Number Bases. Ioan Despi. University of New England. August 4, 2013
|
|
|
- Priscilla Dean
- 6 years ago
- Views:
Transcription
1 Number Bases Ioan Despi University of New England August 4, 2013
2 Outline Ioan Despi AMTH140 2 of 21 1 Frequently Used Number Systems 2 Conversion to Numbers of Different Bases 3 Repeated Division Algorithm 4 Conversion of the decimal part of a number
3 Frequently Used Number Systems Ioan Despi AMTH140 3 of 21 The numbers used most frequently in our daily life are written in the standard decimal number system, e.g., 7086 =
4 Frequently Used Number Systems Ioan Despi AMTH140 3 of 21 The numbers used most frequently in our daily life are written in the standard decimal number system, e.g., 7086 = The base number in the decimal system is 10, whose special role is obvious in the above example, implying there are exactly 10 digits required for the system.
5 Frequently Used Number Systems Ioan Despi AMTH140 3 of 21 The numbers used most frequently in our daily life are written in the standard decimal number system, e.g., 7086 = The base number in the decimal system is 10, whose special role is obvious in the above example, implying there are exactly 10 digits required for the system. These 10 elementary digits are 0, 1, 2,..., 9
6 Frequently Used Number Systems Ioan Despi AMTH140 3 of 21 The numbers used most frequently in our daily life are written in the standard decimal number system, e.g., 7086 = The base number in the decimal system is 10, whose special role is obvious in the above example, implying there are exactly 10 digits required for the system. These 10 elementary digits are 0, 1, 2,..., 9 In general, a nonnegative number in a base p number system is denoted by (a n a n 1... a 1 a 0 b 1 b 2... b N...) p, representing the value of a n p n + a n 1 p n a 1 p 1 + a 0 p 0 + b 1 p 1 + b 2 p b N p N +...
7 Frequently Used Number Systems Ioan Despi AMTH140 4 of 21 We note that a 0,..., a n and b 1, b 2... are all just digits, coming from a set of p elementary digits for the number system.
8 Frequently Used Number Systems Ioan Despi AMTH140 4 of 21 We note that a 0,..., a n and b 1, b 2... are all just digits, coming from a set of p elementary digits for the number system. In other words,
9 Frequently Used Number Systems Ioan Despi AMTH140 4 of 21 We note that a 0,..., a n and b 1, b 2... are all just digits, coming from a set of p elementary digits for the number system. In other words, If base number is p, we use p elementary digits to represent all numbers in base p.
10 Frequently Used Number Systems Ioan Despi AMTH140 4 of 21 We note that a 0,..., a n and b 1, b 2... are all just digits, coming from a set of p elementary digits for the number system. In other words, If base number is p, we use p elementary digits to represent all numbers in base p. The elementary digits represent values from 0 to (p 1), respectively.
11 Frequently Used Number Systems Ioan Despi AMTH140 4 of 21 We note that a 0,..., a n and b 1, b 2... are all just digits, coming from a set of p elementary digits for the number system. In other words, If base number is p, we use p elementary digits to represent all numbers in base p. The elementary digits represent values from 0 to (p 1), respectively.
12 Frequently Used Number Systems Ioan Despi AMTH140 4 of 21 We note that a 0,..., a n and b 1, b 2... are all just digits, coming from a set of p elementary digits for the number system. In other words, If base number is p, we use p elementary digits to represent all numbers in base p. The elementary digits represent values from 0 to (p 1), respectively. Example 1. When base number p = 2, the system is called the binary number system. Two digits are needed, and these two digits are chosen to be 0 and 1, consistently representing the values of 0 and 1 respectively. For example, (1011.1) 2 = = (11.5) 10, i.e., is equal to 11.5 in decimal. Note that the brackets can be dropped.
13 Examples Ioan Despi AMTH140 5 of 21 Example 2. When base p = 8, the system is called the octal number system. Naturally, we still conveniently use the existing first eight digits in the decimal system, i.e., 0, 1, 2,..., 7, to denote the corresponding values for the new number system. For example, = = , i.e., represents the decimal number
14 Examples Ioan Despi AMTH140 5 of 21 Example 2. When base p = 8, the system is called the octal number system. Naturally, we still conveniently use the existing first eight digits in the decimal system, i.e., 0, 1, 2,..., 7, to denote the corresponding values for the new number system. For example, = = , i.e., represents the decimal number Example 3. When p = 10, the system is our standard decimal number system. Thus, for instance, (36429) 10 is just
15 Examples Ioan Despi AMTH140 6 of 21 Example 4. When p = 16, the system is called the hexadecimal number system. 16 digits are needed in this system, representing values from 0 to 15. We shall still make use of the ten digits 0, 1,..., 9 from the decimal number system, and use A, B, C, D, E, F as symbols to make up for the rest of the digits. Hence the digits for the hexadecimal system are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F representing (in base 10) 0 to 15, respectively. For example, A7F 16 = A F 16 0 = = 2687
16 Conversion to Numbers of Different Bases Ioan Despi AMTH140 7 of 21 Recall that for all integers a and b (a, b Z), we say that b divides a if and only if (iff) there exists an integer c ( c Z) such that a = bc. b a iff c Z, a = bc
17 Conversion to Numbers of Different Bases Ioan Despi AMTH140 7 of 21 Recall that for all integers a and b (a, b Z), we say that b divides a if and only if (iff) there exists an integer c ( c Z) such that a = bc. b a iff c Z, a = bc When b divides a, we call b a divisor of a.
18 Conversion to Numbers of Different Bases Ioan Despi AMTH140 7 of 21 Recall that for all integers a and b (a, b Z), we say that b divides a if and only if (iff) there exists an integer c ( c Z) such that a = bc. b a iff c Z, a = bc When b divides a, we call b a divisor of a. If b divides a (a = bc), then so does c so divisors always come in pairs.
19 Conversion to Numbers of Different Bases Ioan Despi AMTH140 7 of 21 Recall that for all integers a and b (a, b Z), we say that b divides a if and only if (iff) there exists an integer c ( c Z) such that a = bc. b a iff c Z, a = bc When b divides a, we call b a divisor of a. If b divides a (a = bc), then so does c so divisors always come in pairs. If a and b are positive integers, then if a divides b and b divides a, then a = b. (prove it!)
