Why Imaginary Numbers Are Real and How They Power the Modern World

In the entire history of mathematics, no concept has suffered from worse public relations than the imaginary number. The name itself, coined in 1637 by the French philosopher and mathematician René Descartes, was intended as an explicit insult:

"For any equation, one can imagine as many roots as its degree declares, but in many cases no quantity exists which corresponds to what one imagines. Hence we may call them imaginary."
— René Descartes, La Géométrie (1637)

Descartes’ derogatory label stuck. For generations of students ever since, the term "imaginary" has reinforced an unfortunate suspicion: that real numbers—1, 2, 3.14, −7—are concrete, solid, and physically genuine, while √−1 is a bizarre, make-believe trick conjured up by mathematicians who refused to admit that certain equations simply have no answer.

The standard high school explanation does little to dispel this suspicion. Students are introduced to the quadratic equation:

They are told: "You cannot multiply a number by itself and get a negative number, because a positive times a positive is positive, and a negative times a negative is positive. But imagine if you could! Let's invent a symbol, i = √−1, so that i² = −1."

To a skeptical mind, this feels like outright cheating. If an equation has no solution, why not simply accept that it has no solution? When a parabola sits entirely above the horizontal axis, it never crosses the line y = 0. Why invent an entire universe of fictitious phantoms just to give an empty curve two imaginary crossing points?

               THE PARABOLA DOES NOT CROSS THE AXIS
                             y
                             │       │
                             │   *   │   *
                             │    *  │  *     y = x² + 1
                             │     * │ *      (Vertex at (0,1))
                       ──────┼───────┼─────── x
                             │       │

Here lies the greatest historical plot twist in mathematics: imaginary numbers were not invented to solve quadratic equations like x² + 1 = 0.

Mathematicians had no problem accepting that x² + 1 = 0 was impossible. For centuries, they simply said: this problem has no geometric meaning; it has no solution.

Imaginary numbers forced their way into human consciousness because of a completely different problem: the cubic equation. In 16th-century Italy, Renaissance algebraists discovered that when an equation had three real, undeniable, physically measurable solutions, the algebraic formula for finding those real answers broke down unless you passed directly through the square roots of negative numbers.

The imaginary numbers were not a gate to nowhere. They were a bridge between real numbers that could not be crossed in any other way.

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1. The Renaissance Duel: The Secret of the Cubic

To understand why imaginary numbers are inevitable, we must travel to Bologna, Venice, and Milan in the 1530s and 1540s. At this time, European universities held public mathematical tournaments. Scholars staked their salaries, reputations, and university chairs on duels where each contestant gave the other thirty problems to solve.

For centuries, mathematicians knew how to solve quadratic equations (ax² + bx + c = 0) using geometric completing-the-square techniques dating back to ancient Babylon and the Persian scholar al-Khwarizmi (see How Algebra Turned Patterns into Equations). But the cubic equation—involving x³—had resisted every civilization for three thousand years. The ancient Greeks, Persians, and Arabs had failed to find a general algebraic formula for it.

Around 1515, Scipione del Ferro, a professor of mathematics at the University of Bologna, discovered the secret. Del Ferro realized that any general cubic equation ax³ + bx² + cx + d = 0 can be transformed (by substituting x = y − b3a) into a depressed cubic, which eliminates the squared term:

Del Ferro kept the formula a closely guarded trade secret to defend his academic chair in intellectual duels. On his deathbed in 1526, he passed the secret to his student, Antonio Maria Fior.

Fior, confident in his inherited weapon, challenged an impoverished, self-taught Venetian mathematics teacher named Niccolò Fontana, known as Tartaglia ("The Stammerer", having suffered a saber slash to the jaw during the 1512 French sack of Brescia).

Tartaglia solved all thirty problems in two hours; Fior solved none. The news of Tartaglia’s triumph electrified Europe.

In Milan, the brilliant polymath, physician, and compulsive gambler Gerolamo Cardano begged Tartaglia to reveal his method. In March 1539, after months of pressure, Tartaglia swore Cardano to secrecy under a solemn religious oath: Cardano could know the formula, but must never publish it under pain of divine damnation.

Cardano, aided by his brilliant teenage assistant Lodovico Ferrari, studied Tartaglia's method and pushed it further, cracking the general cubic and even discovering the solution to the fourth-degree equation (the quartic). But while doing so, Cardano stumbled into an intellectual nightmare that Tartaglia had hidden.

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2. Cardano's Formula and the Casus Irreducibilis

Let us trace Tartaglia and Cardano’s algebraic insight from first principles. How do you solve x³ = px + q?

