Math C Revision

Descripción

Mapa Mental sobre Math C Revision, creado por Shannon Hancock el 22/03/2015.
Shannon Hancock
Mapa Mental por Shannon Hancock, actualizado hace más de 1 año
Shannon Hancock
Creado por Shannon Hancock hace más de 9 años
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Resumen del Recurso

Math C Revision
  1. Groups
    1. Abelian Groups
      1. Commutative

        Nota:

        • Symmetrical about the leading diagonal (OR a(o)b=b(o)a
      2. Modulo Arithmetic

        Nota:

        • When they're is a limit to how high the number can reach before the counting restarts.
        • How many are left over when the number is divided by the mod e.g. mod 12 0=0 1=1 12=0 13=1
        1. Properties
          1. Closure

            Nota:

            • when numbers in the set operated together, the product is also part of the set
            1. Assocciativity

              Nota:

              • The order of the operations in relation to brackets are irrelevant. e.g.  2+(3+4)=(2+3)+4 2+7=5+4 9=9
              1. Identity Element

                Nota:

                • every number has its own identity (u) that a(o)u = u(o)a=a e.g. 2x1=2x1=2
                1. Inverse

                  Nota:

                  • all numbers have an inverse (i) that when operated with equals the identity element a(o)i=i(o)a=u e.g. 2 2x0.5=0.5x2=1
                2. Cayley Table

                  Nota:

                  • http://openi.nlm.nih.gov/imgs/512/14/3586959/3586959_1742-4682-9-54-9.png
                3. Sequences and Series
                  1. Geometric

                    Nota:

                    • a set of numbers where the ratio between succeeding terms is the same e.g. 1, 2, 4, 8, 16
                    • a=first number r=common ratio n=term number tn=the nth term sn= sum of terms up to nth
                    1. Sum of Terms

                      Nota:

                      • Sn = [a(r{n}-1)]/[r-1] {to the power of}
                      1. Finding Terms

                        Nota:

                        • tn=ar{n-1} - to work out tn {to the power of} log(tn/a)=(n-1)xlog(r) to work out n
                      2. Arithmetic

                        Nota:

                        • a set of numbers where the difference between succeeding terms is the same e.g. 1, 2, 3, 4, 5, ...
                        • a=first term l=last term d=difference between succeeding terms tn=the nth term sn=the sum of terms up to nth
                        1. Sum of Term

                          Nota:

                          • Sn=n/2(a+l) OR Sn=n/2{2a+(n-1)d]
                          1. Finding Terms

                            Nota:

                            • tn=a+(n-1)d
                        2. Real Numbers
                          1. Rational

                            Nota:

                            • Can be written as a fraction (ratio) a/b
                            1. Recurring decimals as fractions

                              Nota:

                              • simultaneous equation i.e. 0.53 (both 5 and 3 are recurring) let x=0.53 ->1 let 100x=53.53 ->2 2-1 99x=53 x=53/99
                            2. Irrational

                              Nota:

                              • non-terminating, non-reccuring decimals, pi
                              1. Surds
                                1. Division/Multiplication

                                  Nota:

                                  • Division = √a/√b=√a/b Multiplication = √ax x √b=√ab 
                                  1. Rationalising

                                    Nota:

                                    • rationalized where there are no surds in denominator. i.e. simple 2/√4 2√4/4 √4/2 conjugate3/√3-4 (√3+4)3√3+12/3-163√3+12/-13-3√3-12/13
                                    1. Proof by Contradiction

                                      Nota:

                                      • Assuming a surd is rational to prove it isn't e.g. √2  Assume √2 is rational and can be expressed as a/b in simplest form where a and b have no common factors √2=a/b 2=a{2}/b{2} 2b{2}=a{2} a{2} is an even number (multiple of 2) and also a is an even number (multiple of 2) a=2r a{2}=4r{2} however a{2}=2b{2} 2b{2}=4r{2} b{2}=2r{2} b{2} is an even number (multiple of 2) and also b is an even number (multiple of 2) both a and b are even numbers (multiple of 2) which contradicts the original assumption of √2=a/b where a and b have no common factors √2 is not rational √2 is irrational
                                2. Modulus (Absolute Values)

                                  Nota:

                                  • distance from the origin  |-50| = 50 |4| = 4
                                  1. Solving Modulus

                                    Nota:

                                    • 2 cases, where the equation is positive, and when the equation is negative
                                  2. Inequations

                                    Nota:

                                    • reverse the equation sign when you multiply or divide by a negaitve e.g  -2x>4 x<-2
                                    • 2 cases, when y>0 and when x<0 (x can be whole equations) however, check the answers to ensure they meet the original statement
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