The most beautiful equation in math, explained visually [Euler's Formula]

In March 1712, Gottfried Wilhelm Leibniz (calculus co-founder) wrote Johann Bernoulli claiming logarithms of negative numbers were impossible. Bernoulli flatly rejected this, arguing derivatives proved log(x) equals log(-x), creating a mathematical crisis threatening calculus’s foundations.

Complex numbers combine real and imaginary components, typically written as a + bi where a is real and b is imaginary. Mathematicians visualize these numbers as points on a two-dimensional plane rather than positions on a single number line.

Complex numbers can be expressed in polar form using magnitude (distance from origin) and angle (measured from positive real axis) rather than separate real and imaginary components. This representation reveals multiplication’s geometric meaning.

Euler confronted the puzzle of what it means to raise numbers to imaginary powers, particularly expressions like 2^(bi) where b is real and i is the imaginary unit. This question threatened mathematics by lacking clear definition despite exponentials’ importance in calculus.

Functions with imaginary exponents exhibit counterintuitive behavior compared to real exponents. While $2^x$ grows without bound as $x$ increases, $2^{bi}$ remains confined to the unit circle regardless of $b$‘s value.

Henry Briggs developed a computational approach in 1617 while constructing prolific base-10 logarithm tables. His method handles fractional and decimal exponents by zooming into exponential curves until they appear linear.

The mathematical constant $e \approx 2.71828$ emerges naturally from exponential function definitions through Briggs’ zooming-in procedure. By choosing this specific base, the linearized slope in tiny neighborhoods equals exactly $1$, simplifying calculus operations.

Leonhard Euler developed this formula in 1747 while resolving fundamental contradictions about logarithms of negative numbers that threatened calculus’s foundations. His solution connects exponential functions with trigonometry through imaginary numbers.

Euler proposed that logarithms of negative numbers possess infinitely many solutions rather than single values, resolving contradictions that threatened calculus’s foundational consistency. This radical solution initially faced controversy but eventually gained acceptance.

Expanding familiar real-number functions to accept complex inputs and outputs creates higher-dimensional surfaces revealing unexpected deep connections. Exponential and trigonometric functions, seemingly unrelated in real analysis, emerge as projections of unified complex structures.