Intuitive Maths

Counting Insights Vol 1.2 Applications of Pigeon-Hole Principle Guarantees in a Decimal Expansion A rational number with decimal expansion of the form has a non-empty list of digits ( ) repeating, hence is denoted by For example, A careful study of long division method shows that digits in decimal expansion repeat because remainder repeats itself in the long division method. Therefore, exactly the same sequence of quotients and remainders are generated as before, causing the sequence of digits to repeat (in decimal expansion). A natural question to ask is, why does the remainder repeat? Theorem 2. is rational has repeating decimal expansion. Proof: Let for some and . Without loss of generality, we can assume . By division algorithm, for every integers with unique integers and such that , where . If , we have repeating zeros in decimal expansions. If , then there are only possible non-zero values that can have. Considering the next steps...

Counting Insights Vol 1.1 Pigeon-Hole Principle In Vol 1.0 of -to- Functions, we considered the Devotee-Footwear problem , in which we asked: what is the minimum possible count of devotees present in the temple (exclude the priest)? Now a question arises that why there exists a minimum count (lower-bound) on the number of devotees? Can we prove mathematically the existence of such a lower-bound? Theorem 1. Let be -to- function, and let be finite. Then, set is finite with size . Proof : Since is given to be finite, let . Since is -to- . . To show upper-bound on size of , assume to be surjective. at most pre-images in . But at most elements in . Thus, . Note: achieves its maximum size only when is a surjection and every has number of pre-images. Let us now recall, = set of all footwears lying outside the temple entrance, and = set of all devotees present inside the temple, excluding the priest. We defined actual mapping that assignes e...

Counting Insights Vol 1.0 K-to-1 Functions Let us ask ourselves a basic question: How do we count ? Suppose we have a collection of items. We assign to each item a number from in increasing order (of course starting from 1, then 2, then 3 and so on). The last number assigned is the value of the count (final answer). Notice that we can not assign the same number from to two different items. Thus, if we consider our assignment policy as a mapping from the set of items to , then it is injective in nature. Thus, we can conclude: When we count the number of elements in a set S, we are essentially establishing an injection from set S to . So, when confronted with the questions like: “ how many of the students in your class share birthday (day in week) with Ramesh?”, we can be certain that this counting needs an injective map from some set to . What that set is, how to arrive at this mapping? - before being able to answer these questions, we need to transform the prob...