Congruence Explained

Congruence Explained

According to, congruence, from the Latin congruentia, is the coherence or logical relationship. It is a characteristic that is understood from a link between two or more things. For example: “It is not consistent that you want to give a gift to the person with whom you have a legal dispute”, “The judge detected several inconsistencies between the defendant’s statements and the evidence”, “Each part of this system is consistent with the others ”.

For mathematics, congruence is the algebraic expression that expresses the equality of the remainders of the divisions of two congruent numbers by their modulus (a natural number other than 0). This expression is represented by three horizontal lines between the numbers and, if we assign them the variables a and b, it is read as follows: a is congruent with b modulo m.

Mathematical congruence, therefore, refers to two whole numbers that have the same remainder when divided by a natural number other than zero (the modulus).

On the other hand, for mathematical identity, the concept of congruence can refer to Fermat’s little theorem (one of the most prominent in relation to divisibility), which presents the following formula: if we have the prime number p, then for everything natural number a is given that a raised to p is congruent with a modulus p.

This same theorem is usually presented differently, although both formulas are equivalent: if we have the prime number p, then for all a, a relative prime natural number with p, a raised to p -1 is congruent with 1 modulus p. In other words, if we subtract a from the result of raising that number to p, we get a number divisible by p.

Furthermore, the term congruence is used to express an equation with a minimum of one unknown; in this case, it is intended to know if there is a solution, or more than one.

It is worth mentioning that several of the properties of congruence are also found in equality; Let’s see some examples:

* when the module is fixed, the congruence represents an equivalence, since it is possible to check the reflexivity (a is congruent with a module m), the symmetry (if a is congruent with b module m, then b is congruent with a module m) and transitivity (if a is congruent with b modulo m and b is congruent with c modulo m, then a is congruent withc modulus m);

* if a is a relative prime to m and a is congruent to b modulo m, then it is correct to say that b is a relative prime to m ;

* if a is congruent with b modulo m and there is an integer k, then it is correct to state that: the sum of a and k is congruent with the sum of b and k modulo m ; the product of k by a is congruent with the product of k by b modulo m ; a raised to k is congruent with b raised to k modulo m, provided that k is greater than 0.

The congruence between polygons, on the other hand, is the one-to-one correspondence between their vertices such that the angles are congruent (that is, they have the same measure), as are their sides (that have the same length).

In the field of law, consistency is the conformity between the pronouncements of a judgment and the claims that the parties had made during the trial.

As a rational method of conflict resolution, the judicial process must achieve concordance between the plaintiff’s claim, the defendant’s opposition, the evidence and the court’s decision. This concordance is what is known as congruence.

In religion, finally, congruence is the efficacy of God’s grace, with its ability to work without interfering with the freedom of the human being.

Congruence Explained