Tartaglia imagined breaking the unknown x into the difference of two new unknowns:

Cubing both sides:

Rearranging the middle terms:

Since u − v = x, this simplifies to:

Now, compare this equation directly with the depressed cubic x³ = px + q:

    x³   =   [ 3uv ] · x   +   [ u³ - v³ ]
    │            │                   │
    ▼            ▼                   ▼
    x³   =   [   p   ] · x   +   [    q    ]

For the two equations to match, we simply require:

We now have two equations with two unknowns: the difference of u³ and v³ is q, and their product is (p3)³. This is equivalent to solving a quadratic equation for u³ and −v³. Solving it yields:

Taking cube roots and recalling that x = u − v, Cardano obtained the famous Cardano-Tartaglia Formula:

For many cubics, this formula worked like magic. Consider x³ = 6x + 6, where p = 6, q = 6:

Plugging x ≈ 2.8473 back into x³ = 6x + 6 checks out perfectly.

The Breakdown

Then Cardano tested an innocent-looking equation:

Here, p = 15 and q = 4. Let us evaluate the terms inside the square root:

Substituting this into the formula produces a startling result:

Cardano stopped in his tracks. What was √−121? No number in the Renaissance universe could be multiplied by itself to equal −121. If you squared +11, you got +121. If you squared −11, you got +121. A square root of a negative quantity seemed like nonsense.

Naturally, one might assume that this equation has no solution. But look at the original equation again:

Try plugging in x = 4:

x = 4 is an undeniable, integer solution! In fact, graphing y = x³ − 15x − 4 reveals that it crosses the horizontal axis at three distinct real points: x = 4, x = −2 + √3 ≈ −0.268, and x = −2 − √3 ≈ −3.732.

               THE THREE REAL ROOTS OF x³ - 15x - 4 = 0
                             y
                             │        *
                             │       * 
                 *           │      *
               *   *         │     *    x = 4 (Real root!)
             ─*─────*────────┼────*────── x
           x ≈ -3.73 │ x ≈ -0.27 │
                     │ *         │
                     │  *        │
                     │   *       │

This was the crisis of the casus irreducibilis ("the irreducible case"). When a quadratic equation has a negative discriminant, it has zero real roots. But when a cubic equation has a negative discriminant, it has three real roots!

The algebraic formula for finding those three completely real, physical solutions required taking the square root of a negative number.

Cardano described calculating with these quantities in Chapter 37 of Ars Magna (1545) as a process of "mental tortures", dismissing the results as "as subtle as it is useless." He was trapped: his formula worked, it yielded true answers, but its intermediate mechanism required operating with algebraic phantoms.

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3. Rafael Bombelli's "Wild Thought": Ghost Arithmetic

For nearly thirty years, European algebraists stared at Cardano’s paradox in confusion. Then, in 1572, an Italian hydraulic engineer and architect from Bologna named Rafael Bombelli published his masterpiece, L'Algebra.

Bombelli decided to treat the square root of minus one not as a terrifying impossibility, but as an operational entity that obeys strict, consistent logical rules. As Bombelli wrote:

"It seemed to me more of a dream than a reality, and yet I persevered until I found the rule."

Bombelli hypothesized: what if the expression ³√2 + √−121 could be broken down into the sum of an ordinary real number and a "ghost" quantity?

Let us write √−121 = 11√−1. Bombelli postulated that:

for some integers a and b.

If this were true, look at what happens when we substitute them back into Cardano’s formula:

The two imaginary quantities +b√−1 and −b√−1 have opposite signs. They cancel out completely:

The imaginary ghosts would evaporate into thin air, leaving behind a purely real number!

Verifying the Ghost Arithmetic

To prove this was not wishful thinking, Bombelli had to define how these numbers multiplied. He invented the formal arithmetic of signed imaginary numbers:

• più di meno (+i): positive imaginary unit
• meno di meno (−i): negative imaginary unit
• (+i) × (+i) = −1
• (+i) × (−i) = +1
• (−i) × (−i) = −1

Now, Bombelli tested values for a and b. Guessing that a = 2 and b = 1, he calculated (2 + √−1)³ using standard binomial expansion:

Let us evaluate each term step by step:

1. 2³ = 8
2. 3(4)(√−1) = 12√−1
3. 6(√−1)² = 6(−1) = −6
4. (√−1)³ = (√−1)² · √−1 = (−1)√−1 = −√−1

Combine the real and imaginary parts:

It matched perfectly:

Likewise, cubing (2 − √−1):

Now, plug both cube roots back into Cardano's formula:

The answer was exactly 4.

