#### The totient function.

Maths Coursework The totient function ?(n), also called Euler's totient function, is defined as the number of positive integers <= n which are relatively prime to (i.e., do not contain any factor in common with) n, where 1 is counted as being relatively prime to all numbers. Since a number less than or equal to and relatively prime to a given number is called a totative, the totient function ? (n) can be simply defined as the number of totatives of n. For example, there are eight totatives of 24 (1, 5, 7, 11, 13, 17, 19, and 23), so ?24=8. The totient function is implemented in Mathematics as EulerPhi [n]. In this part of the coursework will be investigating the phi function of: ) ?(p) 2) ?(m x n) = ?(m) x ?(n) 3) ?(pn) 4) Other methods of calculating the Phi values of an integer that I found from the net. Find the value of: (i) ?(3): = 2 3 = 1,2 The number 3 only has 2 positive co-prime integers they are the numbers 1 and 2. (ii) ?(8): = 4 8 = 1,3,5,7 There are 4 positive co-prime integers for the number 8 (iii) ?(11): = 10 1 = 1,2,3,4,5,6,7,8,9,10 The number 11 has 10 positive co-prime integers, and they are shown above. (iv) ?(24): = 8 24 = 1,5,7,11,13,17,19,23 The number 24 has 8 positive co-prime integers, and they are shown above. Part 1 In this part I will try to investigate on ?(n). But before we can start we need to experiment on certain numbers.

#### Beyond Pythagoras

Intro I will be investigating Pythagoras triples using the Pythagoras theorem. A triple is a set of 3 whole numbers where the rule is a2 + b2 = c2. I will be working out the values of the sides of the triangle. The shortest side of the triangle will be represented by the letter 'a', the second longest side will be represented by the letter 'b', and the longest side, the hypotenuse, will be represented by the letter 'c'. Odds nth Length of shortest side (a) Length of middle side (b) Length of longest side (c) Perimeter (a+b+c) Area (a x b)/2 3 4 5 2 6 2 5 2 3 30 30 3 7 24 25 56 84 4 9 40 41 90 80 5 1 60 61 32 330 6 3 84 85 82 546 7 5 12 13 240 840 8 7 14 45 306 224 To begin with, I will draw up a table containing the first eight Pythagorean triples with 'a' being an odd number, and shows the lengths of the sides, their perimeters and their areas. Then I will be able to see if there are any connections or relationships between the numbers, and I will be able to find the nth term for each side, area and perimeter. When I put the numbers into the table like this, I realised two things, firstly that 'a' increases by two, and secondly that 'c' is b+1 on both of the terms. As I currently only have two numbers in the sequence for 'a', my assumption about 'a' increasing by two each time is not necessarily correct, because

#### The Phi Function Investigation

For any positive integer n, the Phi Function o (n) is defined as the number of positive integers less than n which have no factor (other than 1) in common (are co-prime) with n.xml:namespace prefix = o ns = "urn:schemas-microsoft-com:office:office" /> Part one (a) Find the value of: I) o (3) , 2, 3 = 2 o (8) , 2, 3, 4, 5, 6, 7, 8 = 4 o (11) , 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 = 10 o (24) , 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24 = 8 What did I notice? The Phi of both 3 and 11, which are both prime numbers is themselves minus one. So when n is a prime number o n = n - 1 e.g. I would predict that the phi of 17 = 16 o(17) , 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 = 16 My formula has proved successful. PART ONE (b) Obtain the Phi function for at least 5 positive integers of your own choice i) o(10) I am using 10 as it is an even number.... 1, 2, 3 ,4 ,5 ,6 , 7, 8 , 9, 10 = 4 ii) o(17) I am using 17 as it is a prime number.... 1, 2, 3 , 4, 5, 6 , 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 = 16 iii) o(15) I am using 15 as it is a odd number.... 1, 2, 3 , 4, 5, 6 , 7, 8, 9, 10, 11, 12, 13, 14, 15 = 8 iv) o(16) I am using 16 as it is square number.... 1, 2, 3 , 4, 5, 6 , 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 = 8 v) o(27) I am using 27 as it is a cubed number.... , 2, 3, 4, 5, 6, 7, 8, 9,

#### Describe Aristotle's teachings about the differences between the final cause and the other sorts of causes.

Describe Aristotle's teachings about the differences between the final cause and the other sorts of causes In Aristotle's teachings there are different types of causes. There is the material cause, the efficient cause, the formal cause and the final cause. The material cause is the matter from which something is made (like the bricks used for the building of the house). The efficient cause is the agent that brings something about (like a builder that builds a house). The formal cause is the form of something (the building being done). The final cause is the goal or purpose that something wants to achieve (work towards), or the reason why it the way it is (the final building). The material cause is what something is made up of. Without the material cause there would be no other causes, because you need a substance etc, for anything to happen. Using the example above, you cannot build a brick house without any bricks, (the formal cause could not exist without a material cause). Aristotle believed that not only is material cause just the substance something is made from, but also the means by which a thing is brought about (how we see it and can tell what it is). The efficient cause is the means in which something actually becomes something, which includes the maker, the tools and the skills being used. The efficient cause along with all the causes play an important part in

