• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month
Page
  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
  10. 10
    10
  11. 11
    11
  12. 12
    12
  13. 13
    13
  14. 14
    14
  15. 15
    15
  • Level: GCSE
  • Subject: Maths
  • Word count: 2134

The phi function.

Extracts from this document...

Introduction

Charlene Asare

Mathematics Coursework 2

Charlene Asare

Ashbourne College

Mathematics Coursework

April 2003

The phi function.

Ф


The phi function.

Introduction.

Phi is a letter in the Greek alphabet.  In higher mathematics:

  • The upper case form of phi, (Φ) means an angle function
  • The lower case form, φ , means angle mathematics and golden ratio mathematics

For any positive integer n, the Phi Function f (n) is defined as the number of positive integers less than n, which has no factor (other than 1) in common (are co-prime) with n.

In this piece of work I will be investigating several instances involving phi and prime numbers. Patterns between numbers that are prime, co prime involving phi will be found.

PART 1.

  1. Find the values of:
  1. Φ (3)
  2. Φ (8)
  3. Φ (11)
  4. Φ (24)

Solution

  1. Φ (3) = 2, 1.

Φ (3) = 2

This is because there are only 2 positive integers less than 3 which have no common factors with 3 other than 1.

  1. Φ (8) = 7, 5, 3, 1.

Φ (8) = 4

This is because there are 4 positive integers less than 8 which have no common factors with 8 other than 1.

  1. Φ (11) = 10, 9, 8, 7, 6, 5, 4, 3, 2, 1.

Φ (11) = 10

This is because there are 10 positive integers less than 11 which have no common factors with 11 other than 1.

  1. Φ (24) = 23, 19, 17, 13, 11, 7, 5, 1.

Φ (24) = 8

This is because there are 8 positive integers less than 24 which have no common factors with 24 other than 1.

...read more.

Middle

Φ (3 x 7) = x Φ (3) x Φ (7)

Φ (3 x 7)

= Φ (21)

Φ (21) = 20, 19, 17, 16, 13, 11, 10, 8, 5. 4, 2, 1.

= Φ (21) = 12

Φ (3)

= Φ (3) = 2,1.

Φ (3) = 2

Φ (7)

= Φ (7) = 6, 5, 4, 3, 2, 1.

Φ (7) = 6

As the answer gotten from Φ (21) is equal to that of Φ (3) x Φ (7), the equation Φ (n x m) = Φ (n) x Φ (m), where n = 3 and m = 7 is correct.

12 integers = 2 integers x 6 integers

12 integers = 12 integers.

Solution

For my third choice I will use the numbers 8 and 4 where n = 8 and 4= m.

Φ (n x m) = Φ (n) x Φ (m)

Φ (8 x 4) = Φ (8) x Φ (4)

Φ (8 x 4)

=  Φ (32)

Φ (32) = 31, 29, 27, 25, 23, 21, 19, 17, 15, 13, 11, 9, 7, 5, 3, 1.

Φ (32) = 16

Φ (8)

Φ (8) = 7, 5, 3, 1

= Φ (8) = 4

Φ (4)

Φ (4) = 3, 1

Φ (4) = 2

As the answer gotten from Φ (32) is not equal to that of Φ (4) x Φ (8), the equation Φ (n x m) = Φ (n) x Φ (m), where n = 8 and m = 4 is incorrect.

16 integers ≠ 4 integers x 2 integers

16 integers ≠ 8 integers

PART 3.

In some cases Φ (n x m) = Φ (n) x Φ (m) whilst in other cases this is not so. Investigate this situation.

Investigation:

From the results gotten in Part 2 its very obvious that in cases where the variables n and m have common factors the equation:

Φ (n × m) = Φ (n) × Φ (m);

does not work. In cases in which the variables n and m had no common factors the equation:

Φ (n × m) = Φ (n) × Φ (m);

worked. With this information I can draw a very accurate prediction.

Prediction:

...read more.

Conclusion

Analysis

A multiplicative function is a function f such that f(a × b) = f(a) × f(b). Is phi multiplicative?To an extent it is as in all the sets of numbers that had common factors the equation Φ (n × m) = Φ (n) × Φ (m) did not work.

Conclusion

With all the problems and examples I have solved in relation to this I can conclude that;

Φ (n x m) = Φ (n) x Φ (m)

where n is a prime number and m is any number at all.

PART 4.

