# Definition:Imaginary Number/Historical Note

## Historical Note on Imaginary Number

Gerolamo Cardano was one of the first to accept negative numbers, and possibly the first to consider their square roots, which he did in his *Ars Magna*.

When considering the roots of $x^2 + 40 = 10 x$, and determining that they are $5 \pm \sqrt {-15}$, he concluded:

*These quantities are "truly sophisticated" and that to continue working with them would be "as subtle as it would be useless".*

As negative numbers were even then considered "false" and "fictitious", no wonder the square roots of negative numbers would be named **imaginary**.

John Wallis, while happy to accept negative numbers, wrote of complex numbers:

*These Imaginary Quantities (as they are commonly called) arising from the Supposed Root of a Negative Square (when they happen) are reputed to imply that the Case proposed is Impossible.*

Leonhard Paul Euler used $\sqrt {-1}$ with confidence, and published what is now known as Euler's Identity:

- $e^{i \pi} + 1 = 0$

and introduced the letter $i$ to mean $\sqrt {-1}$.

Subsequently Caspar Wessel, Jean-Robert Argand and Carl Friedrich Gauss all (independently) had the idea of plotting the real part and imaginary part of a complex number on the plane.

Final acceptance of imaginary numbers was complete when Gauss interpreted a complex number as an ordered pair and defined its properties axiomatically.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $-1$ and $i$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $-1$ and $i$