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EPISODE 7 IS OUT 🎉

This week on From Zero to Zeta:

We ask the really important mathematical questions, like: *“When does normal lead straight into homo-…morphisms?” (Yes, both are math terms. Yes, the pun was irresistible.)

We dig into: • Lagrange’s Theorem • symmetric groups and why they’re chaotic • alternating groups and why they’re slightly less chaotic • what “normal subgroup” actually means • why homomorphisms naturally create layers you can collapse • and how kernels show you exactly what a map can’t see

It’s group theory with maximum drama and maximum structure.

Can Uniqueness Survive Forever? In this combinatorial deep dive, we explore a question with a surprisingly inevitable answer:

Fix a nonempty set S of positive integers. Say that n has a unique representation as a sum of an odd number of distinct elements from S. Show that there are infinitely many n that don’t.

The proof shows why — through parity and combinatorial reasoning — uniqueness can’t last.

🎥 Watch the full video 📚 Problem originally from Actual Education (YouTube/@actual.education) 💬 Join the community: https://discord.gg/wYdSJ2HsZp

Two planes in ℝ³. Clearly geometric. Clearly intersecting. But what exactly is the intersection — and how do you describe it?

This one’s all about structure: reducing a system of linear equations, interpreting the result geometrically, and expressing the solution set as a parametric line in space.

The Short gives you the problem and leaves you thinking. This longer version shows how to break it down cleanly.

🔗 Short (1 min): https://youtube.com/shorts/E6nejFGW7-I 🔗 Full walkthrough (5 min): https://youtu.be/3vzmKQdx0dc

In this tutoring livestream, we explore a deceptive factorial equation involving integers n and m:

((n-1)! − n)(n-2)! = m(m-2), with n and m both greater than 2.

At first, it feels like there should be some clever pair of integers that make it work — but as the reasoning unfolds, every case fails in a surprising way. In the end, it turns out there’s no solution at all.

It’s a perfect example of how math problems that look “solvable” at first glance can reveal elegant impossibilities once you dig in.

Two worlds, one structure.

What if "even and odd" weren’t just number categories, but parallel universes of arithmetic? In the newest episode of From Zero to Zeta, we explore: → how equivalence relations give rise to equivalence classes → how subgroups carve those into cosets → and how modular arithmetic turns those slices into a whole new group.

It’s where abstraction meets symmetry — and where modular math stops being a trick with remainders and starts being a structural truth about the universe of numbers.

🎥 Watch now on YouTube: https://youtu.be/YLLP9beJd-I

We started at 0… and ended up beyond infinity.

In this first episode of Math Wikipedia Roulette, I let Wikipedia decide where math takes us — and in just three clicks, we went from the concept of 0, through zero morphisms and the category of sets, all the way to inaccessible cardinals. From nothingness to the unreachable edges of infinity, all in under ten minutes.

🎥 Watch the full episode: https://youtu.be/KSRnsjtkdpE

🗳️ Vote in the poll for Episode 2: I’ve narrowed it down to three directions, and you get to decide where we go next: 🔹 Regular cardinal — https://en.wikipedia.org/wiki/Regular_cardinal 🔹 Gödel’s Second Incompleteness Theorem — https://en.wikipedia.org/wiki/G%C3%B6del%27s_second_incompleteness_theorem 🔹 Large cardinal — https://en.wikipedia.org/wiki/Large_cardinal You can find the poll here: https://www.youtube.com/post/Ugkx2bfMLkOJNojJXZelrJvNRLW_MlHWciCd

This one’s got everything:

• A solid of revolution • An improper integral • An antiderivative that turns into arctangent • And a final step that makes you think about where arctangent actually goes at the extremes

Direct integration works — but only if you’ve internalized the key formulas and can recognize where inverse tangent shows up.

📚 The Language of Groups — Group Theory, Episode 5

In this new episode of From Zero to Zeta, I move from intuition to precision by building the language of group theory.

We’ll explore:

Set-builder notation, common symbols, and logical shorthand

Functions (domain, codomain, image, inverse, preimage)

Injective, surjective, and bijective functions

Standard group notation: identity, inverses, generated subgroups, orders of groups and elements

Bijections as the foundation for isomorphisms

And then we’ll practice with Cayley tables and Cayley graphs, two complementary ways of seeing group structure at a glance.

This episode equips you with the vocabulary to start speaking group theory. Next time, we’ll use that language to study cosets and equivalence relations — opening the door to theorems like Lagrange’s.

One GRE question tests how well you know your calculus theorems — continuity, integration, differentiability, all in one.

The other looks complicated — vector spaces, matrices, linear transformations — but it collapses with a single line of reasoning if you know what to look for.

Two very different problems. One minute. No calculator.

Three GRE-style questions in 60 seconds:

• A geometry problem on maximizing area with limited fencing • A number theory question: what’s the units digit of 7^25? • A polynomial root-counting challenge — no graphing required

All three come fast — but each one tests something different: reasoning, pattern recognition, and algebraic structure.

Three questions. One minute. No calculator.

1️⃣ A derivative graph — where is the function largest? 2️⃣ An approximation — how close can you get to √1.5 × (266)^(3/2)? 3️⃣ A matrix — can you spot the guaranteed eigenvector?

Each one uses a different kind of reasoning, and all three go by fast — but you can pause if you want to dig in.

Three definite integrals. None of them have a closed-form solution. But the question still asks: Which one is biggest?

This GRE problem looks computational — but the solution is all about function behavior. I compare the derivatives instead of evaluating the integrals directly.

If you want to follow the logic, you might want to pause a few times — the idea is simple, but the reasoning moves fast.

📺 More GRE Shorts → 🌐 Website + tutoring info 💬 Join the Discord

🧠 GRE Math: You Know This — If You Don’t Overthink It

Two GRE-style questions in under 60 seconds:

1️⃣ A definite integral with absolute value — clean split, clean answer. 2️⃣ A triangle inscribed in a semi-circle — what's the maximum area it can have?

The second one feels tricky, but it really isn’t… if you trust your intuition. I included a full geometric proof on-screen, going through the broad strokes audibly — pause if you want to walk through its full detail.

You’ve got 59 seconds. Can you get both GRE questions right?

First one’s quick — just a parabola and a point. Second one? Not so fast. It’s you vs. the circle. How many times can a circle intersect a parabola?