20 Conversion to Numbers of Different Bases Conversion of a number in one base system to another base system can be done from the definitions, e.g., (45) 6 = = 29 (first converted to the decimal system) = = = 4 (4 + 1) = = (131) 4 Ioan Despi AMTH140 8 of 21
21 Conversion to Numbers of Different Bases Conversion of a number in one base system to another base system can be done from the definitions, e.g., (45) 6 = = 29 (first converted to the decimal system) = = = 4 (4 + 1) = = (131) 4 What we were doing was find how many 4 s were hiding in 29 and it involved a form of repeated division. Ioan Despi AMTH140 8 of 21
22 Conversion to Numbers of Different Bases Conversion of a number in one base system to another base system can be done from the definitions, e.g., (45) 6 = = 29 (first converted to the decimal system) = = = 4 (4 + 1) = = (131) 4 What we were doing was find how many 4 s were hiding in 29 and it involved a form of repeated division. However, the procedure in the above example doesn t seem systematic and may be cumbersome to perform in general to say the least. Ioan Despi AMTH140 8 of 21
23 Conversion to Numbers of Different Bases Conversion of a number in one base system to another base system can be done from the definitions, e.g., (45) 6 = = 29 (first converted to the decimal system) = = = 4 (4 + 1) = = (131) 4 What we were doing was find how many 4 s were hiding in 29 and it involved a form of repeated division. However, the procedure in the above example doesn t seem systematic and may be cumbersome to perform in general to say the least. Hence we shall introduce below an algorithm, involving repeated division, to convert numbers from one base system to another. Ioan Despi AMTH140 8 of 21
24 Conversion to Numbers of Different Bases Ioan Despi AMTH140 9 of 21 Theorem (Remainder Theorem) For any integers p and q with p > 0, there exist integers s and r such that q = sp + r, 0 r < p (1)
25 Conversion to Numbers of Different Bases Ioan Despi AMTH140 9 of 21 Theorem (Remainder Theorem) For any integers p and q with p > 0, there exist integers s and r such that q = sp + r, 0 r < p (1) Proof. Let s start by proving that if s and r exist, then they are unique. So we know that q = sp + r, and 0 r < p for some s and r in Z. Then q p = s + r p and 0 r p < 1 Therefore, the definition of flooring implies that s = q p. Thus, if (1) holds, then s is uniquely determined as the floor of q/p. And from (1) it follows that r = q sp is also unique.
26 Conversion to Numbers of Different Bases Proof. Now we prove the existence of s and r satyisfying (1). We set q q s := and r := q sp := q p p p Obviously, s and r, so defined, are integers and they satisfy equation (1). It only remains to show that r satisfies the inequalities in (1). We know that q q q p p < 1 + p Multiplying both inequalities by p > 0, we get q q p q < p + p p p then by multiplying the first inequality by 1 we get q 0 q p := r, so 0 r p and from the second one we get q r := q p < p, so r < p p Ioan Despi AMTH of 21
27 Repeated Division Algorithm The Remainder Theorem is used in the repeated division algorithm to convert a decimal integer a N to a number of base p. It produces the digits r 0, r 1, r 2,... of the number a in base p. Ioan Despi AMTH of 21
28 Repeated Division Algorithm The Remainder Theorem is used in the repeated division algorithm to convert a decimal integer a N to a number of base p. It produces the digits r 0, r 1, r 2,... of the number a in base p. Ioan Despi AMTH of 21
29 Ioan Despi AMTH of 21 Repeated Division Algorithm The Remainder Theorem is used in the repeated division algorithm to convert a decimal integer a N to a number of base p. It produces the digits r 0, r 1, r 2,... of the number a in base p. Algorithm 3 Repeated Division by p (1) a = pq 0 + r 0, 0 r 0 < p, Remainder Theorem on a (2) q 0 = pq 1 + r 1, 0 r 1 < p, R.T. on q 0 (3) q 1 = pq 2 + r 2, 0 r 2 < p, R.T. on q 1.. (m + 1) q m 1 = pq m + r m, 0 r m < p, and q m = 0 Termination Condition is the representation in the base p number sys- Then a = (r m r m 1 r 1 r 0 ) p tem.
30 Ioan Despi AMTH of 21 Conversion of the decimal part of a number To convert a non-integer number b with 0 < b < 1 to a base p number system, the procedure is as follows Algorithm 4 Conversion of the decimal part of a number (1) bp = s 1 + t 1, s 1 = bp, s 1 is the integer part of bp (2) t 1 p = s 2 + t 2, s 2 = t 1 p, s 2 is the integer part of t 1 p (3) t 2 p = s 3 + t 3, s 3 = t 2 p, s 3 is the integer part of t 2 p.. (m) t m 1 p = s m + t m, s m = t m 1 p, procedure terminates at m if t m = 0.. otherwise continue forward is the representation in the base p number sys- Then b = (0.s 1 s 2 s m ) p tem.
31 Conversion Ioan Despi AMTH of 21 We note that while
32 Conversion We note that while the algorithm for converting an integer into a different base will always terminate in finite number of steps, Ioan Despi AMTH of 21
33 Conversion Ioan Despi AMTH of 21 We note that while the algorithm for converting an integer into a different base will always terminate in finite number of steps, the conversion of a non-integer may not terminate at all, although we usually can still stop in finite number of steps after observing a repeating pattern.
34 Conversion Ioan Despi AMTH of 21 We note that while the algorithm for converting an integer into a different base will always terminate in finite number of steps, the conversion of a non-integer may not terminate at all, although we usually can still stop in finite number of steps after observing a repeating pattern. The proofs of the above two algorithms are essentially no more than the re-interpretation of a = r m p m + r m 1 p m r 1 p + r 0, b = s 1 p + s 2 p s m p m +.
35 Examples Ioan Despi AMTH of 21 Example 5. Convert 45 to base 4. (1) 45 = (2) 11 = (3) 2 = Hence 45 = (231) 4. stop now because it s become 0 That is, we first divide 45 by 4.
36 Examples Ioan Despi AMTH of 21 Example 5. Convert 45 to base 4. (1) 45 = (2) 11 = (3) 2 = Hence 45 = (231) 4. stop now because it s become 0 That is, we first divide 45 by 4. The quotient 11 is then divided by 4, with the new quotient 2 being again divided by 4.
