …
1 - 0 = 11 - 1 = 0
Remember when borrowing in base 10:
– We decrease the symbol to the left by one
– Remember when we borrow 1 from the symbol to our left, it has a value that’s “ten more” (or the base amount)
2-25
Binary SubtractionBinary Subtraction
• Let’s subtract the following numbers:
1 0 0 0 0 0 1 1 - 0 1 0 0 0 1 0 0 ------------------
2-26
Binary SubtractionBinary Subtraction
• We need to borrow for the third term, but the eighth is the closest term with something to borrow from!
1 0 0 0 0 0 1 1 - 0 1 0 0 0 1 0 0 ------------------ 1 1
2-27
Binary SubtractionBinary Subtraction
• So, we borrow 2ten from the first place we can
0 2 1 0 0 0 0 0 1 1 - 0 1 0 0 0 1 0 0 ------------------
2-28
Binary SubtractionBinary Subtraction
• Keep borrowing two
1 0 2 2 1 0 0 0 0 0 1 1 - 0 1 0 0 0 1 0 0 ------------------ 1 1
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Binary SubtractionBinary Subtraction
• Keep borrowing 2 . . .
1 1 0 2 2 2 1 0 0 0 0 0 1 1 - 0 1 0 0 0 1 0 0 ------------------ 1 1
2-30
Binary SubtractionBinary Subtraction
• Keep borrowing 2 . . .
1 1 1 0 2 2 2 2 1 0 0 0 0 0 1 1 - 0 1 0 0 0 1 0 0 ------------------ 1 1
2-31
Binary SubtractionBinary Subtraction
1 1 1 1 0 2 2 2 2 2 now, we’re able to subtract here
1 0 0 0 0 0 1 1 - 0 1 0 0 0 1 0 0 ------------------ 1 1 1
2-32
Binary SubtractionBinary Subtraction
• The rest of the subtractions are easy . . .
1 1 1 1 0 2 2 2 2 2 1 0 0 0 0 0 1 1 - 0 1 0 0 0 1 0 0 ------------------ 0 0 1 1 1 1 1 1
So our answer is -> 00111111Let’s convert to decimal to check our work
2-33
Binary SubtractionBinary Subtraction
• converting binary to decimal we get. . .
1 0 0 0 0 0 1 1 -> 128+2+1 = 131 - 0 1 0 0 0 1 0 0 -> 64+4 = 68------------------ 0 0 1 1 1 1 1 1 -> 32+16+8+4+2+1 = 67
131 – 68 = 67
2-34
Binary NumbersBinary Numbers
• Binary numbers actually have some other neat properties as well . . .
– QUESTION: Can you multiply a binary number by two (decimal) and give the result in binary quickly?
• Multiply the number 00110 by two (and give the answer in binary)
– Why does this work?
22222 01234
124816
0 1 1 00
25
32
0