SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Numbers

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1 SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mathematics Numbers. Factors 2. Multiples 3. Prime and Composite Numbers 4. Modular Arithmetic 5. Boolean Algebra 6. Modulo 2 Matrix Arithmetic 7. Number Systems Dr Calum Macdonald

2 Numbers In this section we will briefly discuss basic number theory along with some number systems.. Factors A factor of a given number is a number that divides exactly into that number. Example (i) The numbers, 2, 3, 4, 6 and 2 are all factors of 2 as they each divide exactly into 2. (ii) The numbers, 2, 4, 5, and 2 are all factors of 2 as they each divide exactly into 2. (iii) The number 3 is not a factor of 2 as 3 does not divide exactly into Multiples A multiple is the result of multiplying a number by an integer. Example 2 (i) 2 is a multiple of 6 since (ii) 6 is a multiple of 6 since 6 6 (iii) 22 is not a multiple of Prime and Composite Numbers A prime number is a positive integer greater than that has exactly two factors these factors are the number itself and. In other words, a prime number can be divided only by and by itself. Example 3 (i) 7 is a prime number, since the only factors it has are and 7. (ii) 3 is a prime number, since the only factors it has are and 3. The prime numbers less than 25 are 2, 3, 5, 7,, 3, 7, 9, 23. Note: and are not prime numbers. Note: 2 is the only even number that is a prime number. A composite number is a positive integer which is not prime, i.e., it has at least one more factor than and itself. The composite numbers less than 25 are 4, 6, 8, 9,, 2, 4, 5, 6, 8, 2, 2, 22, 24. 2

3 a) Sieve of Eratosthenes Eratosthenes of Cyrene lived approximately BC and is best known for determining a very good approximation of the Earth's circumference and for inventing the "Sieve of Eratosthenes", a method of identifying prime numbers. The procedure is demonstrated below where the prime numbers between and are identified Proceed as follows:. Cross out as it is not a prime number. 2. Starting from 2, circle 2 and cross out every multiple of 2, i.e. every even number. 3. Starting with 3, circle 3, and cross out every multiple of Starting with 5, circle 5, and cross out every multiple of Starting with 7, circle 7 and cross out every multiple of 7. Question: How do we know when to stop? Answer: We can stop at the square root of, i.e.. The reason for this is that any number less than (9, for example), which is divisible by a number greater than the square root of (3, in this example), is also divisible by a number less than the square root of (7, in this example). So, we have already crossed out all such numbers. Note: The numbers that are crossed are not primes, because they are multiples of other numbers. The numbers that are circled are primes. 3

4 b) Prime Factors Example 4: Find all prime factors of 4. Solution: All the factors of 4 are, 2, 4, 5, 8,, 2, 4. However, only 2 and 5 are prime numbers. Therefore all the prime factors of 4 are 2 and 5. c) Prime Factorisation Any integer can be written as a product of prime numbers in a unique way (except for the order). The process is known as prime factorisation. Example 5: Find the prime factorisation of 264. Solution: We first note that 264 is an even number and can therefore be divided by 2, i.e d) Highest Common Factor (HCF) The HCF of two (or more) numbers is the largest number that divides exactly into both numbers. Example 6: Find the HCF of 24 and 32. Solution: For small numbers like 24 and 32 the easiest method is to proceed as follows: Write down the factors of the smaller number, starting from the largest factor: The factors of 24 are 24, 2, 8, 6, 3, 2,. Write down the factors of the smaller number, starting from the largest factor: The factors of 32 are 32, 6, 8, 4, 2,. The first factor of the smaller number that is also a factor of the larger number is a HCF. Hence, the HCF of 24 and 32 is 8. Note: The HCF is also known as the greatest common divisor (GCD). Note: To find the HCF of larger numbers we use prime factorisation. 4

5 e) Lowest Common Multiple (LCM) The LCM of two (or more) numbers is the smallest number that is a multiple of both the numbers. Example 7: Find the LCM of 9 and 2. Solution: For small numbers like 9 and 2 the easiest method is to: Write down several multiples the smaller number: Multiples of 9 are: 9, 8, 27, 36, 45, 54,... Write down the multiples of the larger number until one of them is also a multiple of the smaller number: Multiples of 2 are: 2, 24, 36,... Now, 36 is also a multiple of 9 and so the LCM of 9 and 2 is 36. To find the LCM of larger numbers we use prime factorisation. 5

