Systems
Decimal, Number Binary, and Hexadecimal
1
Base-N Number System
Base N N Digits: 0, 1, 2, 3, 4, 5, ..., N-1 Example: 1045N Positional Number System
N LN N N N N d n -1 L d 4 d3 d 2 d1 d 0
n -1
4
3
2
1
0
Digit do is the least significant digit (LSD). Digit dn-1 is the most significant digit (MSD).
2
Decimal Number System
Base 10 Ten Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Example: 104510 Positional Number System
10 L10 10 10 10 10 d n -1 L d 4 d3 d 2 d1 d 0
n -1
4
3
2
1
0
Digit d0 is the least significant digit (LSD). Digit dn-1 is the most significant digit (MSD).
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Binary Number System
Base 2 Two Digits: 0, 1 Example: 10101102 Positional Number System
2 L2 2 2 2 2 bn -1 L b4 b3 b2 b1 b0
Binary Digits are called Bits Bit bo is the least significant bit (LSB). Bit bn-1 is the most significant bit (MSB).
n -1
4 3 2 1 0
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Definitions
nybble = 4 bits byte = 8 bits (short) word = 2 bytes = 16 bits (double) word = 4 bytes = 32 bits (long) word = 8 bytes = 64 bits 1K (kilo or "kibi") = 1,024 1M (mega or "mebi") = (1K)*(1K) = 1,048,576 1G (giga or "gibi") = (1K)*(1M) = 1,073,741,824
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Hexadecimal Number System
Base 16 Sixteen Digits: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F Example: EF5616 Positional Number System
16
n-1
L16 16 16 16 16
0100 0101 0110 0111 4 5 6 7 1000 1001 1010 1011 8 9 A B
4
3
2
1
0
0000 0001 0010 0011
0 1 2 3
1100 1101 1110 1111
C D E F
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Collaborative Learning
Learning methodology in which students are not only responsible for their own learning but for the learning of other members of the group.
7
Think - Pair - Share (TPS) Quizzes
Think Pair Share
Think individually for one time units Pair with partner for two time units Share with group for one and half time units Report results
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Quiz 1-A (Practice)
Assemble in groups of 4 Question: Convert the following binary number into its decimal equivalent: 110102
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Quiz 1-A (Practice)
THINK
One Unit (e.g. 30 Seconds)
10
Quiz 1-A (Practice)
PAIR Two Units
(e.g. 60 Seconds)
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Quiz 1-A (Practice)
SHARE 1.5 units
(e.g. 45 Seconds)
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Quiz 1-A (Practice) Report
Write names of all group members and the consensus answer on one sheet of paper. All sheets will be collected. One will be picked at random to read to the class. All papers will be graded!
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Quiz 1-A Solution
Convert the following number into base 10 decimal:
110102 = 2 + 2 + 2 = 16 + 8 + 2 = 2610
4 3 1
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Quiz 1-B
Convert the following number into base 10 decimal: 1010116
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Collaborative Learning
Think for 30 seconds Pair for 1 minute Share for 45 seconds Report
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Quiz 1-B Solution
Convert the following number into base 10 decimal:
1010116 = 1164 + 0163 + 1162 + 0161 + 1160 = 164 + 162 + 160 = 65,536 + 256 + 1 = 65,793
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TPS Quiz 2
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Binary Addition
Single Bit Addition Table
0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 10 Note "carry"
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Hex Addition
4-bit Addition
4 + 4 = 8 4 + 8 = C 8 + 7 = F F + E = 1D Note "carry"
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Hex Digit Addition Table
+ 0 1 0 0 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 A A A B B B C C C D D D E E E F F F 10 2 2 3 4 5 6 7 8 9 A B C D E F 10 11 3 3 4 5 6 7 8 9 A B C D E F 10 11 12 4 4 5 6 7 8 9 A B C D E F 10 11 12 13 5 5 6 7 8 9 A B C D E F 10 11 12 13 14 6 6 7 8 9 A B C D E F 10 11 12 13 14 15 7 7 8 9 A B C D E F 10 11 12 13 14 15 16 8 8 9 A B C D E F 10 11 12 13 14 15 16 17 9 9 A B C D E F 10 11 12 13 14 15 16 17 18 A A B C D E F 10 11 12 13 14 15 16 17 18 19 B B C D E F 10 11 12 13 14 15 16 17 18 19 1A C C D E F 10 11 12 13 14 15 16 17 18 19 1A 1B D D E F 10 11 12 13 14 15 16 17 18 19 1A 1B 1C E E F 10 11 12 13 14 15 16 17 18 19 1A 1B 1C 1D F F 10 11 12 13 14 15 16 17 18 19 1A 1B 1C 1D 1E
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TPS Quiz 3
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Complements
1's complement
To calculate the 1's complement of a binary number just "flip" each bit of the original binary number. E.g. 0 1 , 1 0 01010100100 10101011011
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Complements
2's complement
To calculate the 2's complement just calculate the 1's complement, then add 1.
01010100100 10101011011 + 1= 10101011100
Handy Trick: Leave all of the least significant 0's and first 1 unchanged, and then "flip" the bits for all other digits.
