reziduál

Reziduál, also known as remainder, is a fundamental concept in mathematics, particularly in arithmetic and algebra. In the context of division, when a number, called the dividend, is divided by another number, called the divisor, the result is a quotient and a remainder. The remainder is the amount "left over" after the division is performed as many times as possible without resulting in fractions or decimals. For example, when 10 is divided by 3, the quotient is 3 and the remainder is 1, because 3 times 3 is 9, and 10 minus 9 equals 1.

The remainder is always less than the absolute value of the divisor. Mathematically, this relationship is expressed by the division algorithm, which states that for any integers a (dividend) and b (divisor) with b non-zero, there exist unique integers q (quotient) and r (remainder) such that a = bq + r, where 0 ≤ r < |b|. This principle applies to both positive and negative integers.

The concept of remainder is crucial in various areas of mathematics, including modular arithmetic, cryptography, and number theory. Modular arithmetic, for instance, focuses on remainders after division by a specific number (the modulus). This is often denoted using the modulo operator, symbolized by "%" in many programming languages or "mod" in mathematical notation. For example, 10 mod 3 = 1. The remainder's properties are also utilized in algorithms for checking divisibility and finding patterns in sequences of numbers.