Euler’s identity, on one page.
Episode 1 in one picture: i as a quarter turn, π as the walk of a U-turn, e from tiny steps, and why the arrow lands on minus one. It fits on one sheet of paper. Leave your email and it downloads right away.
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What’s on the page
×i is a quarter turn
Multiplying by −1 is a U-turn. Multiplying by i is half of that, so two of them make a U-turn: i × i = −1.
Numbers are arrows
To add, go tip to tail. To multiply, lengths multiply and angles add, so 2i × 2i = −4.
π is how far a U-turn walks
Turn half a circle and the tip walks π radii. That’s why a U-turn is π radians.
e is one width grown in tiny steps
Split the growth into n equal slices and (1 + 1/n)n climbs to e = 2.71828…
Spend iπ in tiny turns
Each slice pushes the arrow sideways, so it turns and barely stretches. Fine enough, it walks π and lands on −1.
+1 lands on zero
One step right from −1 is 0. And noise-cancelling headphones use the same U-turn.