37 Examples Ioan Despi AMTH of 21 Example 5. Convert 45 to base 4. (1) 45 = (2) 11 = (3) 2 = Hence 45 = (231) 4. stop now because it s become 0 That is, we first divide 45 by 4. The quotient 11 is then divided by 4, with the new quotient 2 being again divided by 4. This repeating process is then stopped because the latest quotient has reached 0.
38 Examples Ioan Despi AMTH of 21 Example 5. Convert 45 to base 4. (1) 45 = (2) 11 = (3) 2 = Hence 45 = (231) 4. stop now because it s become 0 That is, we first divide 45 by 4. The quotient 11 is then divided by 4, with the new quotient 2 being again divided by 4. This repeating process is then stopped because the latest quotient has reached 0. The remainders obtained at these division steps, when listed in the reverse order, finally give the desired representation of 45 in base 4.
39 Examples Ioan Despi AMTH of 21 Example 5. Convert 45 to base 4. (1) 45 = (2) 11 = (3) 2 = Hence 45 = (231) 4. stop now because it s become 0 That is, we first divide 45 by 4. The quotient 11 is then divided by 4, with the new quotient 2 being again divided by 4. This repeating process is then stopped because the latest quotient has reached 0. The remainders obtained at these division steps, when listed in the reverse order, finally give the desired representation of 45 in base 4. Probe: (231) 4 = = = 45.
40 Examples Ioan Despi AMTH of 21 Example 6. Convert 123 into binary. ( ) = successive divisions end when quotient becomes zero Hence 123= We illustrate here an alternative approach to number conversions.
41 Examples Ioan Despi AMTH of 21 Example 6. Convert 123 into binary. ( ) = successive divisions end when quotient becomes zero Hence 123= We illustrate here an alternative approach to number conversions. It uses exactly the same algorithm as before, and the only difference lies in the organisation or presentation of the derivation.
42 Examples Ioan Despi AMTH of 21 Example 6. Convert 123 into binary. ( ) = successive divisions end when quotient becomes zero Hence 123= We illustrate here an alternative approach to number conversions. It uses exactly the same algorithm as before, and the only difference lies in the organisation or presentation of the derivation. Obviously you don t have to bother with this approach if you don t like it: the approach adopted in the previous example is already neat enough.
43 Examples Ioan Despi AMTH of 21 Example 7. Convert 0.1 to base 2. (1) = (2) = (3) = Hence 0.1 = ( ) 2 = ( ) 2. (4) = (5) = is repeating the outcome of step (1) integer part/portion
44 Blockwise Conversion Ioan Despi AMTH of 21 If p = q k and p, q and k are all positive integers, then the conversion of (a n... a 1 a 0 b 1 b 2...) p into base q can be done digitwise, with each digit a i or b j being mapped to a block of k consecutive digits in the base q number system.
45 Blockwise Conversion Ioan Despi AMTH of 21 If p = q k and p, q and k are all positive integers, then the conversion of (a n... a 1 a 0 b 1 b 2...) p into base q can be done digitwise, with each digit a i or b j being mapped to a block of k consecutive digits in the base q number system. This is called the blockwise conversion.
46 Examples Ioan Despi AMTH of 21 Example 8. Convert (7A.1) 16 into binary format. Solution. Since 16 = 2 4, 1 digit in hexademical will correspond to exactly 4 digits in binary. Since 7 16 = , A = , 1 16 = , make sure they are all full 4-bit blocks we have (7A 1) 16 = ( ) 2.
47 Examples Ioan Despi AMTH of 21 Example 9. Convert the binary number ( ) 2 to base 8. Solution. Since 8 = 2 3 implies every 3 consecutive digits, going left or right starting from the fractional point, in binary will correspond to a single digit in base 8. By grouping consecutive digits into blocks of 3, appending proper 0 s if necessary, we have ( ) 2 = ( ) 2 = (172.04) 8 because = 1 8, = 7 8, = 2 8 and = 4 8.
48 Examples Ioan Despi AMTH of 21 Basic direct operations such as additions and subtractions can be performed in any number systems in a similar way to what we would do in the standard decimal system. Example Find i.e., = = 7 = (12) 5
49 Examples Ioan Despi AMTH of 21 Basic direct operations such as additions and subtractions can be performed in any number systems in a similar way to what we would do in the standard decimal system. Example Find i.e., = = 7 = (12) 5 We note that the addition is performed digitwise from the rightmost digit towards the left.
50 Examples Ioan Despi AMTH of 21 Basic direct operations such as additions and subtractions can be performed in any number systems in a similar way to what we would do in the standard decimal system. Example Find i.e., = = 7 = (12) 5 We note that the addition is performed digitwise from the rightmost digit towards the left. On the rightmost digit, = 7 10 = (12) 5 gives the resulting digit 2 and the carry 1 to the next digit on the left.
51 Examples Ioan Despi AMTH of 21 Basic direct operations such as additions and subtractions can be performed in any number systems in a similar way to what we would do in the standard decimal system. Example Find i.e., = = 7 = (12) 5 We note that the addition is performed digitwise from the rightmost digit towards the left. On the rightmost digit, = 7 10 = (12) 5 gives the resulting digit 2 and the carry 1 to the next digit on the left. On the 2nd rightmost digit, we have thus (carry) 1 5 = 6 10 = (11) 5, resulting to digit 1 plus a new carry 1.
52 Examples Ioan Despi AMTH of 21 Basic direct operations such as additions and subtractions can be performed in any number systems in a similar way to what we would do in the standard decimal system. Example Find i.e., = = 7 = (12) 5 We note that the addition is performed digitwise from the rightmost digit towards the left. On the rightmost digit, = 7 10 = (12) 5 gives the resulting digit 2 and the carry 1 to the next digit on the left. On the 2nd rightmost digit, we have thus (carry) 1 5 = 6 10 = (11) 5, resulting to digit 1 plus a new carry 1. As the last step, we then have = 5 10 = (10) 5.
