"A day without sunshine is like... well, night."
There is a strange creature in mathematics, called an axiom (or, in some texts, a postulate). Axioms are usually not mentioned much, if at all, in math courses at the lowerdivision level; however, I think it is important to know where math comes from so that we may know the road on which we tread.
DEFINITION  Axiom
An axiom is a selfevident or universally recognized truth. It is accepted as true, without proof, as the basis for argument.
An axiom is a selfevident or universally recognized truth. It is accepted as true, without proof, as the basis for argument.
Like definitions, the truthfulness of any axiom is taken for granted; however, axioms do not define things – instead they describe a basic, underlying quality about something. The opening quote of this section can be considered an axiom. It is not defining sunshine; it is just stating something that is obvious.
An axiom has the feel of something that should be justifiable or proved. Oddly enough, though, axioms cannot be proved\(^1\). They are jumping points for future logical deductions, but nothing exists to state initially that the axiom itself is true.
As we shall see later on in this textbook, the foundation of everything we know in mathematics comes from a simple set of axioms. 
Mathematical Interest 
EXAMPLE 1  Everyday axioms
Each of the following statements is an example of an axiom.
Each of the following statements is an example of an axiom.
 "If I am wearing a red shirt and blue jeans, then I am wearing something that is red."
 "If I have $30 and you have $30, then we have the same amount of money."
 "It is possible to draw a line between any two points."
The first statement is almost trivial. There is no arguing that I would be wearing red because I already told you I was wearing red. The second is also obvious. Finally, it's easy to see that the third statement is a universally recognized truth. It is one of those statements that students, and most professors, would likely say, “It's true because… well… it's obvious.”
It takes some extremely creative thinking to come up with axioms; however, that is not the goal of this section nor will we even approach that conversation in this textbook. When we begin our exploration of numbers, we will introduce the ten basic axioms upon which almost all of mathematics is built. It is enough for now that you know there is an object in mathematics called an axiom.
It takes some extremely creative thinking to come up with axioms; however, that is not the goal of this section nor will we even approach that conversation in this textbook. When we begin our exploration of numbers, we will introduce the ten basic axioms upon which almost all of mathematics is built. It is enough for now that you know there is an object in mathematics called an axiom.
\(^1\) The idea of a proof is coming up soon.

