maths:algebra

# a brief summary of algebra

## Field laws

• 11 field laws of algebra for real numbers (set of R):
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
• 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; 