“Lions and tigers, and bears, oh my!” ~ Dorothy in Wizard of Oz

Or should we say axioms, corollaries, lemmas, postulates, conjectures and theorems, oh my!

## Axiom

There are certain elementary statements, which are self evident and which are accepted without any questions. These are called **axioms.**

**Axiom 1: Things which are equal to the same thing are equal to one another.**

For example:

Draw a line segment AB of length 10cm. Draw a second line CD having length equal to that of AB, using a compass. Measure the length of CD. We see that, CD = 10cm.

We can write it as, CD = AB and AB = 10cm implies CD = 10cm.

Arif, View. 2016. “Axioms, Postulates And Theorems – Class VIII”. *Breath Math*. https://breathmath.com/2016/02/18/axioms-postulates-and-theorems-class-viii/.

A statement that is taken to be true, so that further reasoning can be done.

It is not something we want to prove.

Example: one of Euclid’s axioms (over 2300 years ago!) is:*“If A and B are two numbers that are the same, and C and D are also the same, A+C is the same as B+D”*

“Definition Of Axiom”. 2021. *mathsisfun.Com*. https://www.mathsisfun.com/definitions/axiom.html.

In mathematics an axiom is something which is the starting point for the logical deduction of other theorems. They cannot be proven with a logic derivation unless they are redundant. That means every field in mathematics can be boiled down to a set of axioms. One of the axioms of arithmetic is that a + b = b + a. You can’t prove that, but it is the basis of arithmetic and something we use rather often. “Theorems, Lemmas And Other Definitions | Mathblog”. 2011. *mathblog.dk*. https://www.mathblog.dk/theorems-lemmas/.

## Corollary

A theorem that **follows on** from another theorem.

Example: there is a **Theorem** that says: two angles that together form a straight line are “supplementary” (they add to 180°).

A **Corollary** to this is the “Vertical Angle Theorem” that says: where two lines intersect, the angles opposite each other are equal (a=c and b=d in the diagram).

Proof that a=c:

Angles a and b are on a straight line, so:

⇒ angles a + b = 180° and so a = 180° − b

Angles c and b are also on a straight line, so:

⇒ angles c + b = 180° and so c = 180° − b

So angle a = angle c

“Corollary Definition (Illustrated Mathematics Dictionary)”. 2021. *mathsisfun.com*. https://www.mathsisfun.com/definitions/corollary.html.

A corollary of a theorem or a definition is a statement that can be deduced directly from that theorem or statement. It still needs to be proved, though.

A simple example: Theorem: The sum of the angles of a triangle is pi radians.

Corollary: No angle in a right angled triangle can be obtuse.

Or: Definition: A prime number is one that can be divided without remainder only by 1 and itself.

Corollary: No even number > 2 can be prime.

A corollary is a theorem that can be proved from another theorem. For example: If two angles of a triangle are equal, then the sides opposite them are equal . A corollary would be: If a triangle is equilateral, it is also equiangular.

“What Are The Examples Of Corollary In Math? – Quora”. 2021. *quora.com*. https://www.quora.com/What-are-the-examples-of-corollary-in-math.

## Lemma

There is not formal difference between a theorem and a lemma. A lemma is a proven proposition just like a theorem. Usually a lemma is used as a stepping stone for proving something larger. That means the convention is to call the main statement for a theorem and then split the problem into several smaller problems which are stated as lemmas. Wolfram suggest that a lemma is a short theorem used to prove something larger.

Breaking part of the main proof out into lemmas is a good way to create a structure in a proof and sometimes their importance will prove more valuable than the main theorem.

“Theorems, Lemmas And Other Definitions | Mathblog”. 2011. *mathblog.dk*. https://www.mathblog.dk/theorems-lemmas/.

Like a Theorem, but not as important. It is a minor result that has been proved to be true (using facts that were already known). “Definition Of Lemma”. 2021. mathsisfun.com. https://www.mathsisfun.com/definitions/lemma.html.

## Postulate

**Postulates** in geometry are very similar to axioms, self-evident truths, and beliefs in logic, political philosophy and personal decision-making.

Geometry postulates, or axioms, are accepted statements or facts. Thus, there is no need to prove them.

