Bases

Humans started crossing every fifth mark. Faster to read, but still one mark at a time.

Add a block.

Keep adding. A second carry builds another rod. A third, and you'll see the hundreds place: a flat.

You have 0 blocks. Try setting the base to 5.

The rightmost column is ones. Each step left is worth the base times more. The line below shows the decomposition.

Same 0 blocks. Switch between bases to see them rearrange. The quantity never changes.

Some numbers are clean in base 2. Try a few.

What happens if the base is negative?

In base −2, positions alternate sign: +1, −2, +4, −8, +16...

Digits are still 0 and 1. But alternating signs mean every integer has a representation. No minus sign needed.

Example: 7 in base −2:

What is 5 in base −2?

The slider now reaches below zero. Drag it to −2 and use + / − to explore.

Roman numerals use subtractive notation: a smaller symbol before a larger means subtraction.

Base negative two alternates +1, −2, +4, −8... Base negative one is simpler: positions just alternate +1, −1, +1, −1. Representations grow fast.

A base doesn't have to be an integer. In base 1.5, positions are worth 1, 1.5, 2.25, 3.375... Digits are 0 and 1, but most integers need remainders.

Every positive integer is a sum of distinct, non-consecutive Fibonacci numbers. That's Zeckendorf's theorem. It gives each integer a unique base-φ representation.

Other irrational bases work the same way, but most integers don't fit cleanly. Type a number to see the remainder.

Multiplying by −1 rotates a number 180° on the line: 3 becomes −3.

What rotation, done twice, gives 180°? 90°. Call that operation “multiply by i.” Two rotations by i make 180°, so i² = −1.

Click the plane to place a point.

Every point is a + bi. The horizontal axis is real, the vertical is imaginary.

The unit circle marks distance 1 from the origin.

The powers of 2i spiral around this plane: 1, 2i, −4, −8i, 16... Every integer, even complex ones, has a unique representation using digits 0, 1, 2, 3.

The unit circle has radius 1. A point on it at angle θ sits at coordinates (cos θ, sin θ). Drag the angle.

Look familiar?

Multiplying by i rotates 90°. So √i should rotate 45°.

At 45° on the unit circle: cos 45° + i·sin 45° = 1/√2 + i/√2.

Every polynomial equation has solutions in the complex numbers. Roots that don't exist on the number line live on this plane.

All controls unlocked.

The Human Kiloannum Clock is five place-value columns in base 10.

Each cube is worth 10× the one to its right. The leftmost column is ten-thousands.