               THE IMAGINARY BRIDGE ACROSS REAL NUMBERS
               
    Cardano Formula ──► √[2 + 11i]  +  √[2 - 11i] 
                               │            │
                               ▼            ▼
    Bombelli Expansion ──► (2 + i)    +   (2 - i)
                               │            │
                               └─────┬──────┘
                                     ▼
                               x = 2 + 2 = 4  (Real Solution!)

This was the intellectual turning point. You could no longer dismiss √−1 as meaningless nonsense. If you started with an equation with real coefficients (x³ = 15x + 4), and you ended with a real, verifiable answer (x = 4), and the only way to travel from start to finish was by stepping through √−1, then √−1 was doing real work.

It was an algebraic catalyst: entering the reaction, performing the necessary rearrangement, and leaving behind a pure real product.

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4. The Geometric Epiphany: Multiplication as Rotation

Despite Bombelli’s triumph, mathematicians remained deeply uneasy for the next two centuries. What was this quantity? You could not measure √−1 feet of lumber or pour √−1 pints of wine.

The breakthrough came when mathematicians stopped asking "What amount does i represent?" and started asking "What geometric transformation does i perform?"

The One-Dimensional Cage

For millennia, humans thought of numbers as points along a one-dimensional line:

       Negative Numbers                     Positive Numbers
  ◄───────────────────────────┬───────────────────────────►
   -3        -2        -1     0     +1        +2        +3

On this line, what does multiplication mean?

• Multiplying by +1 leaves a number unchanged (an identity transformation).
• Multiplying by −1 flips a number across zero to the opposite side:

Geometrically, multiplying by −1 is a 180^∘ rotation around the origin!

                         Multiplying by -1
                       ┌──────────────────┐
                       │  180° Rotation   │
                       ▼                  │
  ◄────────────────────●──────────────────●──────────►
                      -1        0        +1

Now, consider the definition of the imaginary unit:

Multiplying by i twice produces a multiplication by −1. Therefore, performing the operation i twice is equivalent to a 180^∘ rotation.

If two identical steps produce a 180^∘ turn, what must a single step produce?

Multiplying by i is not a scalar stretch along the line. Multiplying by i is a 90^∘ counter-clockwise rotation!

                         THE 90-DEGREE ROTATION OF i
                                    +i (Vertical Axis)
                                     │   ▲
                                     │  / 90° Turn
                                     │ /
  ◄──────────────────────────────────┼─●────────────────►
   -1                                0  +1 (Real Axis)
                                     │
                                     │
                                    -i

The Argand Plane and Gauss

This profound geometric insight was discovered independently by three people:

1. John Wallis (1685), an English mathematician who first suggested that imaginary numbers could be represented on a vertical axis perpendicular to the real axis.
2. Caspar Wessel (1797), a Norwegian surveyor who presented the idea to the Royal Danish Academy.
3. Jean-Robert Argand (1806), a Parisian bookkeeper who published an essay titled Essai sur une manière de représenter les quantités imaginaires dans les constructions géométriques.

It was popularized and formalized into rigorous mathematics by Carl Friedrich Gauss in 1831, who gave the plane its modern name: the Complex Plane (C).

In the complex plane:

• The horizontal axis represents the Real numbers (Re).
• The vertical axis represents the Imaginary numbers (Im).
• Every complex number z = a + bi is not an impossible entity; it is simply a two-dimensional point or vector with coordinates (a, b).

                             Imaginary (Im)
                                    ▲
                                    │       z = a + bi = (a, b)
                                  b ┼───────●
                                    │      /│
                                    │  r  / │
                                    │    /  │
                                    │   / θ │
                                    └───┴───┴────────► Real (Re)
                                       0    a

By adding a second dimension, all the mystery of imaginary arithmetic disappears into elementary geometry:

1. Addition is Vector Translation:

You simply place the vectors tip-to-tail, exactly like adding forces in physics.

2. Multiplication is Rotation and Scaling: Every complex number can be expressed in polar form (r, θ), where r = √a² + b² is its distance from the origin (magnitude) and θ = arctan(b/a) is its angle with the real axis (argument).

   When you multiply two complex numbers, you multiply their lengths and add their angles:

Let us test this rule with i:

• The magnitude of i is 1. Its angle is 90^∘.
• i × i: length is 1 × 1 = 1; angle is 90^∘ + 90^∘ = 180^∘. The point at distance 1 and angle 180^∘ is −1.
• i × i × i = i³: angle is 270^∘, which is −i.
• i × i × i × i = i⁴: angle is 360^∘ ≡ 0^∘, which is +1.

