RESIDUE CLASS Meaning and
Definition
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A residue class, often referred to as a congruence class, is a fundamental concept in number theory and abstract algebra. It is a subset of integers that are all congruent modulo a given integer.
In more precise terms, let's assume we have an integer "a" and a positive integer "n". The residue class of "a" modulo "n" is the set of all integers that leave the same remainder when divided by "n". This set can be denoted as [a]ₙ or simply as [a].
For instance, if "n" is 7 and "a" is any integer, the residue class [a]₇ would include all integers that have the same remainder as "a" when divided by 7. Therefore, [a]₇ would consist of the numbers {..., a-14, a-7, a, a+7, a+14, ...}.
Residue classes have several important properties. Firstly, every integer belongs to a unique residue class modulo "n". Secondly, the corresponding residue classes are equal if and only if the integers in these classes are congruent modulo "n". This allows us to perform arithmetic operations on residue classes.
Residue classes find various applications, particularly in number theory, algebraic structures, and cryptography. They are useful in solving problems related to equations, systems of linear congruences, modular arithmetic, and constructing finite fields. Overall, residue classes provide a powerful framework for studying the behavior of integers within a given modulus.
Common Misspellings for RESIDUE CLASS
- eesidue class
- desidue class
- fesidue class
- tesidue class
- 5esidue class
- 4esidue class
- rwsidue class
- rssidue class
- rdsidue class
- rrsidue class
- r4sidue class
- r3sidue class
- reaidue class
- rezidue class
- rexidue class
- redidue class
- reeidue class
- rewidue class
- resudue class
- resjdue class
Etymology of RESIDUE CLASS
The word "residue class" is a mathematical term that originated from the field of number theory.
The term "residue" comes from the Latin word "residuum", which means "remainder" or "that which is left behind". In number theory, a residue refers to the remainder obtained when a number is divided by another number. The idea of residues can be traced back to the ancient Greeks, who studied the properties of remainders in division.
The term "class" in this context refers to a set or group of objects that share a common characteristic or property. In mathematics, a class usually represents an equivalence class, where elements within the class are considered to be equivalent or indistinguishable in some sense.
Combining the two terms, "residue class" refers to a set of numbers that are equivalent to each other when divided by a given modulus.
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