Why do we call numbers 'real' and 'imaginary'?
Why do we call numbers 'real' and 'imaginary'?
To the modern student, the terms "real" and "imaginary" numbers sound like a philosophical judgment on whether these mathematical objects actually exist. However, these labels are the result of historical accidents, early skepticism, and branding by some of history's greatest mathematicians.
The Roots of the Confusion
For centuries, mathematicians could easily visualize "real" numbers. They represented physical quantities: lengths, weights, amounts of money, or positions on a line. Positive numbers made intuitive sense.
Even negative numbers faced a long period of doubt before being accepted. But when mathematicians encountered equations like:
$$x^2 + 1 = 0 \implies x = \pm\sqrt{-1}$$
They hit a wall. There is no real number that, when multiplied by itself, yields a negative result.
René Descartes and the Insult
In the 17th century, the philosopher and mathematician René Descartes coined the term imaginary in his 1637 treatise La Géométrie. However, he did not mean it as a helpful mathematical classification; he used it as an insult.
Descartes was expressing his skepticism about these strange roots. By calling them "imaginary," he meant that they were fictitious, impossible, and merely products of the human imagination—constructs that had no place in rigorous geometry.
Leonhard Euler and Legitimacy
Despite Descartes' dismissive label, mathematicians kept finding that these "impossible" numbers were remarkably useful for solving real cubic and quadratic equations.
In the 18th century, Leonhard Euler revolutionized the field by introducing the symbol $i$ for $\sqrt{-1}$ and formalizing their use. Slowly, the mathematical community realized that imaginary numbers weren't useless fantasies at all.
Are Imaginary Numbers Real?
Today, we know that the naming convention is entirely misleading:
- They are consistent: Complex numbers (numbers with both a real and an imaginary part, like $3 + 4i$) follow rigorous, consistent mathematical rules.
- They are applicable: Imaginary numbers are essential in quantum mechanics, electrical engineering (AC circuits), signal processing, and fluid dynamics. Without them, modern physics and engineering would collapse.
If we were to rename them today, mathematicians might prefer terms like "lateral numbers" (as suggested by mathematician John Wallis) because they extend the number line into a two-dimensional plane rather than existing on a single line. But the historical labels of "real" and "imaginary" stuck, and they remain with us to this day.