Document ID: V_2_22
Section: V_Mathematics_Information / V2_Pure_Mathematics
Keywords: imaginary numbers, complex numbers, √-1, i, Cardano, Bombelli, Descartes, Euler, Gauss, Argand, Wessel, Cauchy, Riemann, complex analysis, Schrödinger equation, quantum mechanics, productive fiction, impossible numbers, history of mathematics, complex plane
Category Tags: mathematics, history-of-science, productive-fictions, philosophy-of-mathematics
Cross-References: V_2_03 — History of Algebra · V_2_08 — Mathematical Proof · P_5_01 — Mathematics Discovered or Invented · ZA_1_11 — Weak Measurements · G_3_28 — Phlogiston Theory
Reliability Tier: Tier 1 (primary texts survive; mathematical proofs permanent; physics results independently replicated)
Last Updated: May 29, 2026 | Source Count: 11 | Weighted Score: 24 | Source Confidence: [4/5] | Confidence: High

QUICK SUMMARY

In 1545, the Italian mathematician Girolamo Cardano encountered expressions involving the square root of a negative number while solving cubic equations in his Ars Magna. He used the expression — computed with it, obtained a correct numerical answer — and then described it as "truly imaginary" and declared the result "as subtle as it is useless." He went no further.

From that dismissal in 1545 to full mathematical acceptance around 1800, imaginary numbers took approximately 250 years to move from "impossible and useless" to "legitimate." They spent much of that journey as what Bombelli called "a new type of cube root" and Descartes dismissed as imaginary in the pejorative sense — coining the word in 1637 precisely to marginalize them.

Today, imaginary numbers are not a curiosity but a structural necessity. Quantum mechanics requires them: the Schrödinger equation is irreducibly complex-valued; a 2022 experimental study (Nature, Chen et al.) confirmed that real-valued quantum theory cannot reproduce all predictions of standard quantum mechanics. Complex numbers are also foundational to electrical engineering (AC circuit analysis), signal processing (Fourier transforms), control theory, fluid dynamics, and special relativity. An object that was declared useless in 1545 now underpins the physics of the universe.

The imaginary number story is the clearest mathematical case of a productive fiction — an "impossible" object that, when accepted as if real, unlocked mathematical and physical domains that would otherwise have remained closed. The key move was Bombelli's: pretend it exists and see what happens.

1. VERIFIED CLAIMS (Tier 1 — Primary Texts / Independently Verified)

1.1 Cardano and the First Use (1545)

• Girolamo Cardano (1501–1576): In Ars Magna, sive de regulis algebraicis (Nuremberg, 1545), Cardano published the general solution for cubic equations (credit disputed with Tartaglia; see V_2_03 §1.4).
• In the process, he encountered a problem requiring the reader to divide 10 into two parts whose product is 40. The algebraic solution requires evaluating:

(5 + √−15)(5 − √−15) = 25 − (−15) = 40

• The intermediate expressions involve the square root of a negative number — something with no real-number interpretation. Yet the calculation works: the imaginary parts cancel and the real answer (40) is correct.
• Cardano's reaction: he described these quantities as vere sophistica ("truly imaginary") and called the result "as subtle as it is useless" (tam est subtilis quam inutilis). He performed the computation but declined to investigate further.
• Primary Source: Cardano, Girolamo. Ars Magna, sive de regulis algebraicis. Nuremberg: Petreius, 1545. English trans. T. Richard Witmer. Cambridge, MA: MIT Press, 1968.

1.2 Bombelli's Key Move: "Pretend They Exist" (1572)

• Rafael Bombelli (1526–1572): In L'Algebra (Bologna, 1572), Bombelli became the first mathematician to work out systematic rules for arithmetic with complex numbers.
• Bombelli's motivation was the "casus irreducibilis" of cubic equations: cases where Cardano's formula requires computing cube roots of complex numbers, even when all three roots of the cubic are real numbers. Ignoring these expressions meant losing real solutions.
• Bombelli's decision: rather than dismiss the expressions, he decided to treat them as if they obeyed consistent rules and derived those rules — establishing how to add, subtract, multiply, and work with expressions of the form a + b√(−1).
• He identified the key rules: √(−1) · √(−1) = −1; (−√(−1)) · √(−1) = 1.
• He did not claim these numbers "existed" in any philosophical sense. He simply established that proceeding as if they did gave consistent and correct results for real cubic equations.
• The productive-fiction move: Bombelli's contribution is methodologically identical to what the philosopher Vaihinger would later call Als-Ob reasoning ("As-if" philosophy): treat the fiction as real, derive consequences, verify results.
• Primary Source: Bombelli, Rafael. L'Algebra. Bologna: Giovanni Rossi, 1572.