6 4. Modular Arithmetic Modular arithmetic, or clock arithmetic as it is sometimes known, was first studied by Gauss in the late 8th century. It is a special type of arithmetic involving integers and features in branches of mathematics such as number theory and abstract algebra. Modular arithmetic is the central mathematical concept in cryptography and in computing the arithmetic operations performed by most computers use it. In everyday life we encounter modular arithmetic, even though we may not realise it, when we tell the time. Hence the term clock arithmetic. In modulo m arithmetic all integers are replaced by their remainders after division by m. For example, if 8 is divided by 6 the remainder is 2. Here 6 is called the modulus and we write 8 (mod 6) 2. We can perform this calculation for any number: (mod 6) (mod 6) 2 (mod 6) 2 3 (mod 6) 3 4 (mod 6) 4 5 (mod 6) 5 6 (mod 6) 7 (mod 6), etc. Every time we reach a multiple of 6 we start counting from again. The set of integers modulo 6 is {,, 2, 3, 4, 5}. In general, the set of integers modulo m is defined as {,, 2, 3,..., m - }. 6

7 a) Arithmetic Calculations Modular arithmetic allows standard mathematical calculations such as addition, subtraction and multiplication along with division by certain numbers. To illustrate addition and multiplication we will construct the addition and multiplication tables for Ζ 6. Recall that we only use the integers,, 2, 3, 4 and 5 and every time we reach a multiple of 6 we return to. Addition () Add the numbers together to obtain their sum. (2) Divide the sum by the modulus to obtain the remainder which is the answer. Example 8 (i) (3 ) mod 6 4 mod 6 4, (2 4) mod 6 6 mod 6 (ii) (4 4) mod 6 8 mod 6 2, (5 2) mod 6 7 mod Addition Table for Ζ 6 7

8 Multiplication () Multiply the numbers together to obtain their product. (2) Divide the product by the modulus to obtain the remainder which is the answer. Example 9 (i) (2 3) mod 6 6 mod 6, (3 5) mod 6 5 mod 6 3 (ii) (4 5) mod 6 2 mod 6 2, (5 5) mod 6 25 mod Multiplication Table for Ζ 6 Note: These tables are often called Cayley tables after the British Mathematician Arthur Cayley (82-895). Clock Arithmetic: Note that on the 24 hour clock 7: hours corresponds to 5:pm on the 2 hour clock. This is actually obtained using modulo 2 arithmetic, i.e. 7 (mod 2) 5. 8

9 c) Subtraction () Perform the subtraction. (2) Divide the sum by the modulus to calculate the remainder. There are two possible outcomes: A. The answer is positive Example : (8 2) mod 9 6 mod 9 6; (2 5) mod 3 7 mod 3. B. The answer is negative Add the modulus to the answer to get a positive number between and the modulus. Example : (i) (2 5) mod 7-3 mod 7 4 since (ii) (4 5) mod 7 - mod 7 3 since

10 5. Boolean Algebra Boolean Algebra was introduced by the English mathematician George Boole in 854 and has many practical applications in the physical sciences including electrical engineering and computing. Essentially it is algebra suited to two-valued computer logic and enables algebraic manipulation of logical statements that occur in, for example, digital circuit theory. A two element Boolean algebra is a set {, } together with the binary operations sum and product, and the unary operation, complementation (also called negation). The two states, and are sometimes referred to as TRUE (T) and FALSE (F); ON and OFF ; YES and NO ; HIGH and LOW, etc. The logical operators: sum, product and complementation are associated with the OR, AND, and NOT operators in propositional logic. In Boolean algebra, just as TRUE OR TRUE results in TRUE in propositional logic. Boolean addition corresponds to the logical OR function. In Boolean algebra, just as TRUE AND TRUE results in TRUE in propositional logic. Note that Boolean multiplication is identical to standard arithmetic multiplication in that anything multiplied by yields, and anything multiplied by remains unchanged. Boolean multiplication corresponds to the logical AND function. Consider the electric circuits shown in the schematic diagrams below: A B S S 2 For current to flow from A to B we need to have both switches CLOSED, i.e. we need S AND S 2 CLOSED. S A B For current to flow from A to B we can have either one switch CLOSED or both switches CLOSED, i.e. we need S OR S 2 CLOSED. S 2

