# Imaginary Numbers: The 16th-Century Secret Behind i

*Science & Technology · Priya Nair · 2026-09-06*

## TL;DR

Imaginary numbers were first encountered in 1545 by Girolamo Cardano while solving cubic equations, but it was Rafael Bombelli in 1572 who developed systematic rules for working with them. The term "imaginary" was coined by René Descartes in 1637 as an insult. For over two centuries, mathematicians treated them with suspicion, until Leonhard Euler and Carl Friedrich Gauss gave them a solid foundation in the 18th and 19th centuries.

## Key facts

- In 1545, Girolamo Cardano encountered square roots of negative numbers in his work Ars Magna, though he dismissed them as useless.
- French philosopher René Descartes coined the dismissive term "imaginary" for these numbers in his 1637 work, La Géométrie.
- Swiss mathematician Leonhard Euler introduced the symbol i for the square root of negative one in the 18th century.
- Caspar Wessel presented a geometric interpretation of complex numbers in 1797, and Jean-Robert Argand independently developed a similar one in 1806.
- By the early 20th century, imaginary numbers became essential tools in AC circuit analysis and, later, in quantum mechanics.

## The Puzzle That Launched Imaginary Numbers

Can you think of a number that, when multiplied by itself, gives a negative result? For real numbers, the answer is a firm no. A positive number squared stays positive, and a negative number squared also turns out positive. That means the innocent-looking equation x² = −1 has no solution among the real numbers at all. It looks like a dead end, yet this single impossibility cracked open an entire branch of mathematics that would take centuries to understand.

Here is the strange part of imaginary numbers history: mathematicians kept bumping into these impossible expressions while solving perfectly ordinary problems. In the 16th century, scholars working on cubic equations discovered something unsettling. They could reach real, correct final answers, but only by passing through square roots of negative numbers in the middle of their calculations. The destination made sense, but the road signs along the way seemed absurd.

Girolamo Cardano, an Italian polymath, ran straight into this puzzle in his 1545 work *Ars Magna*, one of the most influential algebra texts of the era. He encountered square roots of negative numbers while working through equations, but he brushed them off as "sophistic," meaning clever but ultimately useless. To him, these quantities were not meaningful mathematical objects, just curiosities that appeared and then vanished.

Then came Rafael Bombelli, another Italian mathematician, who took the opposite approach. In his 1572 book *L'Algebra*, Bombelli decided to stop dismissing these strange roots and instead figure out how to work with them systematically. He developed rules for adding, subtracting, and multiplying these seemingly impossible quantities. What had been a dead end became a calculable system, even if nobody yet understood what these numbers really represented.

The names mathematicians gave these quantities reveal how uncomfortable they felt. They called them "imaginary," "impossible," even "absurd." These were harsh words for a discipline built on certainty. Yet the numbers kept producing correct results, and that stubborn fact refused to go away. The more mathematicians used them, the more they seemed to work, even as their own intuition screamed that something must be wrong.

## Why Mathematicians Fought Over a Symbol

The fight was never really about the letter *i* itself. It was about what the square root sign actually meant, and two brilliant mathematicians could look at the exact same expression and walk away with different answers.

René Descartes poured fuel on the fire in the 17th century when he coined the term "imaginary" in his 1637 work *La Géométrie*. The word was meant as an insult, a label suggesting these numbers were fake, impossible, even absurd. The name stuck, and with it came centuries of suspicion.

Leonhard Euler tried to clean things up in the 18th century. He introduced the symbol *i* for √−1, giving mathematicians a tidy shorthand. But a new symbol alone could not settle the deeper confusion about how these numbers behaved.

Here is where things get genuinely strange. Take the rule √a × √b = √ab. It works beautifully for positive numbers. Try it with two negative radicands, though, and the whole system collapses. Multiply √−1 × √−1 and you should get i² = −1. But blindly apply the old rule and you get √1 = 1. Suddenly you have proven that −1 equals 1, which is nonsense.