53 Examples Ioan Despi AMTH of 21 Basic direct operations such as additions and subtractions can be performed in any number systems in a similar way to what we would do in the standard decimal system. Example Find i.e., = = 7 = (12) 5 We note that the addition is performed digitwise from the rightmost digit towards the left. On the rightmost digit, = 7 10 = (12) 5 gives the resulting digit 2 and the carry 1 to the next digit on the left. On the 2nd rightmost digit, we have thus (carry) 1 5 = 6 10 = (11) 5, resulting to digit 1 plus a new carry 1. As the last step, we then have = 5 10 = (10) 5. This completes our addition.
54 Ioan Despi AMTH of 21 Example 11. Find Examples Hence = The subtraction is also performed digitwise from the rightmost digit towards the left.
55 Ioan Despi AMTH of 21 Example 11. Find Examples Hence = The subtraction is also performed digitwise from the rightmost digit towards the left. On the rightmost digit, = 0 2 gives the resulting digit 0.
56 Ioan Despi AMTH of 21 Example 11. Find Examples Hence = The subtraction is also performed digitwise from the rightmost digit towards the left. On the rightmost digit, = 0 2 gives the resulting digit 0. On the 2nd rightmost digit, = 1 2 gives the result 1.
57 Ioan Despi AMTH of 21 Example 11. Find Examples Hence = The subtraction is also performed digitwise from the rightmost digit towards the left. On the rightmost digit, = 0 2 gives the resulting digit 0. On the 2nd rightmost digit, = 1 2 gives the result 1. On the 3rd rightmost digit, we borrow 1 from the digit on the left; actually we have to borrow 1 further on the left because the digit on the immediate left is 0 and therefore can t be borrowed.
58 Ioan Despi AMTH of 21 Example 11. Find Examples Hence = The subtraction is also performed digitwise from the rightmost digit towards the left. On the rightmost digit, = 0 2 gives the resulting digit 0. On the 2nd rightmost digit, = 1 2 gives the result 1. On the 3rd rightmost digit, we borrow 1 from the digit on the left; actually we have to borrow 1 further on the left because the digit on the immediate left is 0 and therefore can t be borrowed. The subtraction on the 3rd rightmost digit thus becomes (10) = = 1 10 = 1 2.
59 Ioan Despi AMTH of 21 Example 11. Find Examples Hence = The subtraction is also performed digitwise from the rightmost digit towards the left. On the rightmost digit, = 0 2 gives the resulting digit 0. On the 2nd rightmost digit, = 1 2 gives the result 1. On the 3rd rightmost digit, we borrow 1 from the digit on the left; actually we have to borrow 1 further on the left because the digit on the immediate left is 0 and therefore can t be borrowed. The subtraction on the 3rd rightmost digit thus becomes (10) = = 1 10 = 1 2. Finally, the 4th rightmost digit, after borrowed 1 from the left and being borrowed 1 from the right, is left with the remaining 1.
60 The subtraction is also performed digitwise from the rightmost digit towards the left. On the rightmost digit, = 0 2 gives the resulting digit 0. On the 2nd rightmost digit, = 1 2 gives the result 1. On the 3rd rightmost digit, we borrow 1 from the digit on the left; actually we have to borrow 1 further on the left because the digit on the immediate left is 0 and therefore can t be borrowed. The subtraction on the 3rd rightmost digit thus becomes (10) = = 1 10 = 1 2. Finally, the 4th rightmost digit, after borrowed 1 from the left and being borrowed 1 from the right, is left with the remaining 1. Ioan Despi This AMTH140 completes the subtraction. 21 of 21 Example 11. Find Examples Hence =
MATH Dr. Halimah Alshehri Dr. Halimah Alshehri
MATH 1101 haalshehri@ksu.edu.sa 1 Introduction To Number Systems First Section: Binary System Second Section: Octal Number System Third Section: Hexadecimal System 2 Binary System 3 Binary System The binary
Conversions between Decimal and Binary
Conversions between Decimal and Binary Binary to Decimal Technique - use the definition of a number in a positional number system with base 2 - evaluate the definition formula ( the formula ) using decimal
Numbering Systems. Contents: Binary & Decimal. Converting From: B D, D B. Arithmetic operation on Binary.
Numbering Systems Contents: Binary & Decimal. Converting From: B D, D B. Arithmetic operation on Binary. Addition & Subtraction using Octal & Hexadecimal 2 s Complement, Subtraction Using 2 s Complement.
ENGIN 112 Intro to Electrical and Computer Engineering
ENGIN 112 Intro to Electrical and Computer Engineering Lecture 2 Number Systems Russell Tessier KEB 309 G tessier@ecs.umass.edu Overview The design of computers It all starts with numbers Building circuits
12/31/2010. Digital Operations and Computations Course Notes. 01-Number Systems Text: Unit 1. Overview. What is a Digital System?
Digital Operations and Computations Course Notes 0-Number Systems Text: Unit Winter 20 Professor H. Louie Department of Electrical & Computer Engineering Seattle University ECEGR/ISSC 20 Digital Operations
Week No. 06: Numbering Systems
Week No. 06: Numbering Systems Numbering System: A numbering system defined as A set of values used to represent quantity. OR A number system is a term used for a set of different symbols or digits, which
3 The fundamentals: Algorithms, the integers, and matrices
3 The fundamentals: Algorithms, the integers, and matrices 3.4 The integers and division This section introduces the basics of number theory number theory is the part of mathematics involving integers
Menu. Review of Number Systems EEL3701 EEL3701. Math. Review of number systems >Binary math >Signed number systems
Menu Review of number systems >Binary math >Signed number systems Look into my... 1 Our decimal (base 10 or radix 10) number system is positional. Ex: 9437 10 = 9x10 3 + 4x10 2 + 3x10 1 + 7x10 0 We have
Recurrence Relations
Recurrence Relations Simplest Case of General Solutions Ioan Despi despi@turing.une.edu.au University of New England September 13, 2013 Outline Ioan Despi AMTH140 2 of 1 Simplest Case of General Solutions
14:332:231 DIGITAL LOGIC DESIGN. Why Binary Number System?
