Erika Duan 2023-07-22
Summary
This tutorial introduces basic algebra concepts and illustrates some algebraic tricks, such as factorisation and working with inequalities, that are useful for solving mathematical problems related to calculus and statistical mathematics in later tutorials.
Elementary algebra, also referred to as just algebra, is commonly
associated with tasks like Find x if
.
However, it is more useful to think of algebra as a form of expression
that allows us to represent infinitely possible terms using a finite
one.
For example, the terms
,
and
are equivalent and infinite variations of this expression exist, with
the simplest being
.
When we read the term
,
we can intuit that 2 parts of
and 1 part of
are always required
(usually to make up another quantity).
Mathematics involves being precise with descriptions, and it is much
easier to write
than to write ‘all possible values where we have 2 parts of one
component and one part of a different component’.
An algebraic term can therefore be decomposed into three components:
- Variable(s): a variable is a varying quantity of an entity, usually
represented by concise symbols such as
,
,
or
where
.
- Operator(s): the arithmetic operation applied to variables. For
example, in additive models, the relationship between parameters
is additive and the dependent variable
for per unit increase in
.
- Relative quantity of variable(s): For example, let
describes a 1:1 ratio of eggs to sugar whereas the term
describes a 2:1 ratio of eggs to sugar and will result in a very different taste.
A few rules of algebraic manipulation are:
- We can simplify product terms using product expansion. For example,
.
- We cannot further simplify a term if it is the input of another
mathematical operator. For example,
(2x + 5y)^2 = 3z$.
- We can add or subtract fractions by multiplying the fractions to form
a common denominator. For example,
.
::: panel-tabset ## R
# Solve algebraic term in R ----------------------------------------------------
4 <= 4
#> [1] TRUE
4 < 4
#> [1] FALSE
class(4 < 4)
#> [1] "logical"Factorisation is the reverse process to product expansion and can be
thought of as breaking down a fully expanded algebraic term into the
product of its factors. For example, the factors of
are
and
as
.
The reason why factorisation is useful is that it allows us to solve for special function properties, for example to identify whether a quadratic function intersects the x-axis.
Quadratic equations with the form
can be simplified through factorisation using:
Algebraic terms using inequalities are common when we want to prove the
existence of an upper or lower bound. For example, if A is an event in
the probability space, we know that the probability of event A occurring
is between 0 and 1 inclusive i.e.
.
There are three rules for manipulating inequalities:
- Adding or subtracting the same quantity from both sides of an inequality leaves the inequality symbol unchanged.
- Multiplying or dividing both sides of an inequality by a positive number leaves the inequality symbol unchanged.
- Multiplying or dividing both sides of an inequality by a negative number reverses the inequality symbol.
In R, inequality statements are outputted as Boolean values i.e. TRUE
or FALSE.
# Compute inequality in R ------------------------------------------------------
4 <= 4
#> [1] TRUE
4 < 4
#> [1] FALSE
class(4 < 4)
#> [1] "logical"In Python, inequality statements are also outputted as Boolean values
i.e. True or False.
# Compute inequality in Python -------------------------------------------------
4 <= 4
#> True type(4 <= 4)
#> <class 'bool'> In Julia, inequality statements are also outputted as Boolean values
i.e. true or false.
# Compute inequality in Julia --------------------------------------------------
4 <= 4
#> true
typeof(4 <= 4)
#> true
#> Bool
a = 1
b = 2
c = 3
a < b, a + c < b + c
#> (true, true) - Entry on algebra from the Stanford Encyclopedia of Philosophy.
- Khan academy YouTube series on algebra basics.
- A factsheet on manipulating inequalities from the Uk Maths Centre.