God's Equation by Leonhard Euler (1707–1783)
e^{iπ} + 1 = 0
This identity unites five fundamental constants; e (base of natural logarithm), i (imaginary unit), π, 1, and 0; in one elegant stroke.
Derived from Euler’s formula e^{ix} = cos x + i sin x by setting x = π, it reveals cos(π) = –1 and sin(π) = 0. The diagram beautifully illustrates the unit circle in the complex plane, showing e^{iφ} = cos φ + i sin φ.
Often called “God’s equation,” it symbolizes the profound harmony of mathematics, bridging exponentials, trigonometry, and complex numbers.
The limit as h → 0... Watch the secant line collapse into the exact tangent at that point.
This is the single most beautiful moment in all of calculus. The definition of the derivative, live.
The Basics of Electromagnetic Waves:
Electricity and magnetism can sit still, like static electricity in your hair or a magnet stuck to your fridge. But when they move and change, they actually create each other. Together, they team up to form invisible ripples of energy called electromagnetic waves.
Unlike ocean waves or sound waves, which need water or air to ripple through, electromagnetic waves don't need any material at all. They can easily travel through the completely empty vacuum of space.
Maxwell's Big Idea:
In the 1860s and 1870s, a Scottish scientist named James Clerk Maxwell figured out how this works. He wrote down the math showing exactly how electricity and magnetism link together to make these travelling waves. Today, scientists call his famous rules Maxwell's Equations.
Hertz Proves It:
Later, a German physicist named Heinrich Hertz took Maxwell's ideas and brought them to life. He was the first person to actually create and catch radio waves. To honour his work, we use the word hertz to measure how fast a wave vibrates (one cycle per second).
Hertz's experiments proved two massive ideas:
Radio waves are just invisible light: He showed that radio waves travel at the exact same speed as light, proving that they are actually a form of light we just can't see.
Going wireless: He finally figured out how to detach these energy fields from physical wires, allowing the waves to fly freely through the air exactly as Maxwell had predicted.