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:**- x
^{2}= y ⇒ x = +-sqrt(y), y >= 0;

**Modulus:**- |x| = sqr x
^{2}

## polynomials:

- a polynomial is an algebraic expression of the form:
- P(x) = a
_{n}x^{n}+ a_{n-1}x^{n-1}+ … + a_{1}x + a0, where n, n-1, … are elements of J^{+}; a_{n}, a_{n-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:**- ax
^{2}+ 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: x
^{2}- a^{2}= (x - a)(x + a) - x
^{2}+ a^{2}= no real factors - x
^{3}- a^{3}= (x - a)(x^{2}+ ax + a^{2}) - x
^{3}+ a^{3}= (x + a)(x^{2}- ax + a^{2})

**Some basic expansions:**- (x - a)
^{2}= x^{2}- 2ax + a^{2} - (x + a)
^{2}= x^{2}+ 2ax + a^{2} - (x + a)
^{3}= x^{3}+ 3x^{2}a + 3xa^{2}+ a^{3} - (x - a)
^{3}= x^{3}- 3x^{2}a + 3xa^{2}- a^{3}

### Binomial theorem:

- if n is set of N:
- (x+a)
^{n}= x^{n}+ nx^{n-1}a + n(n-1)x^{n-2}a^{2}/2 +…..+ [n(n-1)(n-2)..(n-r+1)]x^{n-r}a^{r}/ r! + .. + a^{n} **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 - r^{n})/(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 *pi
^{2}/ 6

## Expansions:

- sinx = x - (x
^{3}/3!) + (x^{5}/5!) - (x^{7}/7!) + …; - cosx = 1 - (x
^{2}/2!) + (x^{4}/4!) - (x^{6}/6!) + …; - ln(1+x) = x - (x
^{2}/2) + (x^{3}/3) - (x^{4}/4) + …; - (1+x)
^{n}= 1 + nx + n(n-1)x^{2}/2! + n(n-1)(n-2)x^{3}/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=x
^{2}, xER }

**Mapping Notation:**- eg. f: X → Y where f(x)=x
^{2}- 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) = x
^{2}and g(x) = 2x+4 then (f+g)(x) = x^{2}+ 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) = x
^{2}and g(x) = 2x+4 then gof(x) = 2x^{2}+ 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