1. Chapter 6 Section 6.1 What is Group Theory?

2. Patterns in Mathematical Objects and Numbers The study of different patterns that can occur in numbers, shapes and movement of shapes is called Group Theory. How different ways of combining numbers (operations) or shapes (motions) form different patterns is of great interest in solving many problems that do occur in the world. Abstract Formulas & Algebra Abstract numerical formulas play an important role in describing patterns that exist in numbers. When written in terms of algebra we call it a general expression. If you add together a number, double that number and triple the number you get six times the number. This number pattern has an abstract numerical formula, which is really an algebraic equation that can describe it instead of writing it in words. If we let the variable x stand for any number. We get a general expression for this pattern. x + 2 x + 3 x = 6 x (General expression) If we replace x by a specific number we call it a specific instance of the formula. 1 + 2·1 + 3·1 = 6·1Specific instance with x = 5 + 2·5 + 3·5 = 6·5Specific instance with x = 12+ 2·12 + 3·12 = 6·12Specific instance with x = 1 5 12

3. Working Backwards From Specific Instances We can work backwards from several specific instances to a general expression. We need to be clever when it comes to noticing a pattern. xy addsubmultsquare x square y subSpecific Instance 32 3+2 3-2 (3+2)(3-2) 3232 2 3 2 -2 2 (3+2)(3-2)=3 2 -2 2 5 3 5 1 5 9 4 5 5=5 5+3 5-3 (5+3)(5-3) 5252 3232 5 2 -3 2 (5+3)(5-3)=5 2 -3 2 8 2 16 25 9 16 16=16 6 1 6+1 6-1 (6+1)(6-1) 6262 1212 6 2 -1 2 (6+1)(6-1)=6 2 -1 2 7 5 35 36 1 35 35=35 For any two numbers if we multiply together their sum and difference we get the difference between their squares of their squares! General Expression: ( x + y )( x – y ) = x 2 – y 2

4. Clock Arithmetic If we picture the numbers 1 through 12 arranged around a clock and add the number of hours together the clock position we end up in is called the clock sum. We use the operation symbol  to stand for this. Here are some examples. 12 6 1 2 3 4 5 7 8 9 10 11 4 hours + 5 hours = 9 hours 4  5 = 9 12 6 1 2 3 4 5 7 8 9 10 11 7 hours + 6 hours = 1 hour 7  6 = 1 12 6 1 2 3 4 5 7 8 9 10 11 11 hours + 9 hours = 8 hours 11  9 = 8 An easier way to do this without drawing a clock each time is to add the numbers like you normally would, if the number you get is 12 or less you are fine use that number, if it is bigger than 12 subtract 12 from it. 6  4 = ?6 hours + 4 hours = 10 hours6  4 = 10 (i.e. 10 ≤ 12) 7  8 = ?7 hours + 8 hours = 15 hours – 127  8 = 3 (i.e. 15 > 12)

5. Different Clock Arithmetic Sometimes it is useful to work with a "clock" that has a different group of numbers around it. The clock to the right uses only the numbers {0,1,2,3}. The way you combine the numbers is just like before except if you go over 3 you need to subtract 4. Example: 2  1 = 3 and 3  2 = 1 0 1 2 3 The pattern that exists can be expressed using an operation table. Organizing this helps us identify different patterns that exist not only in the table but between two different tables.  0123 00123 11230 22301 33012 This is especially useful in recognizing patterns that exist in different shapes. If we consider the situation of rotating a shape 90° at a time.

6. 0° rotation (original position) 90° rotation 180° rotation 270° rotation If we combine any two of these rotations (by doing one right after the other) we get the same effect as just doing one of them. The table to the right shows the pattern that develops. Notice the similarity with the table for the clock arithmetic with the numbers {0,1,2,3}. *0°0°90°180°270° 0°0°0°0°90°180°270° 90° 180°270°0°0° 180° 270°0°0°90° 270° 0°0°90°180°