Most math content on the internet is either tutorials on calculus or celebrity oddities like the Fibonacci sequence. Derek Muller's piece stands out because it introduces a genuinely foreign concept: numbers that behave in ways our standard number system cannot accommodate — and shows why mathematicians actually need them.
Muller opens with what feels like a magic trick. "Take the number five and square it, you get 25," he writes. "Now take 25 and square it, you get 625... so does this pattern continue?" The pattern does continue — into infinity — and produces something unexpected: a number that is its own square.
This isn't a party trick. It's the gateway into an entire number system that most people, including those who finished math degrees, have never encountered. The piece's real argument comes when Muller asks: "does it even make sense to talk about numbers that have infinite digits going off to the left of the decimal point? I mean, isn't that just Infinity?" His answer is yes — and these numbers solve problems that regular numbers simply cannot.
A number will have infinite digits, and if you square it, you'll get back that same number. The number is its own square.
The claim is bold: these "10-adic" numbers (written in base 10) allow you to do addition, subtraction, and multiplication exactly as you'd expect. They contain fractions without using the division symbol. They include negative numbers "without having to use additional symbols." The structure of the number itself encodes negatives — a property Muller demonstrates by showing how all 9s equals negative one.
This is where the piece becomes genuinely compelling. Because these numbers behave differently, they break something mathematicians rely on: the ability to factor and solve equations by factoring. "If several terms multiplied together equal zero," Muller explains, "you know at least one of those terms must be zero." This tool — fundamental to solving polynomial equations — fails with 10-adic numbers.
The reason isn't arbitrary; it's structural. "We're working in base 10 and 10 is a composite number — it's not a prime, it's 5 times 2," Muller writes. The fix is elegant: use a prime base instead. This creates p-adic numbers (where P stands for Prime), which restore the property that if two numbers multiply to zero, one of them must be itself entirely zero.
In the three-adics, the only way they multiply to zero is if one of the three-adic numbers is itself entirely zero — and this works for any prime base.
The connection to Fermat's Last Theorem emerges naturally here. Muller recounts how in 1637, Pierre de Fermat wrote a statement that "will go down in history as one of the most infamous of all time": x^n + y^n = z^n has no solutions in integers for any n greater than 2. "I have a truly marvelous proof of this fact," Fermat famously wrote in the margin of his copy of Diophantus's Arithmetica, "but it's too long to be contained in the margin."
The solution required new numbers — p-adic numbers — precisely because they restore the ability to factor and solve equations that work with prime bases. The modular arithmetic approach Muller describes allows mathematicians to work out coefficients "one at a time by first solving the equation mod 3, then mod 9, then mod 27."
Why This Matters Outside Abstract Math
The piece makes something genuinely difficult accessible: it shows readers that when they encounter equations with no solutions in ordinary numbers, the problem isn't the equation — it's the number system. "They were even involved in cracking one of the most legendary math problems of all time," Muller notes about p-adic numbers.
This framing is effective because it reframes what seems like abstract nonsense as practical infrastructure. The numbers aren't invented to be weird; they're invented because regular numbers fail at certain tasks. It's a subtle but powerful shift — from "math is strange" to "math is the right tool for certain jobs."
Critics might note that the piece moves quickly through several complex concepts without fully connecting them. The jump from 10-adic self-squaring to Fermat's Last Theorem is conceptually dense, and viewers may need to watch twice to catch all the logic. But that's also the point: these ideas are interlocked.
Bottom Line
Muller accomplishes something difficult: making advanced number theory genuinely entertaining through a single visual — infinite digits extending leftward like decimals in reverse. His strongest move is connecting this curiosity to real mathematical breakthroughs, showing that 10-adic and p-adic numbers aren't just weird puzzles but essential tools used by Fields medalists. The piece's vulnerability is its speed — viewers will either grasp the connections or miss them entirely, with little middle ground. Either way, they'll remember the number that's its own square.