Click a cow to send it out to graze (a pebble is added to the pot). Click a grazing cow to bring it home (a pebble is removed). If pebbles remain in the pot at "end of day," a cow is missing!
29 uniform notches carved into a baboon fibula โ possibly a lunar phase counter. Hover over the notches to count them.
One column groups notches into 11, 13, 17, 19 โ the prime numbers between 10 and 20. Explore the pattern:
Brahmagupta was the first to define 0 = a โ a and give explicit arithmetic laws for it. Try any number below and watch the rule apply live.
In the Upanishads and Buddhist thought, ลhลซnyatฤ (emptiness) described a meditative state of stillness. Indian mathematicians โ ฤryabhaแนญa and finally Brahmagupta โ transformed this philosophical "nothingness" into a working mathematical number: 0.
Zero sits exactly at the boundary between wealth and debt on the number line โ the seed for negative numbers, explored fully in the next tab.
Brahmagupta grounded negative numbers in real commerce. Drag the slider to see a point move along the integer line.
| Rule | Example | Meaning |
|---|---|---|
| fortune + fortune = fortune | 5 + 4 = 9 | Wealth adds to wealth |
| debt + debt = debt | (โ5) + (โ4) = โ9 | Debts pile up |
| debt ร fortune = debt | (โ3) ร 4 = โ12 | 4 debts of โน3 |
| debt ร debt = fortune | (โ3) ร (โ4) = 12 | Removing debts enriches you |
You have 4 debts of โน3 each. Someone removes (โ) all 4 of them. Watch what happens to your net worth:
Enter p and q to plot p/q on the number line. The interval between consecutive integers is split into q equal parts.
Between ANY two rational numbers lies another โ their average. Click "Find Midpoint" repeatedly to zoom infinitely between two points.
Rational numbers have no unique representation โ 1/2 = 2/4 = 3/6 = ...
Click through Hippasus's classic proof that โ2 cannot be written as a fraction p/q.
The Square Root Spiral โ each right triangle has one leg of length 1, giving hypotenuses โ2, โ3, โ4, โ5...
By the BaudhฤyanaโPythagoras theorem, a unit square's diagonal has length โ2 โ a length that defies fractions.
Mฤdhava of Sangamagrama (14th century) discovered: ฯ = 4 ร (1 โ 1/3 + 1/5 โ 1/7 + ...). Watch it converge to ฯ as more terms are added.
Enter any denominator q โ we'll predict (via prime factorisation) whether p/q terminates, without doing long division.
Click "Step" to perform long division one digit at a time and watch the remainders cycle โ the moment a remainder repeats, the decimal starts repeating!
Multiply 142857 by 1 through 6 โ the same digits just rotate!
Drag or click each number into the correct nested set: Natural (โ) โ Integers (โค) โ Rational (โ), with Irrational (๐) separate โ together forming Real Numbers (โ).
Even (โ1)ร(โ1) = 1 shows no real number squares to โ1 โ mathematicians invented i to step off the real number line entirely, a story for another year!