If p and q are prime investigate: Φ (pmqn)

The equationΦ (pmqn) = Φ (pm) x Φ (qn) and so with the knowledge gotten from part 3 I can predict that Φ (pmqn) = Φ (pm) x Φ (qn) will be true where p and q are prime. to further illustrate this I will do a few examples.

Example 1.

P= 2, q = 3 ; m = 3, n = 2

Φ (pm x qn)

Φ (pmqn) = Φ (pm)xΦ (qn)

Φ (23 x 32) = Φ (23)xΦ (32)

Φ (8 x 9) = Φ (8)xΦ (9)

Φ (72) = Φ (8) xΦ (9)

24 = 4 x 6

24 = 24

Example 2.

P= 7, q = 5 ; m = 4, n = 6

Φ (pm x qn)

Φ (pmqn) = Φ (pm)xΦ (qn)

Φ (54 x 71) = Φ (54)xΦ (71)

Φ (625 x 7 ) = Φ (625)xΦ (7)

Φ (4375) =  Φ (625) xΦ (7)

3000 = 500 x 6

3000 = 3000

Example 3

P= 5, q = 11 ; m = 1, n = 3

Φ (pm x qn)

Φ (pmqn) = Φ (pm)xΦ (qn)

Φ (51 x 113) = Φ (51)xΦ (113)

Φ (5 x 1331) = Φ (5)xΦ (1331)

Φ (6655) =  Φ (5) xΦ (1331)

4840 = 4 x 1210

4840 = 4840

Example 4

P= 13, q = 2 ; m = 2, n = 4

Φ (pm x qn)

Φ (pmqn) = Φ (pm)xΦ (qn)

Φ (132 x 24) = Φ (132)xΦ (24)

Φ (169 x 16 ) = Φ (169)xΦ (16)

Φ (2704) =  Φ (169) xΦ (16)

1248 = 156 x 8

1248 = 1248

...read more.

This student written piece of work is one of many that can be found in our GCSE Phi Function section.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related GCSE Phi Function essays

  1. Identify and explain the rules and equations associated with the Phi function.

    However I did see that as the powers of 3 increased the number was always 2/3 of the Phi value we are intending to work out. x 3x ?(3x) 1 3 2 2 9 6 3 27 18 We can see that the ? of 3x is 2/3 of 3x.

  2. Binary Integers

    00000000 = Binary integer Denary 27 26 25 24 23 22 21 1 0 0 0 0 0 0 0 0 * 00000000 Binary = 0 = 0 = 0 6. 11111111 = Binary integer Denary 27 26 25 24 23 22 21 1 1 1 1 1 1 1

  1. Investigating the Phi function

    Example (12): (6) = 2 - 2 x 2 =4 - therefore (12) = 4 (500): (250) = 100 - 100x2 = 200 - therefore (500) = 200 I formulated a general equation to show the difference in the mathematical relationship between even and odd integers when they are doubled

  2. The Phi Function

    1,2,7,14 No 15 1,3,5,15 No 16 1,2,4,8,16 No 17 1,17 Yes 18 1,2,3,6,9,18 No 19 1,19 Yes 20 1,2,4,5,10,20 No 21 1,3,7,21 No 22 1,2,11,22 No 23 1,23 Yes From looking at the above table I can tell you that the phi function of 8 is 4.

  1. The totient function.

    By using the same formula above all I did now was just divide the whole equation by two. That is- ?(n) = ?(2 x n) / 2 Check: - ?10 = 4 (My formula) ?10 = 10 x 2= 20, ?20=8, 8 / 2= 4 (proved)

  2. Millikan's theory.

    Given a proper or Normal function F and a biological item B, (1) A is a reproduction of some prior item that, because of the possession of certain reproduced properties, actually performed F in the past, and A exists because of this performance; or (2)

  1. The Phi function.

    We multiply the result by 2 giving us 8. Now we check what 2n is. 2n is 16. So ?(16) should be equal to 8 and ?(16) is equal to 8. We can take another number such as 10. The ?(6) = 2 and then we multiply it by 2 and get 4. The ?(12) should also be 4.

  2. In this coursework I was asked to investigate the Phi Function (f) of a ...

    (14); 1 2 3 4 5 6 7 8 9 10 11 12 13; =6 ? (15); 1 2 3 4 5 6 7 8 9 10 11 12 13 14; =8 ? (16); 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15; =8 ?

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work