11 6. Modulo 2 Matrix Arithmetic We can now extend the concept of matrix multiplication, encountered earlier in the course, to operations on matrices in which the elements are all or with addition and multiplication carried out modulo 2. Example 2 (i). (ii).. (iii). (iv)

12 7. Number Systems We are all familiar with the decimal (Base ) number which uses the digits to 9. Here we will look at the binary (Base 2) and hexadecimal (Base 6) number systems as they are widely used in computing. Here we will briefly define these three number systems and then look at converting numbers between the three systems. To avoid confusion, where appropriate, we will write the base as a subscript to the number. a). Decimal Number System The decimal, or Base, number system uses the ten symbols (digits),,2, 3, 4, 5, 6, 7, 8, 9 to represent numbers. b). BinaryNumber System The binary, or Base 2, number system uses the two symbols (binary digits or bits) and to represent numbers. Binary numbers are closely related to digital electronics. In digital electronics a means that current / electricity is present and a means it is not present. The different parts of a computer communicate through pulses of current (s and s). c). Hexadecimal Number System The hexadecimal, or Base 6, number system uses the sixteen symbols,,, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. to represent numbers. Hexadecimal is very common in computing because each hexadecimal digit represents four binary digits (bits), i.e. two hexadecimal digits code 8 bits ( byte). 2

13 Decimal to Binary to Hexadecimal Look-Up Table The following table shows the numbers to 5 in the decimal, binary and hexadecimal number systems. You should familiarize yourself with these values. Decimal (Base ) Binary (Base 2) A B 2 C 3 D 4 E 5 F Example - Convert Binary to Decimal Convert the binary number to a decimal number. Hexadecimal (Base 6) Solution: [note the ordering!] Hence, Example 2 - Convert Decimal to Binary Convert the decimal number 495 to a binary number. Solution: We start by dividing the original number by 2 and keep the remainder. Then repeat the process until we can no longer perform a division. 495 / 2 247, remainder 247 / 2 23, remainder 23 / 2 6, remainder 6 / 2 3, remainder 3 / 2 5, remainder 5 / 2 7, remainder 7 / 2 3, remainder 3 / 2, remainder / 2, remainder 3

14 Now read the binary number from the bottom to the top:. Hence, [Check this answer by converting the binary back to decimal]. Example 3 - Convert Hexadecimal to Decimal Convert the hexadecimal number 3B2 to a decimal number. 2 Solution: 3B Hence, 3 B Example 4 - Convert Decimal to Hexadecimal Convert the decimal number 4598 to a hexadecimal number. Solution: 4598/6 287 remainder 6 287/6 7 remainder 5 (F) 7/6 remainder /6 remainder Now read from the bottom to the top: F6 Hence, 4598 F6 6. Example 5 - Convert Hexadecimal to Binary Convert the hexadecimal number 3C7D to a binary number. Solution Replace each hexadecimal number with its binary equivalent. Hence, 3 C 7D C 7 D 4

15 Example 6 - Convert Binary to Hexadecimal Convert the binary number to a hexadecimal number. Solution STEP : Starting from the right hand side, binary numbers are split into groups of four for conversion into hexadecimal: STEP2: Replace each hexadecimal number with its decimal equivalent and then replace each decimal number with its hexadecimal equivalent. { 5 { F { 2 { C { 4{ 4 { { Hence, 2 F C46. Note: With a little more practice you will be able to omit the conversion to decimal stage and convert each group of four directly from binary to hexadecimal. You could of course use the table above to carry out this step. converts to 5 which is F in hexadecimal. converts to 2 which is C in hexadecimal. converts to 4. converts to. Example 8 Convert the binary number to a hexadecimal number. Solution We note here that the binary number only contains digits and so cannot be split evenly into groups of four as in the previous example. This is not a problem though as we start from the right hand side and split into groups of four { 3{ 3 { 4{ 4 { { B Hence, 2 34 B6. Note: If you wish you can prefix the two digits in the leftmost group with two zeros to obtain a group of four, i.e. { { { and then convert to hexadecimal. 3{ 4{ { 3 4 B 5