For example:

**Postulate 1.1, Through two points, there is exactly 1 line. Line t is the only line passing through E and F.**

In geometry, “**Axiom**” and “**Postulate**” are essentially interchangeable. In antiquity, they referred to propositions that were “obviously true” and only had to be stated, and not proven. In modern mathematics there is no longer an assumption that axioms are “obviously true”. Axioms are merely ‘background’ assumptions we make. The best analogy I know is that axioms are the “rules of the game”. In Euclid’s Geometry, the main axioms/postulates are:

- Given any two distinct points, there is a line that contains them.
- Any line segment can be extended to an infinite line.
- Given a point and a radius, there is a circle with center in that point and that radius.
- All right angles are equal to one another.
- If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. (The
*parallel postulate*).

A **theorem** is a logical consequence of the axioms. In Geometry, the “propositions” are all theorems: they are derived using the axioms and the valid rules. A “Corollary” is a theorem that is usually considered an “easy consequence” of another theorem. What is or is not a corollary is entirely subjective. Sometimes what an author thinks is a ‘corollary’ is deemed more important than the corresponding theorem. (The same goes for “**Lemma**“s, which are theorems that are considered auxiliary to proving some other, more important in the view of the author, theorem).

A “**hypothesis**” is an assumption made. For example, “If xx is an even integer, then x2x2 is an even integer” I am not asserting that x2x2 is even or odd; I am asserting that if *something* happens (namely, if xx happens to be an even integer) then *something else* will also happen. Here, “xx is an even integer” is the hypothesis being made to prove it.

Difference between axioms, and hypotheses, Gordon Gustafson, and Arturo Magidin. 2010. “Difference Between Axioms, Theorems, Postulates, Corollaries, And Hypotheses”. *Mathematics Stack Exchange*. https://math.stackexchange.com/questions/7717/difference-between-axioms-theorems-postulates-corollaries-and-hypotheses.

In geometry, a postulate is a statement that is assumed to be true based on basic geometric principles. An example of a postulate is the statement “exactly one line may be drawn through any two points.” A long time ago, postulates were the ideas that were thought to be so obviously true they did not require a proof. ^{1}

## Conjecture

A **conjecture** is a mathematical statement that has not yet been rigorously proved. Conjectures arise when one notices a pattern that holds true for many cases. However, just because a pattern holds true for many cases does not mean that the pattern will hold true for all cases. Conjectures must be proved for the mathematical observation to be fully accepted. When a conjecture is rigorously proved, it becomes a theorem.

“Conjectures | Brilliant Math & Science Wiki”. 2022. *brilliant.org*. https://brilliant.org/wiki/conjectures/.

“The Subtle Art Of The Mathematical Conjecture | Quanta Magazine”. 2019. *Quanta Magazine*. https://www.quantamagazine.org/the-subtle-art-of-the-mathematical-conjecture-20190507/.

## Theorem

A result that has been **proved to be true** (using operations and facts that were already known).

Example: The “Pythagoras Theorem” proved that a^{2} + b^{2} = c^{2} for a right angled triangle.

A Theorem is a major result, a minor result is called a Lemma.

“Theorem Definition (Illustrated Mathematics Dictionary)”. 2021. *mathsisfun.Com*. https://www.mathsisfun.com/definitions/theorem.html.

“Theorems, Corollaries, Lemmas”. 2021. *mathsisfun.com*. https://www.mathsisfun.com/algebra/theorems-lemmas.html.

A statement that is proven true using postulates, definitions, and previously proven theorems.

A theorem is a mathematical statement that can and must be proven to be true. You may have been first exposed to the term when learning about the Pythagorean Theorem. Learning different theorems and proving they are true is an important part of Geometry. ^{1}

## References

^{1} “4.1 Theorems and Proofs”. 2022. *CK-12 Foundation*. https://flexbooks.ck12.org/cbook/ck-12-interactive-geometry-for-ccss/section/4.1/primary/lesson/theorems-and-proofs-geo-ccss/.

## Additional Reading

“Basic Math Definitions”. 2021. *mathsisfun.com*. https://www.mathsisfun.com/basic-math-definitions.html.

“Byrne’s Euclid”. 2021. *C82.Net*. https://www.c82.net/euclid/.

THE FIRST SIX BOOKS OF

THE ELEMENTS OF EUCLID

WITH COLOURED DIAGRAMS AND SYMBOLS

A reproduction of Oliver Byrne’s celebrated work from 1847 plus interactive diagrams, cross references, and posters designed by Nicholas Rougeux

“Definitions. Postulates. Axioms: First Principles Of Plane Geometry “. 2021. *themathpage.com*. https://themathpage.com/aBookI/first.htm#post.

“Geometry Postulates”. 2021. *basic-mathematics.com*. https://www.basic-mathematics.com/geometry-postulates.html.