Euler found himself standing right in the middle of this mess, and later mathematicians would accuse him of making a careless error. The disputed calculation went like this: √−4 × √−9 = √36. Under today's convention of principal square roots, √−4 = 2i and √−9 = 3i. Their product is 6i² = −6, a single clean answer.

But in Euler's day, √4 did not automatically mean just 2. Mathematicians of the 18th century routinely treated square roots as having two values, so √4 meant ±2. Apply that same thinking to his calculation and the picture changes. If √−4 = ±2i and √−9 = ±3i, then the product can be either +6 or −6, depending on which signs you choose. And since √36 also meant ±6, Euler's equation held up perfectly under his own rules.

So was Euler wrong? Not necessarily. He was using a different convention for the square root symbol, one that later generations misread as a blunder. The symbols looked identical on paper, but they carried different meanings depending on the era and the mathematician holding the pen.

## Finding a Place for i: The Geometric Breakthrough

By the 18th century, the arguments over imaginary numbers had grown fierce, and Francis Maseres, a mathematician, flatly rejected them. Some of his colleagues even doubted that negative numbers deserved to stand as independent quantities. Yet the deeper puzzle remained stubbornly unsolved: real numbers fit neatly on a number line, so where on earth did √−1 belong?

John Wallis, an English mathematician, was among the first to wonder whether negative numbers and their square roots might have some geometric meaning. He speculated about it, but his ideas never quite crystallized into a workable system. The real breakthrough arrived at the end of the 18th century, when Caspar Wessel presented a paper in 1797 proposing that complex numbers could be understood as directed line segments in a plane. It was a beautiful idea, yet it sank without a trace. Wessel's work went largely unnoticed, and he remained an obscure figure in the story of imaginary numbers history.

A few years later, Jean-Robert Argand independently arrived at the same geometric interpretation in 1806. His version caught on more widely, perhaps because it was published in French and reached a broader audience. The picture was now clear: a number like a + bi could be drawn as a point on a flat surface, with the horizontal axis carrying the real part and the vertical axis carrying the imaginary part. That single insight made everything click.

Then came Carl Friedrich Gauss in the 19th century. Gauss, already celebrated as one of the greatest mathematicians of his era, embraced the geometric view and helped spread it through his lectures and writings. Thanks to his authority, the representation of complex numbers as points on a plane became standard practice, and today we often call that plane the Gauss plane in his honor. Once i had a visual home, it stopped being a suspicious algebraic trick and became something far more powerful: a bridge connecting algebra to geometry, a tool that could turn abstract equations into pictures anyone could trace with a pencil.

## From 'Imaginary' to Indispensable: What Complex Numbers Do Today

When you stream a song, filter a photo, or flip a light switch, you are relying on mathematics that was once dismissed as nonsense. The numbers that 18th-century mathematician Francis Maseres rejected now power the technology we touch daily. Complex numbers, built on that strange √−1, have become essential tools in physics and engineering wherever magnitude and phase matter.

Consider the electricity running through your walls. In alternating current circuits, voltage and current do not always peak at the same moment. That timing offset, called the phase difference, is awkward to track with ordinary arithmetic. Complex numbers express it elegantly, allowing engineers to add, multiply, and divide these quantities without drowning in trigonometric identities.

The same trick shows up in wave phenomena everywhere. Sound, light, radio, even the ripples in a pond, all obey wave mathematics. Euler's formula, e^(iθ) = cos θ + i sin θ, collapses lengthy trigonometric calculations into compact algebraic steps. Instead of juggling sine and cosine identities through a messy derivation, a physicist writes a single exponential, works through the algebra, and extracts the answer at the end.

Quantum mechanics leans on complex numbers at its very foundation. The mathematics of the subatomic world simply cannot be expressed without them. Signal processing depends on them just as heavily. When software separates an audio recording into its frequency components, it relies on the same underlying framework. Image processing and many other modern techniques build on this mathematics as well.