1.3 Descartes Names Them (Pejoratively) and Euler Formalizes Notation

• René Descartes (1596–1650): In La Géométrie (1637), Descartes coined the term "imaginary" (imaginaires) as a dismissal — quantities that were "imagined" but had no geometric reality. He contrasted them with "real" numbers. The word was intended to marginalize, not to describe.
• Primary Source: Descartes, René. La Géométrie. Appendix to Discours de la méthode. Leiden: Jan Maire, 1637. English trans. David Eugene Smith and Marcia L. Latham. Chicago: Open Court, 1925.
• Leonhard Euler (1707–1783): Euler systematized the use of complex numbers throughout the 18th century and introduced the symbol i to denote √(−1), formalizing the notation now universally used. He also derived Euler's identity: e^(iπ) + 1 = 0 — connecting five fundamental constants in one equation. Euler continued to call these "imaginary" while using them as a standard tool.
• Euler's formula (derived c. 1748 in Introductio in analysin infinitorum): e^(iθ) = cos θ + i sin θ — the foundation of complex analysis and indispensable to modern physics and engineering.
• [NOTE]: The precise work and year in which Euler first used the symbol i is debated by historians (range c. 1748–1777); the Introductio (1748) is the most commonly cited source.
• Primary Source: Euler, Leonhard. Introductio in analysin infinitorum. Lausanne: Bousquet, 1748. English trans. John Blanton. New York: Springer, 1988–1990.

1.4 Geometric Interpretation and Full Acceptance (c. 1797–1831)

• The decisive step toward mathematical acceptance was the geometric interpretation of complex numbers as points in a two-dimensional plane:
• Caspar Wessel (1745–1818): Norwegian surveyor; presented the geometric representation to the Royal Danish Academy in 1797, published 1799. His paper was largely unnoticed outside Denmark.
• Jean-Robert Argand (1768–1822): French amateur mathematician; independently published the same geometric interpretation in 1806. The complex plane is sometimes called the "Argand plane" in his honor.
• Carl Friedrich Gauss (1777–1855): provided the geometric interpretation independently (c. 1799, in his doctoral dissertation proving the fundamental theorem of algebra), published in 1831 as "Theoria residuorum biquadraticorum." Gauss's authority secured wide acceptance.
• Once complex numbers could be visualized as points in a plane — with real numbers along the horizontal axis and imaginary numbers along the vertical — the "impossibility" dissolved. A point in a plane is geometrically real. The "imaginary" axis is as physically present as the real axis.
• Primary Source: Gauss, Carl Friedrich. Theoria residuorum biquadraticorum, Commentatio secunda. Göttingen: Dieterich, 1832.
• Augustin-Louis Cauchy (1789–1857): Developed complex analysis systematically in the 1820s–1840s, establishing that complex functions obey powerful analytic constraints (Cauchy-Riemann equations, Cauchy's integral theorem, residue theorem) with no real-number analogues.
• Bernhard Riemann (1826–1866): Extended complex analysis into the Riemann surface framework; the Riemann hypothesis (still unproven) concerns the zeros of the Riemann zeta function — a complex-valued function.

1.5 Physical Necessity: Quantum Mechanics

• Erwin Schrödinger (1887–1961): In "Quantisierung als Eigenwertproblem" (Annalen der Physik, 1926), Schrödinger introduced the wave equation now bearing his name:

iℏ (∂ψ/∂t) = Ĥψ

The symbol i in the Schrödinger equation is not a notational convenience. The wavefunction ψ is irreducibly complex-valued; attempts to reformulate quantum mechanics with purely real-valued functions fail.

