Equations
An equation is a mathematical expression with an equals sign in it.
Possible elements of an equation are:
- numbers
- variables
- binary operators (+, -, * or ÷)
- unary operators (+ or -)
- brackets
- functions (sin(), xy , etc.)
These are explained in following sections.
Solving equations
Once we can solve quadratic equations and simultaneous equations it is tempting to think that we will be able to solve more and more complicated equations by using the same principles but its not always that simple. The quadratic equation:
a x² + b x + c = 0
can be solved using a formula but most polynomial equations don't have formulas to give the solution. Every polynomial with only real or complex coefficients has a complex number solution , this is the fundamental theorem of algebra, but they cannot always be expressed exactly with radicals. To find an approximation to the solution we may have to use numerical methods such as the Newton-Raphson method or Laguerre method.
Generalised reciprocity laws are very complicated algorithms that enable you to get crucial information about some of these more complicated equations. In favorable circumstances, they can be used to prove deep statements about the solution sets of algebraic equations.
An equation does not have to be too complicated to lack a formula, for example:
x5 - x + 1 = 0
Numbers
There are different types of numbers, for instance:
- Z- Integers - whole numbers: ... -3, -2, -1, 0, 1, 2, 3 ... both positive, zero and negative.
- Real
numbers - numbers which are continuous such as when we are representing points along a line - On this site I will sometimes use the term 'Scalar' to mean 'Real' numbers although strictly the term should be used when scaling a vector - In computer programs real numbers have a finite length and may have decimal point and/or exponent this allows us to approximate most real numbers but it is only an approximation. - Q - Rational numbers - Integers and fractions where numerator and denominator are integers.
- Radical Integers - The integers plus any combination of addition, subtraction, multiplication, division and root extraction.
- Qalg - The root of a Z-polynomial - A complex number made up more than just radical integers although it is closed under sum, difference, product, quotient and nth root.
- Modulo 'n' numbers
- C - Complex Numbers - numbers with real and imaginary parts.
- H - Quaternion - Complex number whose elements are complex numbers.
- O - Octonion - Quaternion whose elements are complex numbers.
- Vectors - one dimensional arrays of numbers.
- Matrices - two dimensional arrays of numbers.
and these numbers may be coded in different ways:
- binary
- octal
- decimal
- hexadecimal
Unless otherwise specified we usually assume that numbers are decimal.
Number
theory is a big subject with topics like prime
numbers, Galois theory, Fermats last
theorem containing many active ares of mathematical study.
Variables
Sometimes an equation contains a number but we don't yet know its value, or we may want to apply the equation to a range of values.
An example of the first is using x as the unknown, for example,
x + 1 = 3
An example of the second might be an equation of a line:
y = 2 * x + 3
In general we use:
x, y, z for unknowns.
a, b, c for values which are not yet specified.
Binary Operators
The operators:
- + add
- - subtract
- * multiply (was x but for computers has to be changed to distinguish it from 24th letter of the alphabet)
- ÷ or / divide
take the two numbers on either side and replace it by a single number.
Unary Operators
The operators:
- + plus
- - minus
apply to the number to the right. '-' inverts the number (subtracts from 0) '+' says the number to the right is positive (the default).
Brackets
When we mix + and * then the answer we get depends on the order that we apply them.
For example
2 + 1 * 3
To clarify this we can put brackets around the operation to be applied first:
(2 + 1) * 3 = 9
2 + (1 * 3) = 5
If we don't specify which has precedence by using bracket then by default * and ÷ have precedence over + and -. So,
2 + 1 * 3 = 5
Functions
A function takes one number and uses it to generate another number. For example the function sin() takes an angle as input and returns the ratio of opposite and hypotenuse in a right angled triangle.
