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 is a proposition that, based on certain assumptions or hypothesis, can testably assert a non-self-evident thesis (because in that case it would be a axiom). They are very common within formal languages, like the math wave logic, since they constitute the enunciation of certain formal rules or “game” rules.
Theorems not only propose stable relations between the premises and the conclusion, but also provide the fundamental keys to prove it. The proof of theorems is, in fact, a key part of mathematical logic, since others can be derived from one theorem and thus enlarge the knowledge of the formal system.
However, in the field of mathematical studies, the term "theorem" is used only for propositions of particular interest to the academic community. In contrast, in first-order logic, any provable statement is itself a theorem.
The word “theorem” comes from the Greek theorem, derived from the verb theory, which means "contemplate", "judge" or "reflect", from which the word "theory" also comes.
For the ancient Greeks, a theorem was the result of careful and careful observation and reflection, and it was a term used very 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 theoretical and the second is practical.
Every theorem has three parts:
- Hypothesis either premises. It is the logical content from which the conclusion can be deduced and, 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 deductions or secondary and additional formulations that are obtained from the theorem.
Pythagoras theorem
The Pythagorean Theorem is one of the oldest mathematical theorems known to mankind. It is attributed to the Greek philosopher Pythagoras of Samos (c. 569 – c. 475 BC), although the theorem is believed to be much older, possibly of Babylonian origin, and that Pythagoras was the first to prove it.
This theorem proposes that, given a triangle rectangle (that is, having at least one right angle), the square of the length of the side of the triangle opposite 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 any right triangle, the square of the hypotenuse will be equal to the sum of the squares of the legs.
And with the following formula:
a2 + b2 = c2
Where a Y b equal 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:
a = √c2 – b2
b = √c2 – a2
c = √a2 + b2
The Pythagorean theorem has been proven 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) deals with the geometry of the triangles, as follows:
- Thales' first theorem proposes that if one of the sides of a triangle is continued beyond 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 ratio of two of the sides of the large triangle (A and B) will always be equal to the ratio of the same sides of the small one (C and D).
A/B = C/D
This theorem served, according to the Greek historian Herodotus, 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 circumference whose diameter is AC and center "O" (different from A and C), a right triangle ABC can be formed such that
Two corollaries follow from this:
- In any right triangle, the length of the median corresponding to the hypotenuse is always half the hypotenuse.
- The circumscribed circumference 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's 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 the theory of probability, and is formulated as follows:
This means that it is possible to calculate the probability of an event (A) if we know that it meets a certain necessary condition for its occurrence, inversely to the total probability theorem.
Other known theorems
Other famous theorems are:
- Ptolemy's theorem. It holds that in every cyclic quadrilateral, the sum of the products of the pairs of opposite sides is equal to the product of their diagonals.
- The Euler-Fermat theorem. He maintains that yes a Y n are integers relative cousins, then n divides to aᵩ(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 at (a, b) such that a tangent line at that point is parallel to the secant line through the points (a, F(a)) and (b, F(b)).
- Thomas' theorem. He argues that if people establish a situation as real, that situation becomes real in its consequences.