So the arc is complete. In 1545, Girolamo Cardano stumbled upon square roots of negative numbers while solving cubic equations in *Ars Magna* and called them sophisticated but useless. In 1572, Rafael Bombelli worked out rules for handling them. René Descartes gave them the dismissive name "imaginary" in the 17th century. And now, those same imaginary numbers are the quiet engines behind your phone, your power grid, and your understanding of the subatomic world. The label stuck, but the scorn faded. What was once dismissed as impossible became one of the most powerful frameworks we have for describing waves, vibrations, and the phase relationships that shape physical reality.


## Frequently asked questions

**What are imaginary numbers and why were they invented?**

Imaginary numbers are numbers that give a negative result when squared, with the foundational example being i, where i² = −1. They were invented because equations like x² = −1 have no solution among real numbers, yet they kept appearing naturally while mathematicians in the 16th century worked on solving cubic equations. Even when the final answer was a real number, square roots of negatives showed up in the intermediate steps, forcing mathematicians to take them seriously.

**Who first introduced the concept of imaginary numbers?**

Girolamo Cardano, an Italian mathematician, encountered square roots of negative numbers in his 1545 work Ars Magna , though he dismissed them as useless. It was Rafael Bombelli who changed everything in 1572 with his book L'Algebra , where he developed systematic rules for adding, subtracting, multiplying, and dividing these strange quantities. Bombelli treated them as legitimate algebraic objects rather than mere curiosities.

**Why did mathematicians initially reject imaginary numbers?**

They called these numbers "imaginary," "impossible," and even "absurd," with René Descartes coining the term "imaginary" in the 17th century. The rejection made sense because imaginary numbers broke familiar rules, such as √a × √b = √ab, which fails when both a and b are negative. This created apparent contradictions like −1 = 1, which rightly troubled mathematicians who trusted their established methods.

**How did Euler's use of the square root symbol differ from today's?**

In the 18th century, Leonhard Euler and his contemporaries often treated √4 as meaning ±2, not just the principal root 2 that students learn today. This multi-valued convention meant calculations like √−4 × √−9 could yield both +6 and −6 depending on sign choices, which matched Euler's result of √36 = ±6. Under today's principal root convention, the same product simply equals −6.

**What is the geometric interpretation of imaginary numbers?**

Complex numbers like a + bi correspond to points on a plane, with the horizontal axis showing the real part and the vertical axis showing the imaginary part. Caspar Wessel first presented this idea in 1797, and Jean-Robert Argand independently developed a similar interpretation in 1806. Carl Friedrich Gauss later popularized the approach in the 19th century, helping imaginary numbers gain acceptance as legitimate mathematical objects.

**Why is the rule √a × √b = √ab not valid for negative numbers?**

The rule fails because applying it to negative numbers creates a direct contradiction. If you compute √−1 × √−1, you get i × i = −1 by definition, but the rule would give √(−1 × −1) = √1 = 1. Since −1 cannot equal 1, the rule only holds when at least one of a or b is non-negative. This subtle point confused mathematicians for centuries before the convention of principal square roots was standardized.

**How are complex numbers used in physics and engineering?**

Complex numbers represent phase differences in alternating current circuits, where voltage and current are offset from each other. Euler's formula e^(iθ) = cos θ + i sin θ simplifies wave analysis by converting lengthy trigonometric calculations into shorter algebraic ones. They are also fundamental tools in quantum mechanics and signal processing, from separating audio recordings into frequencies to image processing techniques.

**What role did Gauss play in the acceptance of imaginary numbers?**

Gauss promoted the geometric representation of complex numbers in the 19th century, lending his considerable authority to the idea that they were legitimate mathematical objects. His endorsement helped shift the conversation from whether imaginary numbers existed to how they could be visualized and applied. The complex plane is still sometimes called the Gauss plane in his honor, reflecting his central role in bridging algebra and geometry. Editorial note: Any facts which remain unconfirmed are presented as such, hedged clearly within the narrative above.

Canonical: https://curio.madeda.com/imaginary-numbers-the-16th-century-secret-behind-i/