From TI-84 Craigslist Scores to Conjectures That Keep You Up — A Grad’s Guide to Doing Math Like an Adult

# Dr. Katya Steiner — The Categorical Imperative

You finished college, swapped ramen for a grocery list that occasionally contains vegetables, and suddenly you have the audacity to ask the big questions: which books will actually teach you math, will that half-dead TI-84 be worth thirty bucks, and is your midnight shower thought about digit-sums a Nobel-level idea or just Tuesday? Welcome to doing math as an adult — less attendance, more existential dread, but also more joy if you learn to play smart.

## Start here: books that won’t make you cry (too much)

If you want fundamentals without a PhD-level initiation rite, think balance: clarity, a little rigor, and a pinch of joy. Don’t hoard canonical tomes like they’re museum artifacts. Choose three companions and read them like friends — not like enemies to be conquered.

– Proofs and thinking like a mathematician: pick a gentle primer that teaches logic, quantifiers, and proof techniques. You’re training a muscle: spotting why something is true rather than memorizing formulas. Do the exercises with a pen. Re-do proofs. Repeat until it stops feeling like a foreign language.
– One rigorous subject pick: analysis or linear algebra. The goal isn’t to become an epsilon-descendent overnight; it’s to develop patience and exactness. For linear algebra, get a book that explains vector spaces and transformations before it dives into determinants. Trust me: the conceptual switch from “how to compute” to “what it means” is career-defining.
– A playful text: combinatorics, elementary number theory, or recreational problem-solving. These books rebuild curiosity and problem-solving muscle — the joy of a trick or a cute identity is underrated therapy.

Read proofs slowly. Annotate, question, and try variant versions. If a proof is clever, ask whether the trick generalizes. That curiosity is how you go from student to independent thinker.

## The broken TI-84: yes, you will still be judged by your calculator

I’ve seen grown mathematicians auction off calculators like endangered species. Here’s the cold take: calculators are convenience, not character. A dead screen and a working board are fixable; a fried motherboard is not.

Practical checklist when buying used:

– battery cover and batteries present,
– boot behavior (does it boot? does anything flash?),
– keys intact, no sticky buttons,
– reset works or at least can be invoked.

Alternatives: emulators and laptop graphing software do everything better. But if a class bans laptops or you crave that tactile nostalgia, buy a working unit. If the price is tiny and you like tinkering, buy for parts. Otherwise, don’t romanticize “cheap”; sometimes convenience costs less than regret. Hell, sometimes thirty bucks buys you fewer headaches and more sleep.

## Conjectures: what to do when your shower thought seems deep

You notice digit sequences where sum equals product, and you conjecture the shared value can’t exceed twice the number of digits. Cute, elegant, and exactly the kind of idea that wakes you at 2 a.m. What’s the adult move? Treat it like a kid: gentle testing, then escalation.

How to attack a neat little conjecture:

– Reduce complexity. Digits are 0–9, but if leading zeros are banned your digits are 1–9; ones are neutral for product, twos behave nicely, and digits ≥3 rapidly inflate products. Categorize by counts of 1s, 2s, and “big” digits.
– Casework is your friend. Once you control the number of digits ≥3, the problem often collapses to a finite check. Use inequalities like AM-GM to relate sums and products, or simple bounding to show impossibility beyond a threshold.
– Look for invariants and replacements. If product equals sum, replacing a 1 with two numbers that multiply to 1 (not possible among digits) or replacing a 2 with ones changes the balance — these local moves reveal global constraints.

You probably won’t get a Fields Medal. You will get a proof that sharpens your proof-crafting skills far more effectively than doomscrolling. And you’ll tell a better story at parties.

## Playful structures: the charm of magic squares

Magic squares are pure craft disguised as math: geometry, symmetry, arithmetic and the calming ritual of rearrangement. Treat them as constrained design problems — change the rules slightly and you’ve got new territory.

– Vary the magic total, allow repeated entries, or impose geometric constraints (diagonal shapes, toroidal wrap-arounds).
– Small experiments yield families. Record them, notice symmetries, see which transformations preserve magic. You’ll discover that a tiny tweak can produce dozens of related configurations.

This is the kind of math that doubles as meditation and Instagram content. Don’t undervalue it: playful experiments often seed formal questions.

## When algebra goes weird: computing geometric products in odd metrics

If you learned only orthonormal bases, non-Euclidean metrics feel like your algebra suddenly grew elbows. The inner product becomes a matrix, and the identities you rely on need careful reinterpretation.

Two practical approaches:

– Change the basis. Diagonalize or orthonormalize the metric (Gram–Schmidt if appropriate). Compute in that friendly basis, then translate back. It’s conceptually clean but can be algebra-heavy.
– Work with metric components. Use the rule e_i e_j = g_{ij} + e_i ∧ e_j and expand carefully. It’s messier but keeps you in original coordinates and forces discipline with signs and wedge products.

Both approaches teach a lesson: basis choice matters. When your algebra starts to smell funny, the right change of coordinates usually clears the air.

## Takeaway: be curious, not frantic

Being a math-inclined adult means juggling textbooks, thrift-store calculators, late-night conjectures, and projects that are part craft, part obsession. Don’t collect math like trophies; practice it like a hobby you actually enjoy. Read selectively, do exercises with intention, play with weird examples, and write down your midnight ideas.

Failures count. They’re practice, amusement, and narrative fuel. A proof, a counterexample, or a dead-end all teach you about what questions are interesting and why. The trick is to keep your curiosity calibrated: rigorous where it matters, playful where it helps, and humble enough to ask for help.

So here’s the thing I want you to take home: the adult mathematician’s life is less about prestige and more about process — selecting the right texts, making practical gear choices, and giving small ideas the time they deserve. And when your head buzzes with a new conjecture at 2 a.m., write it down, test small cases, and then either prove it or fold it into your curious ephemera.

Before you go: what small, apparently trivial pattern have you marginally obsessed over that turned out to teach you something deeper?