Intuitive Maths

NT Insights Vol 1.1 Addition on Natural Numbers We have already seen how the 5 axioms together generate the entire set of natural numbers in Vol 1.0 . Now we will define a binary operation, called addition, on in terms of those 5 axioms. Before that, let us give these elements of their usual names, as follows: and so on… An important observation is that the successive natural numbers differ by a single reference to . Let us call a reference to as a unit distance. Then, clearly, the successive natural numbers are unit distance apart. Also, the difference between any two natural numbers is a multiple of this unit distance. Another important observation is that does not involve any reference to function . This is so because all other numbers are successors of , but is not successor of any natural number (Reminds me of Lord Shiva, who is called Anaadi). The magnitude of distance, therefore, can not be assigned to . Whereas, for every non-zero natural number , it...

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...

NT Insights Vol 1.0 Axiomatizing Natural Numbers One can not prove theorems by assuming nothing. But what is more fascinating is that a large number of concepts can be explained by a fixed number of elementary notions, known as axioms . Axioms are assumed to be self-evident truths. One such set of axioms about Natural Numbers was proposed by Peano: Peano’s Axioms Let denote a set. Suppose we are not permitted to see its contents. Instead, we are given the following information about : A mapping , different from indentity map, such that is invariant under , i.e., Pre-image of does not exist under , i.e., there does not exist an such that The mapping is injective, i.e., If with and , then With these axioms alone, we can describe/generate the entire set , without actually seeing its elements. Let us see HOW . Let us start by analyzing what are we provided with. We are sure that there exists an element denoted by in . So is non-empty. Additional...