Number theory

Question

What is the largest number? What does infinity mean?

Answer

Hi Alex! This is a very interesting question. A short answer would be that there is no largest number. Infinity is not a number, but is a concept that means without limit or without end. Even though infinity is not a number, it can be used to answer some "how many" questions. For example, it is correct to say that there are infinitely many numbers because there is no limit on them, for any number there is always a bigger number. Actually, even the question "What is a number?" has caused a surprising amount of discussion and argument among philosophers and mathematicians through history. Numbers are concepts, and there are different types of numbers. The numbers we usually think of are "natural concepts". This means that they arise naturally from noticing and counting different numbers of objects in the world around us. This has happened in very similar ways in different cultures and languages. Once we have the concept of numbers for counting, the concept of fractions follows naturally when we think about dividing objects into different numbers of pieces. It is also possible to extend the concept of number to include negative numbers, all the numbers on a number line (called "real numbers"), and more kinds of numbers. Besides the "natural" concepts of numbers, there are also "formal" concepts of numbers, which are studied by mathematicians in fields like Foundations of Mathematics, Number Theory, and Set Theory. If you find numbers and infinity interesting, introductory books on these subjects can be found in libraries, starting around the high school level.