Euler, Erdős

In the beginning, there was nothing, \( 0 = \emptyset \). Then we realized it was something, \( 1 = \{ \emptyset \} \). Then we wondered, why not have another thing? \(2\). And another thing, \(3\). And another, \(4\), and another, \(5\), and another, \(6\), etc. Just like that, we had the natural numbers!¹ And there was an obvious way of "adding" these numbers,² which is too complicated to get into in this post.³ The natural numbers are what a mathematician would now call a monoid.

A monoid is a set with an associative binary operation (like addition) and an identity element (like zero).

We loved the natural numbers! They represent progress! They always moved forward into the future and never turned back! But we are humans and humans are nostalgic beings. We missed the past and the simplicity of \( 0 \). In the midst of these midlife crisis, we as a people found the perfect solution. We invented the reversed passage of time. We invented the negative number that allowed us to go back to the beginning. We did this knowing that it also meant we could go past the beginning and see what came before the beginning.⁴ So the abelian group called the integers was born.

A group is a monoid where the binary operation can always be reversed (think subtraction). If the order in which the operation is applied (\( a + b \) vs \( b + a \)) doesn't matter, it is known as an abelian group.

The integers were amazing; they knew no limitations. Are you craving to see what the future has in store for you? Do you miss the good ol' days? Would you rather just stay exactly where you are? The integers are just for you! But soon we realized how tedious they could get. No one wanted to take a hundred steps just to get from \( 20 \) to \( 120 \). The world would be a better place if we just took fifty strides that were twice as long. Or twenty leaps that were equivalent to five steps each. Thus we started multiplying numbers and the integers became a ring.⁵

A ring is an abelian group with a new binary operation (called multiplication) that satisfies the distributive law with the group operation (called addition).

This new addition, no pun intended, to our skillset came with a caveat: "A negative number multiplied with another negative number should be a positive number, and asking why is punishable by \( (-10)(-10) \) push-ups. How many push-ups is that? We'll multiply by \( 2 \) for each wrong answer you give!"^{6 7}

We then started asking weird questions like, "If you were \( 55 \) and there was a happy memory at \( 20 \) that you wanted to revisit after \( 3 \) long strides, then how long should each stride be?" To be honest, these were post-modern rhetorical questions that were accidentally taken seriously.⁸ But the damage was done and we did the rational thing, i.e., answered the rhetorical question with fractions and division.⁹ The integers were upgraded from ring to field and renamed as the rational numbers.

A field is a commutative ring with a multiplicative identity where each non-zero element has a multiplicative inverse (basically you can divide too).

Now remember the people who asked a rhetorical question that flew right over everyone else's head? They got back to the drawing board and, after a little critical analysis, concluded that "to really show how absurd numbers are, we need to remove all ties to reality and ask really abstract questions." They figured that purely abstract questions will obviously be interpreted as the rhetorical questions they really are. So they asked, "So you think the rational number system is the answer to everything? Then tell us this, what rational number when multiplied with itself gives the number two? What does that even mean?" To their chagrin, the first question was promptly answered, "Yes!" and the second one had them stumped.¹⁰ No one ever discussed or even thought about the third question.

Since answering questions involving numbers had basically become our civic duty¹¹, we eventually came up with a rather complicated solution.¹² We imagined the integers as equally-spaced dots on a straight-line and the rational numbers as even smaller dots that seemed to fill in the line but in reality didn't! So to truly make a line, we simply filled in all gaps on the line.¹³ So \( \sqrt{2} \) was one such gap which was kinda close to the rational number \( 1.414 \). The new numbers that were used to fill the gap were called the irrational numbers.¹⁴

And together, all numbers were real numbers. We were so proud of our achievement that we expected calling them "real" would put some finality to all the madness. But it was only the beginning. Everything we thought we knew about numbers was about to change forever, and ever, and ever, and ever, and ever, etc.

This is what I imagine the history of real numbers was after reading math textbooks.