Axioms and Postulates

A proof must start somewhere. We cannot derive every statement of geometry from other statements , somewhere we must accept a few things without proof. Those starting points are called axioms and postulates. Euclid carefully separated his into two lists; modern mathematics blurs the line, but the distinction is still useful pedagogically.

Definitions

An axiom (Euclid also calls them common notions) is a self-evident truth, applicable to mathematics generally , not specific to geometry. Examples are about equality and quantity: "things equal to the same thing are equal to one another".

A postulate is a starting assumption specific to geometry: a statement about points, lines, and circles that we accept without proof.

A theorem is a statement not assumed but deduced from axioms, postulates, definitions, and previously proved theorems by logical reasoning.

In modern usage the distinction between axiom and postulate has faded , both words now mean "a starting assumption". But Euclid kept them separate, and so will we for this lesson.

Euclid's five common notions (axioms)

Euclid's list reads simply:

Things equal to the same thing are equal to one another. If $a = c$ and $b = c$ , then $a = b$ .
If equals are added to equals, the wholes are equal. If $a = b$ , then $a + c = b + c$ .
If equals are subtracted from equals, the remainders are equal. If $a = b$ , then $a - c = b - c$ .
Things which coincide with one another are equal to one another. If two geometric figures can be superposed exactly, they have the same length, area, etc.
The whole is greater than the part. A proper sub-figure has smaller area, length, etc., than the whole figure.

These five sound obvious , and that is the point. They are exactly the kinds of statements no one would dispute, and they are the bedrock on which all measurement and equality arguments rest. Almost every geometric proof you will see in the next few chapters uses one of them, often without naming it explicitly.

Some modern axioms about points and lines

In addition to Euclid's common notions, modern geometry uses a handful of axioms about the incidence of points and lines:

A1. Two distinct points determine exactly one straight line. (Euclid's Postulate 1 in modern dress.)
A2. Through one point, infinitely many straight lines can pass. (You can draw any number of distinct lines through a single point.)
A3. Two distinct straight lines intersect in at most one point. (If they share two points, they must be the same line.)
A4. If $P$ is a point and $\ell$ is a line not through $P$ , there is at least one line through $P$ that does not meet $\ell$ . (Existence of a parallel.)

These four axioms describe how points and lines combine. Together with the common notions about equality, they let us start proving geometric statements.

Postulates vs. axioms , what's the practical difference?

Euclid intended his postulates to be geometric truths that one assumes but that might in principle need justification someday. His axioms (common notions) were taken as universally true. Modern mathematics rejects this hierarchy: all unproven starting statements are axioms.

For exam purposes you should know both Euclid's five common notions and his five postulates. The next lesson goes through the postulates in detail.

A quick example of axiom-in-use

Claim. If $A, B, C$ are three points on a line with $B$ between $A$ and $C$ , then $AB + BC = AC$ .

This sounds obvious, but a careful proof goes like this. The whole segment $AC$ consists of the union of the segment $AB$ and the segment $BC$ , joined at $B$ . By Common Notion 5 (the whole is greater than the part), $AC \ge AB$ and $AC \ge BC$ . Their sum equals $AC$ by additivity of length, which is itself a modern axiom (a refinement of Euclid's notions). So the statement reduces to applying a common notion correctly.

We will not always be this careful in class , but recognising that even simple statements rest on assumptions is the heart of mathematical maturity.

Worked examples

Example 1. State whether each is an axiom or a postulate (in Euclid's sense): "If equals are added to equals, the wholes are equal"; "A straight line may be drawn from any point to any other point".

First is a common notion (axiom): about quantities in general. Second is a postulate: about straight lines specifically.

Example 2. Which Euclidean common notion is being used in this argument? "If $A = B$ and $C = D$ , then $A + C = B + D$ ."

Add the second equality to the first: $A + C = B + C = B + D$ . This uses Common Notion 2 twice (adding equals to equals).

Example 3. Suppose two distinct lines $\ell$ and $m$ pass through both $P$ and $Q$ with $P \ne Q$ . What does Axiom A3 imply?

A3 says two distinct lines meet in at most one point. If $\ell$ and $m$ share two distinct points, they cannot be different lines , so $\ell = m$ .

Example 4. What is the difference between an axiom and a theorem?

An axiom is accepted without proof. A theorem is proved using axioms, definitions, and previously proved theorems.

Example 5. Is the statement "the sum of the angles in a triangle is $180^\circ$ " an axiom or a theorem?

A theorem. It is proved using the postulates and definitions.

Try it yourself

State Euclid's five common notions in your own words.
Give an example of an algebraic equation that uses Common Notion 2.
State whether each is an axiom or a theorem: "two distinct points determine a unique line"; "vertically opposite angles are equal"; "the area of a rectangle is base times height".
Why is "If $a = b$ , then $a + 5 = b + 5$ " considered self-evident?
State an axiom that justifies "since $\angle ABC = \angle DEF$ and $\angle DEF = \angle XYZ$ , we have $\angle ABC = \angle XYZ$ ".
Can two different lines have more than one common point? Justify using A3.
What is the role of axioms in mathematics?
Distinguish between a postulate and an axiom as Euclid used the terms.
Why do modern textbooks often merge "axiom" and "postulate"?
Write any common notion (in modern words) and an example of its use.

Pitfalls / Insight

Common Notions are not specific to geometry. They apply to numbers, areas, weights, anything that can be equal to or compared with something else.
Postulates are specific to geometry. They are about straight lines, points, and circles.
Theorems are proved, not assumed. Knowing which is which is fundamental to mathematical writing.

Insight. Axioms are the rules of the game and the moves we are allowed to make. Theorems are the positions we can reach by playing the game. Without clear axioms, no theorem can be proved with certainty; without theorems, axioms are just an isolated list. The next lesson goes through Euclid's five postulates , the specifically geometric rules , in detail.