All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Just Math
Unity Root Count (Posted on 2015-03-09) Difficulty: 3 of 5
Consider sets A = {u: u18 = 1} and B = {v: v48 = 1}, where neither of the sets A and B can contain any purely real element.
Define C = {u*v: u ε A, v ε B}.

Not counting any purely real element, determine the total number of distinct complex elements in C.

No Solution Yet Submitted by K Sengupta    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution solution | Comment 1 of 2
Set A has 18 - 2 = 16 elements, while B has 48 - 2 = 46 elements.

The roots are all located on a unit circle in the Argand plane. The LCM of  18 and 48 is 144, so all products will be at multiples of 1/144 of a full circle (adding angles to do multiplication). Elements of A are multiples of 8/144 of a circle, and of B are multiples of 3/144 of a circle.

This program does the addition of the 144ths and computes the resulting 144ths. 0 and 72 mod 144 correspond to real values and are not used as multiplicands (addends in the additions) or products (sums in the additions).

DefDbl A-Z
Dim member(143), crlf$

Private Sub Form_Load()
 Text1.Text = ""
 crlf$ = Chr(13) + Chr(10)
 Form1.Visible = True
  
  ' count 1/144's of a circle in the Argand plane
  For a = 8 To 144 - 8 Step 8
   If a * 2 <> 144 Then
    For b = 3 To 144 - 3 Step 3
      If b * 2 <> 144 Then
        prodAngle = (a + b) Mod 144
        If prodAngle <> 0 And prodAngle <> 72 Then
          member(prodAngle) = member(prodAngle) + 1
        End If
      End If
    Next
   End If
  Next a
  
  For m = 1 To 143
    If member(m) > 0 Then
      ct = ct + 1
      Text1.Text = Text1.Text & m & Str(member(m)) & crlf
    End If
  Next
  Text1.Text = Text1.Text & ct & crlf

  Text1.Text = Text1.Text & "done"
  DoEvents

End Sub

Listed first are counts of the various possible multiples of 1/144 of a circle in the products angles on the Argand plane:

1 6
2 6
3 4
4 6
5 6
6 4
7 6
8 4
9 4
10 6
11 6
12 4
13 6
14 6
15 4
16 4
17 6
18 4
19 6
20 6
21 4
22 6
23 6
24 2
25 6
26 6
27 4
28 6
29 6
30 4
31 6
32 4
33 4
34 6
35 6
36 4
37 6
38 6
39 4
40 4
41 6
42 4
43 6
44 6
45 4
46 6
47 6
48 2
49 6
50 6
51 4
52 6
53 6
54 4
55 6
56 4
57 4
58 6
59 6
60 4
61 6
62 6
63 4
64 4
65 6
66 4
67 6
68 6
69 4
70 6
71 6
73 6
74 6
75 4
76 6
77 6
78 4
79 6
80 4
81 4
82 6
83 6
84 4
85 6
86 6
87 4
88 4
89 6
90 4
91 6
92 6
93 4
94 6
95 6
96 2
97 6
98 6
99 4
100 6
101 6
102 4
103 6
104 4
105 4
106 6
107 6
108 4
109 6
110 6
111 4
112 4
113 6
114 4
115 6
116 6
117 4
118 6
119 6
120 2
121 6
122 6
123 4
124 6
125 6
126 4
127 6
128 4
129 4
130 6
131 6
132 4
133 6
134 6
135 4
136 4
137 6
138 4
139 6
140 6
141 4
142 6
143 6

In total 142 possibilities are represented. Only the two real values are discounted from all 144 possibilities.

Of course, two of the products are purely imaginary also. Do they count as complex?


  Posted by Charlie on 2015-03-09 17:39:29
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (8)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information