:33:3 DIGITAL LOGIC DESIGN Ivan Marsic, Rutgers University Electrical & Computer Engineering Fall 3 Lecture #: Binary Number System Complement Number Representation X Y Why Binary Number System? Because
NUMBERS AND CODES CHAPTER Numbers
CHAPTER 2 NUMBERS AND CODES 2.1 Numbers When a number such as 101 is given, it is impossible to determine its numerical value. Some may say it is five. Others may say it is one hundred and one. Could it
We say that the base of the decimal number system is ten, represented by the symbol
Introduction to counting and positional notation. In the decimal number system, a typical number, N, looks like... d 3 d 2 d 1 d 0.d -1 d -2 d -3... [N1] where the ellipsis at each end indicates that there
EE260: Digital Design, Spring n Digital Computers. n Number Systems. n Representations. n Conversions. n Arithmetic Operations.
EE 260: Introduction to Digital Design Number Systems Yao Zheng Department of Electrical Engineering University of Hawaiʻi at Mānoa Overview n Digital Computers n Number Systems n Representations n Conversions
4 Number Theory and Cryptography
4 Number Theory and Cryptography 4.1 Divisibility and Modular Arithmetic This section introduces the basics of number theory number theory is the part of mathematics involving integers and their properties.
A group of figures, representing a number, is called a numeral. Numbers are divided into the following types.
1. Number System Quantitative Aptitude deals mainly with the different topics in Arithmetic, which is the science which deals with the relations of numbers to one another. It includes all the methods that
Why digital? Overview. Number Systems. Binary to Decimal conversion
Why digital? Overview It has the following advantages over analog. It can be processed and transmitted efficiently and reliably. It can be stored and retrieved with greater accuracy. Noise level does not
Mathematical Induction
Mathematical Induction Representation of integers Mathematical Induction Reading (Epp s textbook) 5.1 5.3 1 Representations of Integers Let b be a positive integer greater than 1. Then if n is a positive
Chapter 2 (Part 3): The Fundamentals: Algorithms, the Integers & Matrices. Integers & Algorithms (2.5)
CSE 54 Discrete Mathematics & Chapter 2 (Part 3): The Fundamentals: Algorithms, the Integers & Matrices Integers & Algorithms (Section 2.5) by Kenneth H. Rosen, Discrete Mathematics & its Applications,
CSE 241 Digital Systems Spring 2013
CSE 241 Digital Systems Spring 2013 Instructor: Prof. Kui Ren Department of Computer Science and Engineering Lecture slides modified from many online resources and used solely for the educational purpose.
CHAPTER 2 NUMBER SYSTEMS
CHAPTER 2 NUMBER SYSTEMS The Decimal Number System : We begin our study of the number systems with the familiar decimal number system. The decimal system contains ten unique symbol 0, 1, 2, 3, 4, 5, 6,
Binary addition example worked out
Binary addition example worked out Some terms are given here Exercise: what are these numbers equivalent to in decimal? The initial carry in is implicitly 0 1 1 1 0 (Carries) 1 0 1 1 (Augend) + 1 1 1 0
CSEN102 Introduction to Computer Science
CSEN102 Introduction to Computer Science Lecture 7: Representing Information I Prof. Dr. Slim Abdennadher Dr. Mohammed Salem, slim.abdennadher@guc.edu.eg, mohammed.salem@guc.edu.eg German University Cairo,
ENGIN 112 Intro to Electrical and Computer Engineering
ENGIN 112 Intro to Electrical and Computer Engineering Lecture 3 More Number Systems Overview Hexadecimal numbers Related to binary and octal numbers Conversion between hexadecimal, octal and binary Value
REVIEW Chapter 1 The Real Number System
REVIEW Chapter The Real Number System In class work: Complete all statements. Solve all exercises. (Section.4) A set is a collection of objects (elements). The Set of Natural Numbers N N = {,,, 4, 5, }
L1 2.1 Long Division of Polynomials and The Remainder Theorem Lesson MHF4U Jensen
L1 2.1 Long Division of Polynomials and The Remainder Theorem Lesson MHF4U Jensen In this section you will apply the method of long division to divide a polynomial by a binomial. You will also learn to
EDULABZ INTERNATIONAL NUMBER SYSTEM
NUMBER SYSTEM 1. Find the product of the place value of 8 and the face value of 7 in the number 7801. Ans. Place value of 8 in 7801 = 800, Face value of 7 in 7801 = 7 Required product = 800 7 = 00. How
Counting in Different Number Systems
Counting in Different Number Systems Base 1 (Decimal) is important because that is the base that we first learn in our culture. Base 2 (Binary) is important because that is the base used for computer codes
4. Number Theory (Part 2)
4. Number Theory (Part 2) Terence Sim Mathematics is the queen of the sciences and number theory is the queen of mathematics. Reading Sections 4.8, 5.2 5.4 of Epp. Carl Friedrich Gauss, 1777 1855 4.3.
L1 2.1 Long Division of Polynomials and The Remainder Theorem Lesson MHF4U Jensen
L1 2.1 Long Division of Polynomials and The Remainder Theorem Lesson MHF4U Jensen In this section you will apply the method of long division to divide a polynomial by a binomial. You will also learn to
Chapter 5. Number Theory. 5.1 Base b representations
Chapter 5 Number Theory The material in this chapter offers a small glimpse of why a lot of facts that you ve probably nown and used for a long time are true. It also offers some exposure to generalization,
A number that can be written as, where p and q are integers and q Number.
RATIONAL NUMBERS 1.1 Definition of Rational Numbers: What are rational numbers? A number that can be written as, where p and q are integers and q Number. 0, is known as Rational Example:, 12, -18 etc.