• Experimental confirmation (2022): Chen, Xu-Fei, et al. (Nature 601, 2022) demonstrated experimentally that a real-valued quantum theory makes different predictions from standard (complex) quantum theory in certain entanglement scenarios, and that the experimental results agree with complex quantum theory. This is the closest thing to an empirical proof that complex numbers are physically necessary — not merely convenient.
• Primary Source: Chen, M.-C., et al. "Ruling Out Real-Valued Standard Formalism in Quantum Theory." Physical Review Letters 128, no. 4 (2022): 040403. doi:10.1103/PhysRevLett.128.040403
• Other physics applications of complex numbers (all Tier 1):
• Maxwell's equations in complex form (AC electromagnetics, antenna theory)
• The Dirac equation (relativistic quantum mechanics) — requires 4-component complex spinors
• Fourier transforms — decompose any signal into complex exponentials
• AC circuit analysis — impedance is complex-valued; without complex numbers, AC circuit design requires real-valued approximations that are far less tractable
• Special relativity — Minkowski spacetime metric can be expressed with imaginary time coordinates

2. CREDIBLE CLAIMS (Tier 2 — Scholarly, Interpretive Debate)

2.1 The 250-Year Delay Reflects Conceptual Rather Than Mathematical Difficulty

• Historians of mathematics (Nahin, Kline, Bourbaki) note that the mathematics of complex numbers was fully available from Bombelli onward. The 250-year delay in acceptance was not due to mathematical inadequacy but to philosophical/ontological resistance: the absence of a geometric or physical interpretation that would justify saying complex numbers "existed."
• Once Argand and Gauss provided geometric representation (~1800), the philosophical obstacle dissolved almost immediately.
• Primary Source: Nahin, Paul J. An Imaginary Tale: The Story of √−1. Princeton: Princeton University Press, 1998.
• Counter-argument: Ferreirós (2007) argues that mathematical acceptance required more than geometric representation — it required the development of rigorous foundations for analysis (Cauchy, Weierstrass), which took until the mid-19th century.

2.2 The Productive-Fiction Structure in Mathematics

• Bombelli's "pretend they exist and see what happens" is arguably the template for a broader pattern in mathematics: acceptance of formally consistent extensions of number systems, even without initial intuitive justification.
• Negative numbers faced similar resistance before the 17th century (ancient Greek and medieval European mathematics treated them as meaningless or "absurd").
• Non-Euclidean geometry: Gauss, Lobachevsky, and Bolyai independently developed geometries where Euclid's parallel postulate fails — at first dismissed as logically impossible, later shown to be physically relevant (Riemannian geometry is the framework of general relativity).
• Transfinite numbers (Cantor): set theory extended number concepts into infinities of different sizes — initially condemned as theology or pathology.
• Each case follows the same structure: an object declared impossible or meaningless is shown to be formally consistent and eventually physically or mathematically necessary.
• Primary Source: Kline, Morris. Mathematical Thought from Ancient to Modern Times. 3 vols. Oxford: Oxford University Press, 1972.

3. SPECULATIVE CLAIMS (Tier 3 — Plausible, Not Demonstrated)

3.1 Complex Numbers May Be More Fundamental Than Real Numbers

• Some mathematical physicists (Penrose, in The Road to Reality, 2004) suggest that complex numbers are not an extension of real numbers but are, in some sense, more fundamental — real numbers are a degenerate special case of complex numbers, and the universe's quantum substrate is irreducibly complex.
• This remains a philosophical position, not a demonstrated theorem. However, the 2022 experimental result (Chen et al.) provides some empirical weight to the idea that the complex structure is a physical feature of reality, not merely a mathematical convenience.

4. DUBIOUS CLAIMS (Tier 4 — No Credible Support)

4.1 Imaginary Numbers Prove That Math Is "Just Made Up" and Has No Reality

• A popular misreading: if mathematicians "invented" imaginary numbers by fiat, then mathematics is merely conventional and has no claim to describing reality. This conclusion does not follow.
• The imaginary number story actually demonstrates the opposite: objects developed purely through formal reasoning, with no initial physical interpretation, turned out to be necessary for describing quantum mechanics. This is a datum in favor of the surprising effectiveness of mathematics, not against its reality.
• Counter-source: Wigner, Eugene P. "The Unreasonable Effectiveness of Mathematics in the Natural Sciences." Communications in Pure and Applied Mathematics 13, no. 1 (1960): 1–14. doi:10.1002/cpa.3160130102

