MATHEMATICAL INDUCTION Meaning and
Definition
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Mathematical induction is a method used in mathematics to prove statements about positive integers or other mathematical objects that can be structured in a sequence. It is a powerful proof technique that allows us to establish the truth of an infinitely many number of statements by verifying only two crucial properties: a base case and an inductive step.
In mathematical induction, the method starts by proving a statement for the smallest possible value, typically denoted as the base case (often 1 or 0, depending on the context). Once the base case is shown to hold, the inductive step is employed. This step involves demonstrating that if the statement is true for a given value (k), it must also hold for the next value (k+1). This process creates a "domino effect" whereby, if the first domino falls (base case), all subsequent dominos (values) will also fall.
The fundamental idea behind mathematical induction is that if the base case holds, and every subsequent case implies the one after it, then the statement is true for all positive integers (or other objects in a sequence). This technique is commonly used to prove statements involving natural numbers, algebraic expressions, inequalities, divisibility, and other related concepts.
Mathematical induction is frequently applied in various branches of mathematics, such as number theory, combinatorics, algebra, and calculus. It provides a systematic and robust approach to prove statements involving recurring patterns and structures, making it an indispensable tool for mathematical reasoning and theorem proving.
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Etymology of MATHEMATICAL INDUCTION
The word "mathematical induction" has its roots in Latin and Greek.
The term "mathematical" comes from the Latin "mathematicus", which in turn comes from the Greek "mathematikos", meaning "inclined to learn" or "studious". The Greek word "mathema" stands for "science" or "learning".
The term "induction" comes from the Latin "inductio", which means "a leading in" or "introduction". It is derived from the Latin verb "inducere", meaning "to lead or bring in".
When we combine these two terms, "mathematical induction" refers to the method or process of introducing and establishing a mathematical proof by starting with a base case and then proving that if a statement is true for one case, it is also true for the next case, thus leading to the conclusion that the statement is true for all cases.