CISC 1400 Discrete Structures
CISC 1400 Discrete Structures Chapter 2 Sequences What is Sequence? A sequence is an ordered list of objects or elements. For example, 1, 2, 3, 4, 5, 6, 7, 8 Each object/element is called a term. 1 st
SECTION 2.3: LONG AND SYNTHETIC POLYNOMIAL DIVISION
2.25 SECTION 2.3: LONG AND SYNTHETIC POLYNOMIAL DIVISION PART A: LONG DIVISION Ancient Example with Integers 2 4 9 8 1 In general: dividend, f divisor, d We can say: 9 4 = 2 + 1 4 By multiplying both sides
ECE380 Digital Logic. Positional representation
ECE380 Digital Logic Number Representation and Arithmetic Circuits: Number Representation and Unsigned Addition Dr. D. J. Jackson Lecture 16-1 Positional representation First consider integers Begin with
Discrete Mathematics GCD, LCM, RSA Algorithm
Discrete Mathematics GCD, LCM, RSA Algorithm Abdul Hameed http://informationtechnology.pk/pucit abdul.hameed@pucit.edu.pk Lecture 16 Greatest Common Divisor 2 Greatest common divisor The greatest common
Apply basic properties of real and complex numbers in constructing mathematical arguments (e.g., if a < b and c < 0, then ac > bc)
ALGEBRA (SMR Domain ) Algebraic Structures (SMR.) Skill a. Apply basic properties of real and complex numbers in constructing mathematical arguments (e.g., if a < b and c < 0, then ac > bc) Basic Properties
Math Review. for the Quantitative Reasoning measure of the GRE General Test
Math Review for the Quantitative Reasoning measure of the GRE General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important for solving
= 1 2x. x 2 a ) 0 (mod p n ), (x 2 + 2a + a2. x a ) 2
8. p-adic numbers 8.1. Motivation: Solving x 2 a (mod p n ). Take an odd prime p, and ( an) integer a coprime to p. Then, as we know, x 2 a (mod p) has a solution x Z iff = 1. In this case we can suppose
CP Algebra 2. Summer Packet. Name:
CP Algebra Summer Packet 018 Name: Objectives for CP Algebra Summer Packet 018 I. Number Sense and Numerical Operations (Problems: 1 to 4) Use the Order of Operations to evaluate expressions. (p. 6) Evaluate
Chapter 1: Fundamentals of Algebra Lecture notes Math 1010
Section 1.1: The Real Number System Definition of set and subset A set is a collection of objects and its objects are called members. If all the members of a set A are also members of a set B, then A is
Four Important Number Systems
Four Important Number Systems System Why? Remarks Decimal Base 10: (10 fingers) Most used system Binary Base 2: On/Off systems 3-4 times more digits than decimal Octal Base 8: Shorthand notation for working
SEVENTH EDITION and EXPANDED SEVENTH EDITION
SEVENTH EDITION and EXPANDED SEVENTH EDITION Slide 5-1 Chapter 5 Number Theory and the Real Number System 5.1 Number Theory Number Theory The study of numbers and their properties. The numbers we use to
ALGEBRA 1. Interactive Notebook Chapter 2: Linear Equations
ALGEBRA 1 Interactive Notebook Chapter 2: Linear Equations 1 TO WRITE AN EQUATION: 1. Identify the unknown (the variable which you are looking to find) 2. Write the sentence as an equation 3. Look for
Decimal Addition: Remember to line up the decimals before adding. Bring the decimal straight down in your answer.
Summer Packet th into 6 th grade Name Addition Find the sum of the two numbers in each problem. Show all work.. 62 2. 20. 726 + + 2 + 26 + 6 6 Decimal Addition: Remember to line up the decimals before
Section 3-4: Least Common Multiple and Greatest Common Factor
Section -: Fraction Terminology Identify the following as proper fractions, improper fractions, or mixed numbers:, proper fraction;,, improper fractions;, mixed number. Write the following in decimal notation:,,.
14:332:231 DIGITAL LOGIC DESIGN. 2 s-complement Representation
4:332:23 DIGITAL LOGIC DESIGN Ivan Marsic, Rutgers University Electrical & Computer Engineering Fall 203 Lecture #3: Addition, Subtraction, Multiplication, and Division 2 s-complement Representation RECALL
Number Theory: Representations of Integers
Instructions: In-class exercises are meant to introduce you to a new topic and provide some practice with the new topic. Work in a team of up to 4 people to complete this exercise. You can work simultaneously
Chapter 1 CSCI
Chapter 1 CSCI-1510-003 What is a Number? An expression of a numerical quantity A mathematical quantity Many types: Natural Numbers Real Numbers Rational Numbers Irrational Numbers Complex Numbers Etc.
SEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION
CHAPTER 5 SEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION Copyright Cengage Learning. All rights reserved. SECTION 5.4 Strong Mathematical Induction and the Well-Ordering Principle for the Integers Copyright
ELEMENTARY NUMBER THEORY AND METHODS OF PROOF
CHAPTER 4 ELEMENTARY NUMBER THEORY AND METHODS OF PROOF Copyright Cengage Learning. All rights reserved. SECTION 4.5 Direct Proof and Counterexample V: Floor and Ceiling Copyright Cengage Learning. All
CSE 20: Discrete Mathematics
Spring 2018 Summary So far: Today: Logic and proofs Divisibility, modular arithmetics Number Systems More logic definitions and proofs Reading: All of Chap. 1 + Chap 4.1, 4.2. Divisibility P = 5 divides
1 Numbers. exponential functions, such as x 7! a x ; where a; x 2 R; trigonometric functions, such as x 7! sin x; where x 2 R; ffiffi x ; where x 0:
Numbers In this book we study the properties of real functions defined on intervals of the real line (possibly the whole real line) and whose image also lies on the real line. In other words, they map
You separate binary numbers into columns in a similar fashion. 2 5 = 32
RSA Encryption 2 At the end of Part I of this article, we stated that RSA encryption works because it s impractical to factor n, which determines P 1 and P 2, which determines our private key, d, which
Chapter 9 - Number Systems
Chapter 9 - Number Systems Luis Tarrataca luis.tarrataca@gmail.com CEFET-RJ L. Tarrataca Chapter 9 - Number Systems 1 / 50 1 Motivation 2 Positional Number Systems 3 Binary System L. Tarrataca Chapter
Chapter 2 INTEGERS. There will be NO CALCULATORS used for this unit!
Chapter 2 INTEGERS There will be NO CALCULATORS used for this unit! 2.2 What are integers? 1. Positives 2. Negatives 3. 0 4. Whole Numbers They are not 1. Not Fractions 2. Not Decimals What Do You Know?!
SEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION
CHAPTER 5 SEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION One of the most important tasks of mathematics is to discover and characterize regular patterns, such as those associated with processes that
Hakim Weatherspoon CS 3410 Computer Science Cornell University
Hakim Weatherspoon CS 3410 Computer Science Cornell University The slides are the product of many rounds of teaching CS 3410 by Professors Weatherspoon, Bala, Bracy, and Sirer. memory inst 32 register
ABE Math Review Package
P a g e ABE Math Review Package This material is intended as a review of skills you once learned and wish to review before your assessment. Before studying Algebra, you should be familiar with all of the
Math From Scratch Lesson 29: Decimal Representation
Math From Scratch Lesson 29: Decimal Representation W. Blaine Dowler January, 203 Contents Introducing Decimals 2 Finite Decimals 3 2. 0................................... 3 2.2 2....................................