Counter-Arguments & Criticisms

1. Calling imaginary numbers a "fiction" is misleading once accepted. At the point of acceptance, imaginary numbers are just numbers — elements of the complex field with rigorous algebraic definition. Calling them a "fiction" imports the historical prejudice as a permanent label. The productive-fiction framing is useful for the historical arc but may not be the right description of the mathematics as it now stands.
2. The 250-year resistance was not irrational. Without a geometric interpretation, there was no agreed basis for saying what "i" was. The resistance was not mere prejudice but reflected a legitimate demand for interpretive grounding. Formal consistency alone is not always sufficient for mathematical acceptance — the extra demand for "what does it mean?" is philosophically defensible.
3. Real-valued quantum theories are not strictly ruled out. While the 2022 Chen et al. result is important, some physicists (notably Aleksandrov, Finkelstein) argue that real-valued formulations of quantum mechanics can be constructed through various devices, though at the cost of elegance or additional structure. The claim that complex numbers are strictly necessary for quantum mechanics is debated at the technical frontier.

IMAGES

BIBLIOGRAPHY

1. Cardano, Girolamo | 1545 | ∅ | Ars Magna, sive de regulis algebraicis | ∅ | ∅ | Nuremberg: Petreius | ∅ | ∅ | ∅ | ∅ | English trans. T. Richard Witmer, MIT Press, 1968
2. Bombelli, Rafael | 1572 | ∅ | L'Algebra | ∅ | ∅ | Bologna: Giovanni Rossi | ∅ | ∅ | ∅ | ∅ | ∅
3. Descartes, René | 1637 | ∅ | La Géométrie | ∅ | ∅ | Appendix to Discours de la méthode, Leiden: Jan Maire | ∅ | ∅ | ∅ | ∅ | English trans. Smith & Latham, Open Court, 1925
4. Euler, Leonhard | 1748 | ∅ | Introductio in analysin infinitorum | ∅ | ∅ | Lausanne: Bousquet | ∅ | ∅ | ∅ | ∅ | English trans. John Blanton, Springer, 1988–1990
5. Gauss, Carl Friedrich | 1832 | Theoria residuorum biquadraticorum, Commentatio secunda | ∅ | ∅ | ∅ | Göttingen: Dieterich | ∅ | ∅ | ∅ | ∅ | ∅
6. Schrödinger, Erwin | 1926 | "Quantisierung als Eigenwertproblem" | Annalen der Physik | ∅ | 79::361–376 | ∅ | ∅ | doi:10.1002/andp.19263840404 | ∅ | ∅ | ∅
7. Chen, M.-C., et al | 2022 | "Ruling Out Real-Valued Standard Formalism in Quantum Theory" | Physical Review Letters | ∅ | 128.4::040403 | ∅ | ∅ | doi:10.1103/PhysRevLett.128.040403 | ∅ | ∅ | ∅
8. Nahin, Paul J | 1998 | ∅ | An Imaginary Tale: The Story of √−1 | ∅ | ∅ | Princeton: Princeton University Press | ∅ | isbn:9781400833894 | ∅ | ∅ | ∅
9. Kline, Morris | 1972 | ∅ | Mathematical Thought from Ancient to Modern Times | ∅ | ∅ | Oxford: Oxford University Press | ∅ | ∅ | ∅ | ∅ | 3 vols.
10. Wigner, Eugene P | 1960 | "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" | Communications in Pure and Applied Mathematics | ∅ | 13.1::1–14 | ∅ | ∅ | doi:10.1002/cpa.3160130102 | ∅ | ∅ | ∅
11. Needham, Tristan | 1997 | ∅ | Visual Complex Analysis | ∅ | ∅ | Oxford: Oxford University Press | ∅ | isbn:9780198534464 | ∅ | ∅ | ∅

CROSS-REFERENCE INDEX

Productive Fictions series. Created May 29, 2026.

Corrections

• Nahin, Paul J. — invalid ISBN 9780691027951 removed. No verified replacement could be found, and supplying an unverified number would be worse than none. The entry's author, title, publisher and year are unchanged.
• An Imaginary Tale: The Story of √−1 — ISBN corrected from 9780691027951 to 9781400833894, verified against Open Library (Imaginary Tale, Paul J. Nahin). The previous number failed its check digit.