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maths:algebra

a brief summary of algebra

Introduction

Field laws

  • 11 field laws of algebra for real numbers (set of R):
field law addition multiplication
closure a + b subset of R ab subset of R
commutative a + b = b + a ab = ba
associative a + (b + c) = (a + b) + c a(bc) = (ab)c
identity a + 0 = a a x 1 = a
inverse a + -a = 0 a x a-1 = 1, a <> 0
distributive a(b+c) =  ab + ac as for additive

Basic theorems

  • some basic theorems of real numbers:
    • if a + c = b + c then a = b (additive cancellation law)
    • if ac = bc then a = b (multiplicative cancellation law)
    • if ab = 0, then either a = 0 or b = 0 or both equal zero (this helps us solve equations like (x-2)(x-3) = 0)
    • if a > b and b > c then a > c (transitive property)
    • if a > b then a + c > b + c (additive property)
    • if a > b then ac > bc (multiplicative property)
  • some properties of surds:
    • closure law: when two different surds are added, they cannot be represented by another single surd
    • when unlike surds are multiplied, the product is a surd: sqrt(a) x sqrt(b) = sqrt (a x b)
    • however, when surds are multiplied or divided, the result may be a rational number, thus surds do not obey the closure laws
  • Square root:
    • x2 = y ⇒ x = +-sqrt(y), y >= 0;
  • Modulus:
    • |x| = sqr x2

polynomials:

  • a polynomial is an algebraic expression of the form:
    • P(x) = anxn + an-1xn-1 + … + a1x + a0, where n, n-1, … are elements of J+; an, an-1, … are elements of R, & x is a variable
  • remainder theorem:
    • enables one to find a remainder R without actual division
    • if P(x) is divided by (x-a) then the remainder R = P(a)
    • if P(a) = 0 ie. remainder is zero then (x-a) is a factor of P(x) - the factor theorem
  • quadratic polynomials:
    • ax2 + bx + c where a <> 0, which can be expressed as a difference of two squares as long as the discriminant, b2 - 4ac >= 0 

Factorisation and expansion

  • Some basic factors:
    • difference of two squares: x2 - a2 = (x - a)(x + a) 
    • x2 + a2 = no real factors
    • x3 - a3 = (x - a)(x2 + ax + a2)
    • x3 + a3 = (x + a)(x2 - ax + a2)
  • Some basic expansions:
    • (x - a)2 = x2 - 2ax + a2
    • (x + a)2 = x2 + 2ax + a2
    • (x + a)3 = x3 + 3x2a + 3xa2 + a3
    • (x - a)3 = x3 - 3x2a + 3xa2 - a3

Binomial theorem:

  • if n is set of N:
  • (x+a)n = xn + nxn-1a + n(n-1)xn-2a2/2 +…..+ [n(n-1)(n-2)..(n-r+1)]xn-rar / r! + .. + an
  • Linear approx. of (1 + x)n:
    • ~ 1 + nx, if x is small, n < R;

Summation of a series

  • Summation of a finite series:
    • Arithmetic:
      • a + (a + d) + … + (a + (n - 1)d) = (n/2)(2a + (n - 1)d)
    • Geometric:
      • a + ar + ar2 + … + ar(n - 1) = a(1 - rn)/(1 - r)
  • Summation of an infinite series:
    • the Basel lighthouse problem and inverse square law of light:
      • eg. if place a series of lighthouses sequentially further from a subject but each added lighthouse is same distance from the previous lighthouse and is of same intensity a, what is the intensity of light the subject sees?
      • a + a/4 + a/9 + a/16 + … a/(n^2) = a *pi2/ 6

Expansions:

  • sinx = x - (x3/3!) + (x5/5!) - (x7/7!) + …;
  • cosx = 1 - (x2/2!) + (x4/4!) - (x6/6!) + …;
  • ln(1+x) = x - (x2/2) + (x3/3) - (x4/4) + …;
  • (1+x)n = 1 + nx + n(n-1)x2/2! + n(n-1)(n-2)x3/3! +.;

Functions

  • Cartesian Product:
    • eg. 2 sets:
      • X = {2,3}, Y = {4,5}.
      • X * Y = {(x,y): x E X, y E Y}
        • ie. {(2,4),(2,5),(3,4),(3,5)}
  • Relation:
    • a set of ordered pairs: (domain,range)
  • Ordered Pair Notation:
    • eg. {(x,y): y=x2, xER }
  • Mapping Notation:
    • eg. f: X → Y where f(x)=x2
      • ie. X is mapped onto/into Y by the function f
      • X is the domain;
      • Y is the codomain;
      • f(x) is the range;
  • Algebra of Functions:
    • (f+g)(x) = f(x) + g(x)
    • (fg)(x) = f(x) * g(x);
    • (f/g)(x) = f(x)/g(x), g(x) not= 0;
    • domain(D) = d(f+g) = d(fg) = d(f/g) = df intersection dg (NB. except for (f/g)(x) where exclude values x where g(x) = 0)
    • eg. if f(x) = x2 and g(x) = 2x+4 then (f+g)(x) = x2 + 2x + 4
  • Composite Functions:
    • when x is operated on successively by f & then by g, ie. any occurrence of x in g(x) is substituted by f(x)
      • gof(x) = g[f(x)], domain = df,
      • range f must be subset domain g;
      • eg. if f(x) = x2 and g(x) = 2x+4 then gof(x) = 2x2 + 4
  • Identity Functions:
    • foI = Iof = f, eg I(x)=x;
  • Inverse Functions:
    • fof -1 = f -1 = I, r(f -1) = df, rf = d(f -1);
    • f must be a 1 to 1 function;
maths/algebra.txt · Last modified: 2021/07/24 14:33 by gary1