#### Mathematics GCSE Coursework - The Phi Function.

Mathematics GCSE Coursework The Phi Function In this coursework I will be investigating the Phi function. And I am making clear that crossed numbers like 1 are co-prime, but numbers in circle like 1 are not. Part 1 a) ?(3)=2; 1 2 3 ?(8)=4; 1 2 3 4 ?(11)=10; 1 2 3 4 5 6 7 8 9 10 11 ?(24)=8; 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 And I will do five more examples to show my working. b) ?(19)=18; 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 ?(9)=6; 1 2 3 4 5 6 7 8 9 ?(13)=12; 1 2 3 4 5 6 7 8 9 10 11 12 13 ?(5)=4; 1 2 3 4 5 ?(15)=8; 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Part 2 a) ) ?(7 x 4) = ?(7) x (4); ?(7)=6; 1 2 3 4 5 6 7 ?(4)=2; 1 2 3 4 ?(7 x 4)= ?(28)=12; 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 ?(7) x ?(4)=6 x 2=12; So ?(7 x 4) = ?(7) x (4) is true. 2) ?(6 x 4)=?(6) x ?(4); ?(6)=2; 1 2 3 4 5 6 ?(4)=2; I have found ?(4) already. ?(6 x 4)= ?(24)=8; I have found ?(24) already. ?(6) x ?(4)=2 x 2=4; So ?(6 x 4)=?(6) x ?(4) is true. b) ?(5 x 2)=?(5) x ?(2); ?(5)=4; I have found ?(5) already. ?(2)=1; 1 2 ?(5 x 2)= ?(10)=4; 1 2 3 4 5 6 7 8 9 10 ?(5) x ?(2)=4 x 1=4; So ?(5 x 2)=?(5) x ?(2) is true. ?(7 x 3)=?(7) x ?(3); ?(7)=6; I have found ?(7) already. ?(3)=2; I have found ?(3) already. ?(7 x 3)= ?(21)=12; 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

#### The Phi Function

THE PHI FUNCTION The investigation I chose to do is called The Phi Function. Phi Function means "Integers which are less than n and which have no other factor other than 1." My first task was to work out the value of the Phi Functions: ?(3), ?(8), ?(11) and ?(24). Next I have to make some of my own and find the values of them as well. My second task is to check that if these are correct: ?(7x4) = ?(7) x ?(4) and ?(6x4) = ?(6) x ?(4) then create some of my own and check that if they are equal to each other or not. PART 1 (1) ?(3) = 2 The factors of ?(3) are: 1 and 3. The integers, which are less than 3, are 1 and 2. The table below shows the integers, factors and whether it fits into the expression the number of positive integers less than n which have no factor (other than 1) in common (are co-prime) with n. Integers Factors Does it fit into expression? Yes or No yes 2 ,2 yes This table shows you that the number of positive integers less than three and has no other common factor other than 1 is two integers: 1 and 2. Therefore this shows that the phi function of 3 is 2. (2) ?(8) = 4 The factors of 8 are: 1, 2, 4, and 8. The integers, which are less than 8, are 1, 2, 3, 4, 5, 6 and 7. The table below is similar to the one above, but is for the phi function of 8. Integers Factors Does it fit into expression? Yes or No yes 2 ,2 No 3 ,3 Yes 4

#### The Phi-Function.

Maths Coursework by Yasir Al-Wakeel The Phi-Function A Function in mathematics is the term used to indicate the relationship or correspondence between two or more quantities. It was first used in 1637 by a French mathematician by the name of Rene Descartes to designate a power xn of a variable x. Later, Gottfried Wilhelm Leibniz in 1694 applied the term to various aspects of a curve, such as its slope. The most widely used meaning until quite recently was defined in 829 by the German mathematician Peter Dirichlet. Dirichlet conceived a function as a variable y, called the dependent variable, having its values fixed or determined in some definite manner by the values assigned to the independent variable x, or to several independent variables x1, x2, ..., xk. The Phi-Function is a means of breaking down numbers. It is defined as the number of positive integers less than n, where n is a positive integer, which are co-prime with n. Zero is neither a positive integer nor a negative integer, it is on the boundary and so does not come under the notation of n. The phi function of a positive inter, n, is expressed as ?(n). Two terms are co-prime when they have no factor in common other than one. For example 3 and 4 are co-prime or 5, 7, and 8 are all co-prime. When numbers are co-prime they can be written: (n , m)=1 such as (5,7)=1.

