Be able to explain the connection between "long division" and the ``Division Algorithm'' (the long division algorithm is a schematic version of the proof of the Division Algorithm, which is, strictly speaking, misnamed).
Understand why the Euclidean Algorithm takes as input two positive integers a,b and gives their greatest common divisor (a,b) as output. Be able to use the Euclidean Algorithm to calculate greatest common divisors. Be able to use the Euclidean Algorithm (by itself or in the row-reduction form) to express d = (a,b) in the form d = xa + yb, with x and y integers.
Understand the statement and be able to prove this Theorem: For any integer b > 1 any positive integer has a unique representation in base b. Be able to implement the proof explicitly in switching back and forth between decimal (base 10) and binary (base 2) representations of given numbers.
Understand that the standard multiplication and long-division algorithms work for numbers expressed in any base. Be able to implement them in base 2 as well as in base 10.
Understand the definition of ``congruence transformation.'' We also call such a transformation an ``isometry.''
Be able to prove that a translation is an isometry. Be able to prove that the composition of two translations is a translation.
Be able to prove that a rotation is an isometry. Understand how rotations about the origin in R2 are represented by 2 x 2 matrices. Understand how writing the matrix product for the composition of two rotations about the origin leads to the addition formulas for sin and cos. Be able to express a rotation about a point P not the origin in terms of translations and a rotation about the origin. Be able to implement this calculation to give the x,y-coordinates of the image of a point (a,b) after rotation by 45o about the point P = (-1,2) for example.
Be able to prove that a reflection is an isometry. Be able to prove that the composition of two reflections is a rotation (when the lines of reflection meet) or a translation (when the lines are parallel). Be able to implement this calculation to give the x,y-coordinates of the image of a point (a,b) after reflection about the line x-y=3 and then reflection about the line y=2x for example.
November 30 2002