In a first number theory course, there’s typically a proof that the square root of 2 is irrational. The traditional proof uses the properties of divisibility to arrive at a contradiction. We can even swap a 5 into the proof, in place of the 2, to prove the square root of 5 is irrational. But could we do the same with a 9, and “prove” that the square root of 9 is irrational?
Tag: discrete math
Mathematical Security of Complex Passwords
One common complexity requirement for passwords is that they contain at least one uppercase letter, one lowercase letter, one number, and one symbol. In this blog post, we’ll explore what effect that complexity requirement has on the mathematical security provided by a password. Specifically, we’ll calculate how many bits of security a password generated with or without this requirement provides.
The Cardinality of Natural Numbers and Integers
Both the set of natural numbers and the set of integers are infinite in size. Amazingly, though, they’re exactly the same size. In this blog post, we’ll explore a technique for showing that these two infinite-sized sets have equal cardinality: creating a bijection between them. We’ll use this technique to show that, counter to all intuition, there are exactly as many natural numbers as there are integers.
The Ambiguous Subset Symbol
A common question students face on exams is to determine whether one set, A, is a subset of another set, B. Similarly, a question might ask if set A is a strict subset of B. Questions like these often contain symbols that look similar, and one symbol that has an ambiguous meaning altogether. In this blog post, we work to demystify the symbols used for subsets and strict subsets, to help us better understand different textbooks and online resources on the topic.