Quantitative Aptitude
WWW.UPSCMANTRA.COM Quantitative Aptitude Concept 1 1. Number System 2. HCF and LCM 2011 Prelims Paper II NUMBER SYSTEM 2 NUMBER SYSTEM In Hindu Arabic System, we use ten symbols 0, 1, 2, 3, 4, 5, 6, 7,
MHF4U Unit 2 Polynomial Equation and Inequalities
MHF4U Unit 2 Polynomial Equation and Inequalities Section Pages Questions Prereq Skills 82-83 # 1ac, 2ace, 3adf, 4, 5, 6ace, 7ac, 8ace, 9ac 2.1 91 93 #1, 2, 3bdf, 4ac, 5, 6, 7ab, 8c, 9ad, 10, 12, 15a,
Expressions, Equations and Inequalities Guided Notes
Expressions, Equations and Inequalities Guided Notes Standards: Alg1.M.A.SSE.A.01a - The Highly Proficient student can explain the context of different parts of a formula presented as a complicated expression.
Prepared by Sa diyya Hendrickson. Package Summary
Introduction Prepared by Sa diyya Hendrickson Name: Date: Package Summary Defining Decimal Numbers Things to Remember Adding and Subtracting Decimals Multiplying Decimals Expressing Fractions as Decimals
Cool Results on Primes
Cool Results on Primes LA Math Circle (Advanced) January 24, 2016 Recall that last week we learned an algorithm that seemed to magically spit out greatest common divisors, but we weren t quite sure why
The Euclidean Algorithm
MATH 324 Summer 2006 Elementary Number Theory Notes on the Euclidean Algorithm Department of Mathematical and Statistical Sciences University of Alberta The Euclidean Algorithm Given two positive integers
Digital Systems Overview. Unit 1 Numbering Systems. Why Digital Systems? Levels of Design Abstraction. Dissecting Decimal Numbers
Unit Numbering Systems Fundamentals of Logic Design EE2369 Prof. Eric MacDonald Fall Semester 2003 Digital Systems Overview Digital Systems are Home PC XBOX or Playstation2 Cell phone Network router Data
COT 3100 Applications of Discrete Structures Dr. Michael P. Frank
University of Florida Dept. of Computer & Information Science & Engineering COT 3100 Applications of Discrete Structures Dr. Michael P. Frank Slides for a Course Based on the Text Discrete Mathematics
Linear Equations & Inequalities Definitions
Linear Equations & Inequalities Definitions Constants - a term that is only a number Example: 3; -6; -10.5 Coefficients - the number in front of a term Example: -3x 2, -3 is the coefficient Variable -
Number System conversions
Number System conversions Number Systems The system used to count discrete units is called number system. There are four systems of arithmetic which are often used in digital electronics. Decimal Number
Maths Book Part 1. By Abhishek Jain
Maths Book Part 1 By Abhishek Jain Topics: 1. Number System 2. HCF and LCM 3. Ratio & proportion 4. Average 5. Percentage 6. Profit & loss 7. Time, Speed & Distance 8. Time & Work Number System Understanding
Numbers and Arithmetic
Numbers and Arithmetic See: P&H Chapter 2.4 2.6, 3.2, C.5 C.6 Hakim Weatherspoon CS 3410, Spring 2013 Computer Science Cornell University Big Picture: Building a Processor memory inst register file alu
THE LOGIC OF COMPOUND STATEMENTS
CHAPTER 2 THE LOGIC OF COMPOUND STATEMENTS Copyright Cengage Learning. All rights reserved. SECTION 2.4 Application: Digital Logic Circuits Copyright Cengage Learning. All rights reserved. Application:
Sail into Summer with Math!
Sail into Summer with Math! For Students Entering Algebra 1 This summer math booklet was developed to provide students in kindergarten through the eighth grade an opportunity to review grade level math
UNIT 4 NOTES: PROPERTIES & EXPRESSIONS
UNIT 4 NOTES: PROPERTIES & EXPRESSIONS Vocabulary Mathematics: (from Greek mathema, knowledge, study, learning ) Is the study of quantity, structure, space, and change. Algebra: Is the branch of mathematics
Numbers. Dr Hammadi Nait-Charif. Senior Lecturer Bournemouth University United Kingdom
Numbers Dr Hammadi Nait-Charif Senior Lecturer Bournemouth University United Kingdom hncharif@bournemouth.ac.uk http://nccastaff.bmth.ac.uk/hncharif/mathscgs/maths.html Dr Hammadi Nait-Charif (BU, UK)
Maths Scheme of Work. Class: Year 10. Term: autumn 1: 32 lessons (24 hours) Number of lessons
Maths Scheme of Work Class: Year 10 Term: autumn 1: 32 lessons (24 hours) Number of lessons Topic and Learning objectives Work to be covered Method of differentiation and SMSC 11 OCR 1 Number Operations
Fundamentals of Mathematics I
Fundamentals of Mathematics I Kent State Department of Mathematical Sciences Fall 2008 Available at: http://www.math.kent.edu/ebooks/10031/book.pdf August 4, 2008 Contents 1 Arithmetic 2 1.1 Real Numbers......................................................
CSCI 255. S i g n e d N u m s / S h i f t i n g / A r i t h m e t i c O p s.
