Theorem (Meaning and Explanation)

We explain what a theorem is, its function and what its parts are. In addition, the theorems of Pythagoras, Thales, Bayes and others.

What is a theorem?

a theorem It is a proposition that, starting from certain assumptions or hypotheses, can verifiably affirm a non-evident thesis by itself (since in that case it would be an axiom). They are very common within formal languages, such as mathematics or logic, since they constitute the enunciation of certain formal rules or “game” rules.

Theorems not only propose stable relationships between the premises and the conclusion but they also provide the fundamental keys to verify it. The proof of theorems is, in fact, a key part of mathematical logic, since others can be derived from one theorem and thus expand the knowledge of the formal system.

However, in the field of mathematical studies, the term “theorem” is used only for propositions of particular interest among the academic community. On the other hand, in first-order logic any provable statement constitutes a theorem in itself.

The word “theorem” comes from the Greek theoremderived from the verb theoreinwhich means “contemplate”, “judge” or “reflect”, where the word “theory” also comes from.

For the ancient Greeks, a theorem was the result of careful and attentive observation and reflection, and was a term used frequently by many philosophers and mathematicians of the time. From there also comes the academic distinction between the terms “theorem” and “problem”: the first is of a theoretical type and the second of a practical type.

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Every theorem consists of three parts:

Hypotheses or premises. It is the logical content from which the conclusion can be deduced and which, therefore, precedes it.
Thesis or conclusion. It is what is stated in the theorem and that can be formally demonstrated from what is proposed by the premises.
Corollaries. They are those secondary and additional deductions or formulations that are obtained from the theorem.

Pythagorean theorem

The Pythagorean theorem is one of the oldest mathematical theorems known to humanity. It is attributed to the Greek philosopher Pythagoras of Samos (c. 569 – c. 475 BC), although it is believed that the theorem is much older, possibly of Babylonian origin, and that Pythagoras was the first to verify it.

This theorem proposes that, given a right triangle (that is, one that has at least one right angle), the square of the length of the side of the triangle opposite to the right angle (the hypotenuse) will always be equal to the sum of the square of the length of the other two sides (called legs). This is stated as follows:

In every right triangle, the square of the hypotenuse will be equal to the sum of the squares of the legs.

And with the following formula:

to² + b² = c²

Where to and b are equivalent to the length of the legs and c to the length of the hypotenuse. From there three corollaries can also be deduced, that is, derived formulas that have practical application and algebraic verification:

to = √c² –b²
b = √c² – to²
c = √a²+b²

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The Pythagorean theorem has been demonstrated numerous times throughout history: by Pythagoras himself and by other geometers and mathematicians such as Euclid, Pappus, Bhaskara, Leonardo da Vinci, Garfield, among others.

Thales Theorem

Attributed to the Greek mathematician Thales of Miletus (c. 624 – c. 546 BC), this two-part theorem (or these two theorems with the same name) has to do with the geometry of triangles, as follows manner:

Thales' first theorem proposes that if one of the sides of a triangle is continued further by a parallel line, a larger triangle but of the same proportions will be obtained. This can be expressed as follows:

Given two proportional triangles, one large and one small, The quotient of two of the sides of the large triangle (A and B) will always be equal to the quotient of the same sides of the small one (C and D).

A/B = C/D

theorem first such theorem

This theorem, according to the Greek historian Herodotus, helped Thales to measure the size of the pyramid of Cheops in Egypt, without having to use instruments of immense size.

Thales' second theorem proposes that given a circle whose diameter is AC and center “O” (different from A and C), a right triangle ABC can be formed such that

second such theorem

From there two corollaries emerge:

In every right triangle, the length of the median corresponding to the hypotenuse is always half of the hypotenuse.
The circumscribed circle of any right triangle always has a radius equal to half the hypotenuse and its circumcenter will be located at the midpoint of the hypotenuse.

Bayes theorem

Bayes' theorem was proposed by the English mathematician Thomas Bayes (1702-1761) and published after his death in 1763. This theorem expresses the probability of an event “A given B” occurring and its relationship with the probability of an event “B given A”. This theorem is very important in probability theory, and is formulated as follows:

bayes theorem

This means that it is possible to calculate the probability of an event (A) if we know that it meets a certain condition necessary for it to occur, inversely to the total probability theorem.

Other known theorems

Other famous theorems are:

Ptolemy's theorem. He maintains that in every cyclic quadrilateral, the sum of the products of the pairs of the opposite sides is equal to the product of their diagonals.
The Euler-Fermat theorem. He maintains that yes to and n are relatively prime integers, so n divide to to^ᵩ(n)-1.
Lagrange's theorem. He maintains that yes F is a continuous function on a closed interval (a, b) and differentiable on the open interval (a, b), then there exists a point c in (a, b) so that a tangent line at said point is parallel to the secant line that passes through the points (a, F(a)) and (b, F(b)).
Thomas's theorem. He maintains that if people establish a situation as real, said situation becomes real in its consequences.