#### The Phi Function Investigation

The Phi Function For any positive integer n, the Phi Function ?(n) is defined as the number of positive integers less than n which have no factor (other than 1) in common (are co-prime) with n. Part 1 (a) Find the value of: (I) ?(3) (ii) ?(8) (iii) ?(11) (iv) ?(24) (b) Obtain the Phi-Function for at least 5 positive integers of your own choice. (a) (I) ?(3): 1 2 1,2 3 1,3 3 = 1,2 The number 3 only has 2 positive integers they are the numbers 1 and 2. (ii) ?(8): 1 2 1,2 3 1,3 4 1,2,4 5 1,5 6 1,2,3,6 7 1,7 8 1,2,4,8 8 = 1,3,5,7 There are 4 positive integers for the number 8 (iii) ?(11): 1 2 1,2 3 1,3 4 1,2,4 5 1,5 6 1,2,3,6 7 1,7 8 1,2,4,8 9 1,3,9 0 1,2,5,10 1 1,11 1 = 1,2,3,4,5,6,7,8,9,10 The number 11 has 10 positive integers, they are shown above. (iv) ?(24): 1 2 1,2 3 1,3 4 1,2,4 5 1,5 6 1,2,3,6 7 1,7 8 1,2,4,8 9 1,3,9 0 1,2,5,10 1 1,11 2 1,2,4,6,12 3 1,13 4 1,2,7,14 5 1,3,5,15 6 1,2,4,8,16 7 1,17 8 1,2,6,9,18 9 1,19 20 1,2,4,5,10,20 21 1,3,21 22 1,2,11,22 23 1,23 24 1,2,3,4,6,8,12,24 24 = 1,5,7,11,13,17,19,23 The number 24 has 8 positive integers, they are shown above. (b) Obtain the Phi-Function for at least 5 positive integers of your own choice. (I) ?(6): 1 2 1,2 3 1,3 4 1,2,4 5 1,5 6 1,2,3,6 6 = 1 and 5 The

#### Maths Primes and Multiples Investigation

. A)I) ?(3)-1, 2=2 II) ?(8)-1, 2, 3, 4, 5, 6, 7=4 III) ?(11)-1, 2, 3, 4, 5, 6, 7, 8, 9, 10=10 IV) ?(24)-1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23=8 B)I) ?(5)-1, 2, 3, 4=4 II) ?(10)- 1, 2, 3, 4, 5, 6, 7, 8, 9=4 III) ?(15)- 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14=8 IV) ?(20)- 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19=8 V) ?(50)-1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49=20 2. A)I) ?(7x4) = ?(7) x ?(4) 7x4=28 ?(28)= 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27=12 ?(7)=1, 2, 3, 4, 5, 6=6 ?(4)=1, 2, 3=2 2x6=12 12=12, therefore a prime and an even work(non-prime). B) ?(6x4) = ?(6) x ?(4) 6x4=24 ?(24)=8 ?(6)=1, 2, 3, 4, 5=2 ?(4)=2 2x2=4 4=8, therefore two evens don't work. C) ?(5x10)= ?(50) ?(50)=20 ?(10) x ?(5) 4x4=16 16=20, therefore two multiples don't work. ?(13x3)= ?(39) ?(39)= 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38=24 ?(13)=12 ?(3)=2 2x12=24 24=24, therefore two primes work. Having done these examples I below have drawn up a table of my results. When I made the table I saw

#### Millikan's theory.

Representations are for Millikan part of a larger group of entities for which she considers there is no generic name in English, and which would include "[n]atural signs, animals' signs, people's signs, indexes, signals, indicators, symbols, representations, sentences, maps, charts, pictures" [LTOBC, 85]. For want of a better term, Millikan calls these entities signs, and claims that what is common to all signs is their being, to a greater or lesser degree, intentional. That is, what all signs have in common, in a family resemblance way, is their bearing a certain relationship to entities other than themselves - a relationship which is usually characterized as "being about something else", "meaning something else". In what follows, I will consider what the nature of this "being about something else" is according to Millikan. I will pay particular attention to mental, or inner, representations, despite the fact that Millikan believes "articulate conventional signs" - that is, I take it, verbal utterances - to be the paradigm case of signs. For, like Searle, Millikan regards verbal utterances and other external verbal-like modes of representation (such as writing), to have an intentionality derived from the original intentionality of states of mind, and thus explainable in terms of the latter. Intentionality is, according to Millikan, a question of degree: indeed, she rejects