MathIsland
We are all mathematicians.
Mathematics — a new way of seeing the world
A place to experience math hands-on
MathIsland
The interactive labs are currently in Korean — full English versions are on the way. Every lab below has an English overview.
Why before formula
We never open with a definition. Each lab starts from the problem people faced before the concept existed.
Experiments you can touch
Move sliders, run simulations, watch graphs respond in real time — understanding through your hands.
Concepts that connect
Every concept was born from the limits of the previous one. Each lab hands you over to the next.
Concept Labs
Mathematics connects by questions, not by grade levels. Start anywhere on the seven journeys.
The Journey of Numbers
How much is there — and how do we write it down?
Irrational Numbers
The holes in the world of fractions
Why does this exist?
The diagonal of a unit square cannot be written as any fraction. Filling the number line demanded a new kind of number.
Prime Numbers
The atoms of multiplication
Why does this exist?
Every natural number breaks into primes in exactly one way — and that uniqueness quietly protects your credit card.
The Journey of Language
How do we handle what we don't yet know?
Exponents
The explosion of repeated multiplication
Why does this exist?
Fold a sheet of paper 42 times and it reaches the Moon. Linear intuition always loses to repeated multiplication.
Logarithms
The tool that turns multiplication into addition
Why does this exist?
To compare numbers from 1 to 10 billion in a single view, we needed a way to compress them.
Equations
Isolating the unknown while keeping the balance
Why does this exist?
For which value do the two sides agree? 'Moving terms across' is really just operating a balance scale.
Factoring
Breaking an expression back into its blocks
Why does this exist?
Structure invisible in the expanded form appears the moment you factor — suddenly the roots show themselves.
The Journey of Relations
When one thing changes, how does another respond?
Functions
The language of input–output rules
Why does this exist?
Most change in the world is 'something depending on something else'. A function states that relationship precisely.
Linear Functions
Change at a constant speed
Why does this exist?
Base fare plus a per-unit rate — phone plans and taxi meters are all straight lines. Slope is the seed of the derivative.
Quadratic Functions
When the rate of change itself changes
Why does this exist?
Why does a thrown ball trace exactly that curve? It is the curve of every trajectory under gravity — and of every optimum.
Exponential Functions
The curve of everything that explodes
Why does this exist?
Epidemics, compound interest and populations all grow the same way — in proportion to their current size.
Logarithmic Functions
A lens that compresses explosions to human scale
Why does this exist?
Thirty-year stock charts, earthquake magnitudes, pandemics — exploding values only compare fairly on a log scale.
Trigonometric Functions
Unrolling rotation into waves
Why does this exist?
To capture everything that spins — wheels, seasons, sound — triangle ratios had to be set free on a circle.
The Journey of Change
How fast do things change — and what does change add up to?
Sequences
Reading the pattern in a march of numbers
Why does this exist?
To predict what a repeating process does next, you first have to write its pattern as numbers.
Limits
Handling values you approach but never reach
Why does this exist?
Only when 'almost there' is made rigorous can derivatives and integrals stand at all.
Derivatives
Seeing the speed of change, not the result
Why does this exist?
Sometimes you need to know how fast things are changing right now — not on average.
Optimization
The best hides where the slope is zero
Why does this exist?
Biggest profit, least cost, shortest time — most of life's questions are 'find the best point'.
Integrals
Rebuilding the whole from thin slices
Why does this exist?
If all you know is the speed, how far did you travel? Integration computes the accumulation.
Fundamental Theorem of Calculus
Accumulating and differentiating are mirror images
Why does this exist?
The discovery that they undo each other — the greatest shortcut in the history of mathematics.
The Journey of Space
Where is it, what shape is it, which way does it point?
Pythagorean Theorem
The conservation law of right angles
Why does this exist?
The ancestor of every distance formula. GPS, game physics and vector lengths all descend from it.
Trigonometric Ratios
Measuring the unreachable with a single angle
Why does this exist?
