FORMULA TO CALCULATE GREATEST COMMON DIVISOR
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MAE 301 Fall 2002 Review for Midterm II
Understand the basis for the principle of mathematicalinduction. Be able to derive the formulas for the sum of thefirst n numbers and for the sum of the firstn squares. Understand why the "proof" that any set of coins chosen from abox will have the same denomination is not a correct inductionproof. Be able to use induction to prove elementary propertiesof the Fibonacci sequence and to prove theBinomial Theorem.
Be able to explain the connection between "long division"and the ``Division Algorithm'' (the long division algorithm isa schematic version of the proof of the Division Algorithm,which is, strictly speaking, misnamed).
Understand why the Euclidean Algorithm takes as inputtwo positive integers a,b and gives their greatestcommon divisor (a,b) as output. Be able to use theEuclidean Algorithm to calculate greatest common divisors.Be able to use the Euclidean Algorithm (by itself or in therow-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 aunique representation in base b. Be able to implementthe proof explicitly in switching back and forth betweendecimal (base 10) and binary (base 2) representations ofgiven 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. Beable to prove that the composition of two translations isa translation.
Be able to prove that a rotation is an isometry. Understandhow rotations about the origin in R2 arerepresented by 2 x 2 matrices. Understand how writingthe matrix product for the composition of two rotations about the origin leads to the addition formulas for sinand cos. Be able to express a rotation about a pointP not the origin in terms of translations and arotation about the origin. Be able to implement this calculationto give the x,y-coordinates of the image of a point(a,b) after rotation by 45o about thepoint P = (-1,2) for example.
Be able to prove that a reflection is an isometry. Be able toprove that the composition of two reflections is a rotation(when the lines of reflection meet) or a translation (when thelines are parallel). Be able to implement this calculationto give the x,y-coordinates of the image of a point(a,b) after reflection about the line x-y=3and then reflection about the line y=2x for example.
November 30 2002
