maths:algebra
a brief summary of algebra
Introduction
Field laws
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
polynomials:
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:
Binomial theorem:
Summation of a series
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:
Relation:
Ordered Pair Notation:
Mapping Notation:
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:
Identity Functions:
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