Question 510131
Definitions do not need to be proven; they are simply definitions. A theorem is a statement that requires a proof, in which the proof is often based on previously proven theorems or axioms. It is important you know the difference between a definition and a theorem; when you have to prove something you need to make clear definitions and label your steps sequentially. Here are several examples of definitions/theorems from a wide variety of subjects:


Definition: for a given line segment there exists a midpoint.
Theorem: the midpoint divides the segments into two equal segments.

---------------------

Definition: we can define a complex number in the form a + bi where i is the imaginary unit.
Theorem: *[tex \LARGE e^{i \theta} = \cos{\theta} + i \sin{\theta}] (Euler's formula)

---------------------
Definition: given arbitrary non-negative real numbers *[tex \LARGE a_1], *[tex \LARGE a_2], ..., *[tex \LARGE a_n] and *[tex \LARGE b_1], *[tex \LARGE b_2], ..., *[tex \LARGE b_n], we may assume *[tex \LARGE a_i \le a_j] for *[tex \LARGE i \le j] and *[tex \LARGE b_k \le b_m] for *[tex \LARGE k \le m].

Theorem: The sum *[tex \LARGE \sum a_ib_j] is maximized with *[tex \LARGE M = a_1b_1 + a_2b_2 + ... + a_nb_n] and minimized with *[tex \LARGE m = a_1b_n + a_2b_{n-1} + ... + a_nb_1] (Rearrangement inequality).

---------------------
Definition: the altitudes of a triangle meet at a common point called the orthocenter.
Theorem: there exists such a point (this is a tricky one and the proof is not often covered in geometry classes).

---------------------

Definition: z is a root of the polynomial *[tex \LARGE f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_0] if f(z) = 0.
Theorem: Any n-degree polynomial contains exactly n complex roots, including multiple roots (fundamental theorem of algebra).