16 Tutorial Exercises Factors, Multiples and Primes () Find all the factors of: (i) 3; (ii), (iii) 73 (iv) 84. (2) What are the first three multiples of: (i) 7, (ii) 3? (3) If 65 and 9 are two multiples of a particular number, and the number is not, what is the number? (4) Find all the prime factors of 49. (5) Find the prime factorisation of: (i) 84, (ii) 35, (iii) 4. (6) Find the following (i) HCF of 28 and 42, (ii) LCM of 28 and 42. Modular Arithmetic (7) Solve the following: (i) (3 4) mod 5 (ii) (8 9) mod 3 (iii) (9 3) mod 2 (iv) (7 6) mod 2 (v) (5 3) mod 7 (vi) (2 ) mod (vii) (5 6) mod 7 (viii) (2 ) mod 7 (ix) (25 6) mod 73 (8) Solve the following: (i) (5-3) mod 7 (ii) (3-5) mod 7 (iii) (4-8) mod 2 (iv) ( - 6) mod 7 (v) (3-4) mod 5 (vi) ( - ) mod 2 (9) Construct the tables for addition and multiplication modulo 5. () Construct the tables for addition and multiplication modulo 2. 6

17 Modulo 2 Matrix Arithmetic () Where possible evaluate sums and products, using modulo 2 arithmetic, of the following pairs of matrices. (i), (ii),, (iii),. Number Systems (2) Convert the following decimal numbers to (a) their binary representations. (b) their hexadecimal representations. (i) 39 (ii) 73 (iii) (iv) 359 (v) (vi) 733 (vii) 234 (viii) 22 (3) Convert the following binary numbers to: (a) their decimal representations. (b) their hexadecimal representations (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (4) Convert the following hexadecimal numbers to: (a) their decimal representations. (b) their binary representations. (i)bd3 (ii) EEE (iii) 32 (iv) 5 (v) ABC (vi) BBC (vii) 2FC (viii) AA 7

18 Answers Factors, Multiples and Primes () (i) The factors of 3 are:, 2, 3, 5, 6,, 5, and 3. (ii) The factors of are:, 2, 4, 5,, 2, 25, 5,. (iv) The factors of 73 are: and 73 (73 is a prime number). (iv) The factors of 84 are:, 2,3,4, 6, 7, 2, 4, 2, 28, 42, 84. (2) (i) The first three multiples of 7 are: 7, 4 and 2. (ii) The first three multiples of 3 are 3, 262 and 393. (3) 3 (4) 7 (5) (i) , (ii) , (iii) (6) (i) The factors of 28 are 28, 4, 7, 2,. The factors of 42 are 42, 2, 4, 7, 6, 3, 2. The first factor of 28 that is also a factor of 42 is 4. Hence, HCF(28, 42) 4. (ii) Multiples of 28 are: 28, 56, 84, 2,... Multiples of 42 are: 42, 84, 26,... Hence, LCM(28, 42) 84. 8

19 Modular Arithmetic (7) (i) 2, (ii) 4, (iii), (iv), (v), (vi) 9, (vii) 2, (viii) 3, (ix) 65 (8) (i) 2, (ii) 5, (iii) 8, (iv) 2, (v) 4, (vi) 3 (9) Modulo 5 addition table: Modulo 5 multiplication table:

20 2 () Modulo 2 addition table: Modulo 2 multiplication table Modulo 2 Matrix Arithmetic () (i) (ii) (iii) Cannot add these matrices as they have different sizes. Also they are not compatible for multiplication either (why not?)

21 Number Systems (2) (a) Binary representation: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (b) Hexadecimal representation: (i) 27 (ii) 49 (iii) 64 (iv) 67 (v) 6F (vi) 2DD (vii) 4D2 (viii) 7DC (3) (a) Decimal representation: (i) 255 (ii) 43 (iii) 455 (iv) 82 (v) 38 (vi) 237 (vii) 284 (viii) 423 (b) Hexadecimal representation: (i) FF (ii) 2B (iii) C7 (iv) 52 (v) EE3 (vi) 4D5 (vii) 888 (viii) A7 (4) (a) Decimal representation: (i) 327 (ii) 3822 (iii) 5 (iv) 8 (v) 2748 (vi) 4865 (vii) 764 (viii) 7 (b) Binary representation: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) 2

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