Math C Revision

1. Abelian Groups
   1. Commutative

      Annotations:

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

   Annotations:

   • 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
3. Properties
   1. Closure

      Annotations:

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

      Annotations:

      • 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
   3. Identity Element

      Annotations:

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

      Annotations:

      • 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
4. Cayley Table

   Annotations:

   • http://openi.nlm.nih.gov/imgs/512/14/3586959/3586959_1742-4682-9-54-9.png

1. Geometric

   Annotations:

   • 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

      Annotations:

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

      Annotations:

      • 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

   Annotations:

   • 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

      Annotations:

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

      Annotations:

      • tn=a+(n-1)d

1. Rational

   Annotations:

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

      Annotations:

      • 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

   Annotations:

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

         Annotations:

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

         Annotations:

         • 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
      3. Proof by Contradiction

         Annotations:

         • 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

Annotations:

• distance from the origin  |-50| = 50 |4| = 4

1. Solving Modulus

   Annotations:

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

Annotations:

• 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