Measure a river without crossing it, a mountain without climbing it — the magic ruler that similar triangles gave us.
Vectors
Numbers that carry direction
Why does this exist?
For every problem where 'which way' matters as much as 'how much'.
The Journey of Uncertainty
How do we judge when we cannot know for sure?
Counting & Combinatorics
Counting possible futures — no gaps, no repeats
Why does this exist?
Counting is the denominator of probability. Get it wrong and everything built on top collapses.
Probability
A language for uncertainty, not a crystal ball
Why does this exist?
We must judge and decide even when we cannot know. Probability is the language of that judgement.
Conditional Probability
The world shrinks the moment you learn something
Why does this exist?
Why you should switch doors in the Monty Hall problem: new information changes the denominator.
Bayes' Theorem
The formula for updating belief with evidence
Why does this exist?
A 99%-accurate test comes back positive — yet your chance of being sick can be 9%. Bayes repairs broken intuition.
Expected Value
One scale for comparing uncertain choices
Why does this exist?
Lotteries, insurance, investments — rational choice starts with asking what remains on average.
Mean, Median & Mode
Three spokespeople telling different stories
Why does this exist?
Why the 'average salary' never feels like yours: a single outlier drags the mean.
Variance & Standard Deviation
The size of the wobble the average hides
Why does this exist?
Two assets with the same average return can be wildly different beasts. Risk is variance.
Correlation
Moving together is not cause and effect
Why does this exist?
Ice-cream sales and drownings rise together. Jumping from 'correlates' to 'causes' is the data age's biggest fallacy.
Normal Distribution
Bell-shaped order born from countless accidents
Why does this exist?
Heights, test scores and measurement errors all scatter the same way — the foundation of statistical inference.
Sampling & Estimation
Knowing fifty million minds by asking a thousand
Why does this exist?
A full census is impossible. Most of what humanity knows comes from samples.
The Journey of Information
How do machines learn? — where every journey meets
Gradient Descent
How AI studies: one step downhill at a time
Why does this exist?
Machine 'learning' is a hike down the mountain of error — and the compass is the derivative.
Neural Networks
A pattern machine of composed simple functions
Why does this exist?
Nothing mathematically new: functions, matrices, derivatives and probability meet — and it starts to look like intelligence.
Everyday Math Labs
Not calculators. Salaries, loans, taxes, discounts — experiment with the structure behind the numbers you meet every day.
Percent
Every % is a % of something
Up 30% then down 30% is not break-even. Same percentage, different base — the most common trap in daily math.
Discounts
30% + 20% off is not 50% off
Discounts stack by multiplication, not addition. See the true size of a double discount.
VAT
Add 10%, then subtract 10% — you don't get back
To recover the pre-tax price you must divide, not subtract. (Korea's 10% VAT as the running example.)
Compound Interest
Time multiplies money — it doesn't add
Simple and compound interest look identical at first, until multiplication quietly takes over.
Loans
Early payments are mostly interest
Equal monthly payments hide a shifting split between interest and principal. Watch it flip.
Take-home Pay
Salary ÷ 12 is not your paycheck
Between your salary and your bank account sit insurance and taxes. See the structure (Korean payroll model).
Progressive Tax
Crossing a tax bracket never hurts you
'A raise means a tax bomb' is the most widespread money myth — brackets slice your income; they don't replace the rate.
Exchange Rates
Which side of the ratio are you on?
A rate is a ratio between currencies. Flip the direction and you flip your judgement.
Unit Price
The big pack isn't always cheaper
Real value shows only in the price per unit, not the price tag.
Splitting the Bill
Fair settlement is a ratio problem
Splitting unequal payments fairly is applied ratios.
CAGR
50% over 3 years — how much per year?
Turning cumulative returns into annual ones takes roots, not division.
Expected Value
The one number to check before buying a lottery ticket
The most basic tool for comparing uncertain options.
New labs open regularly
MathIsland grows alongside a Korean YouTube channel about the “why” behind mathematics. English translations of every lab are rolling out.
Visit the channel →