Eg: 01010100100 -> 10101011100
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Complements
Note the 2's complement of the 2's complement is just the original number N
EX: let N = 01010100100 2's comp of N = M = 10101011100 2's comp of M = 01010100100 = N
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Two's Complement Representation for Signed Numbers
Let's introduce a notation for negative digits:
d +1 = d, d +1 = d Notice that in binary, 0 + 1 = -0 + 1 = 1 = 0 where d {0,1}, we have: Two's complement notation: 1 + 1 = -1 + 1 = 0 = 1
For any digit d, define d = -d.
To encode a negative number, we implicitly negate the leftmost (most significant) bit:
E.g., 1000 = (-1)000 = -123 + 022 + 021 + 020 = -8
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Negating in Two's Complement
Theorem: To negate - ( X YZ ) = X YZ + 1 2 2 a two's complement number, just complement it and add 1. Proof (for the case of 3-bit numbers XYZ):
- ( X YZ 2 ) = X YZ 2 = X YZ 2 = ( X + 1)YZ 2 = XYZ 2 + 100 2 = XYZ + 112 + 1 = X (Y + 1)( Z + 1) 2 + 1 = X YZ 2 + 1
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Signed Binary Numbers
Two methods:
First method: sign-magnitude
Use one bit to represent the sign
0 = positive, 1 = negative
Remaining bits are used to represent the magnitude Range - (2n-1 1) to 2n-1 - 1
where n=number of digits
Example: Let n=4: Range is 7 to 7 or 1111 to 0111
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Signed Binary Numbers
Second method: Two's-complement
the Use 2's complement of N to represent
-N
Note: MSB is 0 if positive and 1 if negative Range - 2n-1 to 2n-1 -1
where n=number of digits
Example: Let n=4: Range is 8 to 7
Or 1000 to 0111
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Signed Numbers 4-bit example
Decimal 7 6 5 4 3 2 1 0 2's comp 0111 0110 0101 0100 0011 0010 0001 0000 Sign-Mag 0111 0110 0101 0100 0011 0010 0001 0000
Pos 0
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Signed Numbers-4 bit example
Decimal -8 -7 -6 -5 -4 -3 -2 -1 -0 2's comp Sign-Mag 1000 N/A 1001 1111 1010 1110 1011 1101 1100 1100 1101 1011 1110 1010 1111 1001 0000 (= +0) 1000
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Notes:
"Humans" normally use sign-magnitude representation for signed numbers
Eg: Positive numbers: +N or N Negative numbers: -N
Computers generally use two's-complement representation for signed numbers
First bit still indicates positive or negative. If the number is negative, take 2's complement to determine its magnitude
Or, just add up the values of bits at their positions, remembering that the first bit is implicitly negative.
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Example
Let N=4: two's-complement What is the decimal equivalent of
01012 Since msb is 0, number is positive 01012 = 4+1 = +510
What is the decimal equivalent of
11012 =
Since MSB is one, number is negative Must calculate its 2's complement 11012 = -(0010+1)= - 00112 or -310
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Very Important!!! Unless otherwise stated, assume two'scomplement numbers for all problems, quizzes, HW's, etc.
The first digit will not necessarily be explicitly underlined.
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TPS Quizzes 5-7
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Arithmetic Subtraction
Borrow Method
This is the technique you learned in grade school For binary numbers, we have 0 - 0 = 0 1 - 0 = 1
1
1 - 1 = 0 0 - 1 = 1 with a "borrow"
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Binary Subtraction
Note:
A (+B) = A + (-B) A (-B) = A + (-(-B))= A + (+B) In other words, we can "subtract" B from A by "adding" B to A. However, -B is just the 2's complement of B, so to perform subtraction, we
1. Calculate the 2's complement of B 2. Add A + (-B)
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Binary Subtraction - Example
Let n=4, A=01002 (410), and B=00102 (210) Let's find A+B, A-B and B-A A+B
0 1 0 0 (4)10 + 0 0 1 0 (2)10 0 11 0 6
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Binary Subtraction - Example
AB A+ (B)
0 1 0 0 (4)10 - 0 0 1 0 (2)10 0 1 0 0 (4)10 + 1 1 1 0 (-2)10 10 0 1 0 2
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"Throw this bit" away since n=4
Binary Subtraction - Example
BA B + (A)
0 0 1 0 (2)10 - 0 1 0 0 (4)10 0 0 1 0 (2)10 + 1 1 0 0 (-4)10 1110 -2
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1 1 1 02 = - 0 0 1 02 = -210
"16's Complement" method
The 16's complement of a 16 bit Hexadecimal number is just: =1000016 N16
Q: What is the decimal equivalent of B2CE16 ?
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16's Complement
Since sign bit is one, number is negative. Must calculate the 16's complement to find magnitude. =1000016 B2CE16 = ????? We have 10000 - B2CE
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16's Complement
FFF10 - B2CE
4 D3 2
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16's Complement
So,
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Sign Extension
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Sign Extension
Assume a signed binary system Let A = 0101 (4 bits) and B = 010 (3 bits) What is A+B?
To add these two values we need A and B to be of the same bit width. Do we truncate A to 3 bits or add an additional bit to B?
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Sign Extension
A = 0101 and B=010 Can't truncate A!! Why?
A: 0101 -> 101 But 0101 <> 101 in a signed system 0101 = +5 101 = -3
47
Sign Extension
Must "sign extend" B, so B becomes 010 -> 0010 Note: Value...