How the Square Root of 2 Became a Number | Quanta Magazine

🌈 Abstract

The article discusses the historical development of the understanding of irrational numbers, such as the square root of 2, and how mathematicians like Pythagoras, Dedekind, and Cantor contributed to formalizing the concept of irrational numbers.

🙋 Q&A

[01] The First Crisis in the Foundations of Mathematics

1. What was the first crisis in the foundations of mathematics that the article mentions?

• The article states that the ancient Greeks' aspiration to describe the universe using only whole numbers and ratios was undermined when they realized that the length of the diagonal of a square with sides of length 1 could not be written as a fraction. This was the first crisis in what is called the foundations of mathematics.

2. Who is credited with the first proof of this realization, and what is known about this person?

• The first proof of this realization is commonly attributed to Pythagoras, a 6th-century BCE philosopher, even though none of his writings survive and little is known about him.

3. How did the ancient Greeks respond to this crisis?

• Though the ancient Greeks could establish what √2 was not, they did not have a language for explaining what it was. This ambiguity persisted for millennia.

[02] Defining Irrational Numbers

1. How did Dedekind define irrational numbers?

• Dedekind introduced a way to define and construct the irrational numbers using only the rationals. He did this by splitting all the rational numbers into two sets, where one set contains all rationals whose squares are less than 2, and the other set contains all rationals whose squares are greater than 2. The number that plugs the hole between these two sets is defined as √2.

2. How did Cantor's definition of irrational numbers differ from Dedekind's?

• Cantor came up with a different definition of irrational numbers, expressing each in terms of sequences of rational numbers that approached, or "converged" to, a particular irrational value. Though Cantor's irrational numbers initially looked different from Dedekind's, later work proved that they are mathematically equivalent.

3. What did Cantor's work lead him to realize about the nature of infinity?

• Cantor's work led him to ask how many numbers exist. He showed that, paradoxically, though the number of fractions is the same as the number of integers, there are demonstrably more irrational numbers. He was the first to realize that infinity comes in many sizes.

[03] The Significance of Dedekind's Work

1. How did Dedekind's work on irrational numbers impact the development of mathematics?

• Dedekind's cuts are considered the beginning of modern mathematics, as they allowed mathematicians to actually know what they were talking about when discussing irrational numbers and calculus. Dedekind's work enabled mathematicians to better understand sequences and functions, and it is said that "everything is already in Dedekind."

2. How did Dedekind's work open up new horizons for mathematical exploration?

• A formal definition of √2 opened new horizons for exploration beyond the topics in calculus that initially motivated Dedekind. Mathematicians started to realize that they could invent new concepts altogether, and the whole idea of what mathematics is about became much broader and more flexible.