The powers of i are simply four spokes rotating continuously around a circle:

There was never anything "imaginary" about i. Real numbers were merely a single, one-dimensional horizontal line slice of a complete, two-dimensional mathematical landscape. Confining mathematics to real numbers was like exploring the surface of the Earth while forbidding movement North or South.

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5. Euler's Miracle: The Bridge Between Algebra and Trigonometry

In 1748, the Swiss genius Leonhard Euler published Introductio in analysin infinitorum, unveiling what many physicists and mathematicians consider the most beautiful equation in human history.

Euler noticed a stunning pattern when comparing the infinite Taylor series of the exponential function e^x, the cosine function cos x, and the sine function sin x.

Recall the infinite polynomial expansions discovered by calculus (see How Calculus Predicts Change):

Notice the eerie similarity: e^x contains all powers of x with positive signs, while cos x contains only the even powers with alternating signs, and sin x contains only the odd powers with alternating signs.

Euler asked: what happens if we substitute ix into the exponential series?

Let us compute the powers of ix:

• (ix)¹ = ix
• (ix)² = i² x² = −x²
• (ix)³ = i³ x³ = −ix³
• (ix)⁴ = i⁴ x⁴ = +x⁴
• (ix)⁵ = i⁵ x⁵ = +ix⁵
• (ix)⁶ = i⁶ x⁶ = −x⁶

Substitute these into the expansion of e^ix:

Now group the real terms and the imaginary terms together:

Look at the two parenthetical groupings. The first group is identical to cos x. The second group is identical to sin x.

Euler arrived at his miraculous identity:

Euler's Identity

Now, set the angle x = π radians (180^∘):

Since cos(π) = −1 and sin(π) = 0:

Adding 1 to both sides yields Euler’s Identity:

                    EULER'S IDENTITY: THE FIVE CONSTANTS
                                 e^(iπ) + 1 = 0
                                 │  ││    │ │
              Base of Natural ───┘  ││    │ └── Origin of Counting (Zero)
              Logarithms (e)        ││    └──── Multiplicative Identity (One)
                                    │└───────── Ratio of Circle to Diameter (π)
                                    └────────── Imaginary Unit (i)

Euler’s formula revealed that exponential growth and circular oscillation are the same mathematical process, linked through the imaginary axis.

Moving along the real line in e^x causes explosive, runaway exponential growth (1, 2.7, 7.4, 20.1…). But moving along the imaginary direction in e^ix causes a point to march smoothly around the perimeter of the unit circle, generating pure, periodic waves.

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6. The Industrial Lifeline: AC Electricity and Phasors

If imaginary numbers had remained confined to pure mathematics, they would still be regarded as an intellectual triumph. But in the late 19th century, they became something far more tangible: the operating system of electrical civilization.

During the 1890s, the "War of the Currents" was raging between Thomas Edison's direct current (DC) and Nikola Tesla's alternating current (AC). AC won because transformers could step up alternating voltages to hundreds of thousands of volts for long-distance transmission with minimal heat loss, then step them down for safe domestic use.

However, AC engineers faced a horrific mathematical bottleneck. In DC circuits, Ohm's law is trivial:

Voltage equals current multiplied by resistance. But in AC circuits, the voltage and current are continuous sinusoidal waves oscillating at 50 or 60 Hz:

When alternating current passes through resistors, the current and voltage remain in phase. But when it passes through inductors (coils of wire) and capacitors (separated plates), the physics changes drastically:

• An inductor opposes changes in current through Faraday's law of induction:

Current lags voltage by 90^∘ (π/2).

• A capacitor stores electric charge, resisting changes in voltage:

Current leads voltage by 90^∘ (π/2).

Analyzing even a basic electrical circuit containing resistors, capacitors, and inductors required solving sprawling, coupled systems of second-order differential equations. Designing an entire electrical power grid or telephone network using classical calculus was slow, error-prone, and practically impossible.

Charles Steinmetz's Complex Revolution

In 1893, a German-born dwarf hunchback and socialist mathematical prodigy named Charles Proteus Steinmetz presented a paper to the American Institute of Electrical Engineers (AIEE). Steinmetz showed that by using imaginary numbers, all the differential and integral calculus of AC circuits could be replaced with simple high school algebra.

Steinmetz realized that because differentiation of e^{iω t} simply multiplies the function by iω:

and integration divides by iω:

the calculus of waves could be reduced to arithmetic by defining a new generalized resistance called Complex Impedance (Z):

The real part of impedance (R) represents dissipation—electrical energy converted into heat.

The imaginary part of impedance (± i X) represents reactance—energy stored temporarily in magnetic fields (inductors) or electric fields (capacitors) and returned to the circuit 90^∘ out of phase!