Ying.Yang 1.-1 CSCI 255 http://cs.furman.edu In mathematics, negative integers exists -> forced to do it binary In a binary string, the MSB is sacrificed as the signed bit 0 => Positive Value 1 => Negative
STEP Support Programme. Hints and Partial Solutions for Assignment 17
STEP Support Programme Hints and Partial Solutions for Assignment 7 Warm-up You need to be quite careful with these proofs to ensure that you are not assuming something that should not be assumed. For
0,..., r 1 = digits in radix r number system, that is 0 d i r 1 where m i n 1
RADIX r NUMBER SYSTEM Let (N) r be a radix r number in a positional weighting number system, then (N) r = d n 1 r n 1 + + d 0 r 0 d 1 r 1 + + d m r m where: r = radix d i = digit at position i, m i n 1
Part I, Number Systems. CS131 Mathematics for Computer Scientists II Note 1 INTEGERS
CS131 Part I, Number Systems CS131 Mathematics for Computer Scientists II Note 1 INTEGERS The set of all integers will be denoted by Z. So Z = {..., 2, 1, 0, 1, 2,...}. The decimal number system uses the
Notes on arithmetic. 1. Representation in base B
Notes on arithmetic The Babylonians that is to say, the people that inhabited what is now southern Iraq for reasons not entirely clear to us, ued base 60 in scientific calculation. This offers us an excuse
ARITHMETIC AND BASIC ALGEBRA
C O M P E T E N C Y ARITHMETIC AND BASIC ALGEBRA. Add, subtract, multiply and divide rational numbers expressed in various forms Addition can be indicated by the expressions sum, greater than, and, more
Long and Synthetic Division of Polynomials
Long and Synthetic Division of Polynomials Long and synthetic division are two ways to divide one polynomial (the dividend) by another polynomial (the divisor). These methods are useful when both polynomials
Common Core State Standards for Mathematics Bid Category
A Correlation of Pearson to the for Mathematics Bid Category 12-020-20 for Mathematics for Mathematics Ratios and Proportional Relationships 7.RP Analyze proportional relationships and use them to solve
Math 0310 Final Exam Review
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the linear equation and check the solution. 1) 13(x - 52) = 26 1) A) {26} B) {52} C) {50} D)
Mathematics Pacing. Instruction: 9/7/17 10/31/17 Assessment: 11/1/17 11/8/17. # STUDENT LEARNING OBJECTIVES NJSLS Resources
# STUDENT LEARNING OBJECTIVES NJSLS Resources 1 Describe real-world situations in which (positive and negative) rational numbers are combined, emphasizing rational numbers that combine to make 0. Represent
Math 016 Lessons Wimayra LUY
Math 016 Lessons Wimayra LUY wluy@ccp.edu MATH 016 Lessons LESSON 1 Natural Numbers The set of natural numbers is given by N = {0, 1, 2, 3, 4...}. Natural numbers are used for two main reasons: 1. counting,
REAL NUMBERS. Any positive integer a can be divided by another positive integer b in such a way that it leaves a remainder r that is smaller than b.
REAL NUMBERS Introduction Euclid s Division Algorithm Any positive integer a can be divided by another positive integer b in such a way that it leaves a remainder r that is smaller than b. Fundamental
An Invitation to Mathematics Prof. Sankaran Vishwanath Institute of Mathematical Science, Chennai. Unit - I Polynomials Lecture 1B Long Division
An Invitation to Mathematics Prof. Sankaran Vishwanath Institute of Mathematical Science, Chennai Unit - I Polynomials Lecture 1B Long Division (Refer Slide Time: 00:19) We have looked at three things
Division Algorithm B1 Introduction to the Division Algorithm (Procedure) quotient remainder
A Survey of Divisibility Page 1 SECTION B Division Algorithm By the end of this section you will be able to apply the division algorithm or procedure Our aim in this section is to show that for any given
K K.OA.2 1.OA.2 2.OA.1 3.OA.3 4.OA.3 5.NF.2 6.NS.1 7.NS.3 8.EE.8c
K.OA.2 1.OA.2 2.OA.1 3.OA.3 4.OA.3 5.NF.2 6.NS.1 7.NS.3 8.EE.8c Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to Solve word problems that
JUST THE MATHS UNIT NUMBER 1.4. ALGEBRA 4 (Logarithms) A.J.Hobson
JUST THE MATHS UNIT NUMBER 1.4 ALGEBRA 4 (Logarithms) by A.J.Hobson 1.4.1 Common logarithms 1.4.2 Logarithms in general 1.4.3 Useful Results 1.4.4 Properties of logarithms 1.4.5 Natural logarithms 1.4.6
Chapter 1. Pigeonhole Principle
Chapter 1. Pigeonhole Principle Prof. Tesler Math 184A Fall 2017 Prof. Tesler Ch. 1. Pigeonhole Principle Math 184A / Fall 2017 1 / 16 Pigeonhole principle https://commons.wikimedia.org/wiki/file:toomanypigeons.jpg
PRE-ALGEBRA SUMMARY WHOLE NUMBERS
PRE-ALGEBRA SUMMARY WHOLE NUMBERS Introduction to Whole Numbers and Place Value Digits Digits are the basic symbols of the system 0,,,, 4,, 6, 7, 8, and 9 are digits Place Value The value of a digit in
Number Systems. There are 10 kinds of people those that understand binary, those that don t, and those that expected this joke to be in base 2
Number Systems There are 10 kinds of people those that understand binary, those that don t, and those that expected this joke to be in base 2 A Closer Look at the Numbers We Use What is the difference
Numbers. 2.1 Integers. P(n) = n(n 4 5n 2 + 4) = n(n 2 1)(n 2 4) = (n 2)(n 1)n(n + 1)(n + 2); 120 =
2 Numbers 2.1 Integers You remember the definition of a prime number. On p. 7, we defined a prime number and formulated the Fundamental Theorem of Arithmetic. Numerous beautiful results can be presented
We say that a polynomial is in the standard form if it is written in the order of decreasing exponents of x. Operations on polynomials:
R.4 Polynomials in one variable A monomial: an algebraic expression of the form ax n, where a is a real number, x is a variable and n is a nonnegative integer. : x,, 7 A binomial is the sum (or difference)
Radiological Control Technician Training Fundamental Academic Training Study Guide Phase I
Module 1.01 Basic Mathematics and Algebra Part 4 of 9 Radiological Control Technician Training Fundamental Academic Training Phase I Coordinated and Conducted for the Office of Health, Safety and Security
Twitter: @Owen134866 www.mathsfreeresourcelibrary.com Prior Knowledge Check 1) Simplify: a) 3x 2 5x 5 b) 5x3 y 2 15x 7 2) Factorise: a) x 2 2x 24 b) 3x 2 17x + 20 15x 2 y 3 3) Use long division to calculate: