Definition

A precise and unambiguous description of the meaning of a mathematical term. It characterizes the meaning of a word by giving all the properties and only those properties that must be true.

Theorem

A mathematical statement that is proved using rigorous mathematical reasoning. In a mathematical paper, the term theorem is often reserved for the most important results.

Lemma

A minor result whose sole purpose is to help in proving a theorem. It is a stepping stone on the path to proving a theorem. Very occasionally lemmas can take on a life of their own (e.g., Zorn’s lemma).

Corollary

A result in which the (usually short) proof relies heavily on a given theorem (we often say that “this is a corollary of Theorem A”).

Proposition

A proved and often interesting result, but generally less important than a theorem.

Conjecture

A statement that is unproved, but is believed to be true (e.g., Collatz conjecture, Goldbach conjecture, twin prime conjecture).

Claim

An assertion that is then proved. It is often used like an informal lemma.

Axiom/Postulate

A statement that is assumed to be true without proof. These are the basic building blocks from which all theorems are proved (e.g., Euclid’s five postulates, Zermelo-Frankel axioms).

Identity

A mathematical expression giving the equality of two (often variable) quantities (e.g., trigonometric identities, Euler’s identity).

Equation

An equation is a mathematical statement that expresses the equality between two expressions, but this equality may not always hold true for all values of the variables.

• An equation is true only for specific values of the variables.
• An identity is true for all values of the variables.

Paradox

A statement that can be shown, using a given set of axioms and definitions, to be both true and false. Paradoxes are often used to show the inconsistencies in a flawed theory (e.g., Russell’s paradox).

Algebra

Algebra is beautiful because of its universality and elegance in capturing the essence of patterns and relationships. It works consistently irrespective of time, space, or context. Whether you’re on Earth or the Moon, the same algebraic rules apply, making it a universal language that transcends boundaries.

Algebra doesn’t depend on specific values; it helps us understand general relationships between quantities. For example, the formula for the area of a circle (πr²) holds true regardless of the size of the circle or where it is located.

Algebra also reveals hidden patterns in the world around us. By representing complex relationships with simple symbols, it helps us see the underlying structure. Equations like y = 2x + 3 describe not just one point, but an infinite number of points on a line — all encapsulated in a single, neat equation. This ability to represent the infinite in such a compact and clear form is one of the most remarkable aspects of algebra.

In short, algebra is beautiful because it transforms complexity into simplicity, revealing deep patterns and relationships in a way that is elegant, universal, and timeless.

Numbers

A number is a mathematical concept used to represent a quantity, position in a sequence, or value, and can be used to measure, count, or label objects or concepts.

• Measure: Numbers help quantify or assess the size, amount, or extent of something. Example: The length of a table could be measured as 5 feet.
• Count: Numbers are used to determine how many objects are in a set or collection. Example: If you have 3 apples, the number “3” counts how many apples you have.
• Label: Numbers can be used to identify or label objects in a specific order or category, without necessarily implying a quantity. Example: In a race, the runners are assigned the numbers 1, 2, 3, and so on.