maths:algebra
Table of Contents
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