Because multiplying by i represents a 90^∘ rotation in the complex plane, the imaginary unit automatically keeps track of the phase lead and phase lag without any trigonometric formulas.

To calculate the current in any AC circuit, engineers simply sum the complex impedances in series or parallel, just like DC resistors:

Every power line, generator, transformer, electric motor, radio antenna, Wi-Fi router, audio equalizer, and microchip on Earth is engineered using complex impedance. Without the square root of minus one, modern electrical grids and telecommunications could not exist.

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7. Quantum Mechanics: Nature’s Non-Negotiable Complex Reality

In electrical engineering, one could still cling to an ontological escape hatch. A stubborn engineer could say: "The voltage in the wire is actually real. We are just using the complex plane as an ingenious mathematical shortcut, and taking the real part (Re[Ṽ]) at the end."

For three hundred years, physicists believed that imaginary numbers were always just a mathematical convenience. Classical Newtonian mechanics, Maxwellian electromagnetism, and Einstein’s general relativity can all be formulated entirely using real numbers.

Then came Quantum Mechanics.

In 1925, the Austrian physicist Erwin Schrödinger wrote down the fundamental wave equation that describes the behavior of all matter and radiation in the universe:

Look closely at the very first symbol on the left side: i.

In the fundamental equation governing electrons, atoms, molecules, and chemical bonds, the imaginary unit is not a trick added for convenience. It is hard-coded into the bedrock of reality.

               THE SCHRÖDINGER TIME-DEPENDENT WAVE EQUATION
               
                                i · ℏ · (∂Ψ / ∂t)  =  Ĥ · Ψ
                                │   │       │         │   │
           Imaginary Unit ──────┘   │       │         │   └── Quantum State Vector
           Reduced Planck Const. ───┘       │         └────── Hamiltonian Operator
           Rate of Change Over Time ────────┘

Why must i appear in quantum mechanics?

The Wavefunction Phase and Interference

In classical physics, a state is described by real numbers: position x and momentum p. In quantum physics, a particle does not possess a definite position until measured. Instead, it is described by a complex wavefunction Ψ(x, t).

A complex number has two degrees of freedom: an amplitude r and a phase angle θ:

The probability of finding the particle at position x is given by the Born rule—the square of the magnitude of the wavefunction:

If only the magnitude squared determines probability, why does the complex phase e^iθ matter?

Because of quantum interference. In the double-slit experiment, an electron fired through two slits can arrive at the detector screen via two different paths with wavefunctions Ψ₁ and Ψ₂. The total wavefunction is their sum:

The resulting probability is:

The cross-term 2 Re[Ψ₁^* Ψ₂] is the interference term. Depending on the difference between the complex phase angles of Ψ₁ and Ψ₂, this term can be positive (constructive interference, where particles are likely to be found) or negative (destructive interference, where probabilities cancel to zero).

If quantum states were described only by real numbers, wavefunctions could not rotate continuously in phase to produce the exact interference patterns observed in experiments.

The 2021-2022 Experimental Verdict: Real vs. Complex

For nearly a century, physicists debated whether the complex formulation of quantum mechanics was truly indispensable. Could quantum mechanics be reformulated using only real numbers by doubling the number of dimensions or representing state vectors in real Hilbert spaces?

In December 2021, an international team of theoretical physicists led by Marc-Olivier Renou published a landmark paper in Nature, proposing an experimental test based on a multi-party Bell-type entanglement network.

Within months, two separate experimental teams—one led by Zheng-Da Li and Jian-Wei Pan in Hefei, China, and another led by Alexander Chen in Shenzhen—built photonic and superconducting quantum processors to perform the test.

Their results, published in Physical Review Letters (2022), delivered a decisive experimental verdict: standard complex quantum mechanics correctly predicted the correlations between entangled particles, while every possible real-number formulation failed.

Nature does not use complex numbers as a computational shortcut. At the most fundamental subatomic level, the universe itself operates in the complex plane.

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8. Summary: From Impossible Fiction to Fundamental Law

The story of the imaginary number is the ultimate demonstration of how mathematical truth forces itself upon human thought:

When we restrict our minds to numbers that correspond to tangible, physical heaps of objects, we trap ourselves on a one-dimensional line. We can move forward and backward, but we are blind to the vast plane of rotations, oscillations, and wave phases surrounding us.

Imaginary numbers are not imaginary. They are the orthogonal dimension of reality—the hidden axis along which waves oscillate, electrical currents phase-shift, and quantum probabilities interfere to build the physical universe.