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matqkks
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What the best way to introduce congruences in a number theory course? I am looking for something which will have an impact. What are the really interesting applications of congruent mathematics?
Ask your students this: If it rains at midnight, what is the probability that it will be sunny in 72 hours?matqkks said:What the best way to introduce congruences in a number theory course? I am looking for something which will have an impact. What are the really interesting applications of congruent mathematics?
Cute, but "in 72 hours" can reasonably be interpreted as "72 hours from now". Better wording "If it rains at midnight, what is the probability it will be sunny 72 hour later?Evgeny.Makarov said:Ask your students this: If it rains at midnight, what is the probability that it will be sunny in 72 hours?
Modular arithmetic is a type of arithmetic that deals with integers and their remainders when divided by a fixed number called the modulus. It is also known as clock arithmetic because it is often used to solve problems involving time and periodic events.
Congruences are mathematical statements that describe the relationship between two numbers in modular arithmetic. They are written as a ≡ b (mod m), which means that a and b have the same remainder when divided by m.
Modular arithmetic has many real-world applications, such as in cryptography, computer science, and music theory. It is also used in calculating the day of the week, determining leap years, and solving problems related to repeating patterns or cycles.
Modular arithmetic has several properties that make it useful for solving problems. These include the commutative, associative, and distributive properties, as well as the cancellation property and the Chinese Remainder Theorem.
The main difference between modular arithmetic and regular arithmetic is that in modular arithmetic, the numbers "wrap around" when they reach the modulus. This means that the result of any operation will always be a number between 0 and m-1, where m is the modulus. In regular arithmetic, there are no boundaries for the numbers, and they can continue to increase or decrease infinitely.