# Definition:Imaginary Number

## Informal Definition

The quadratic equation $ax^2 + bx + c = 0$ has no solutions in the real number space $\R$ when $b^2 - 4 a c < 0$.

In particular, this applies to the equation $x^2 + 1 = 0$.

In order to be able to allow such equations to have solutions, the concept $i = \sqrt {-1}$ is introduced.

$i$ does not exist on the real number line, but is a completely separate concept.

It can be treated as a number, and combined with real numbers in algebraic expressions.

When $a, b$ are real numbers, we have:

- $a i = i a$
- $a + i = i + a$
- $i a + i b = i \paren {a + b} = \paren {a + b} i = a i + b i$

etc.

In engineering applications, $j$ is usually used instead.

Numbers of the form $a i$ (or $i a$), where $a \in \R$, are known as **imaginary numbers**.

Numbers of the form $a + b i$ are known as complex numbers.

## Historical Note

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

- 1972: Frank Ayres, Jr. and J.C. Ault:
*Theory and Problems of Differential and Integral Calculus*(SI ed.) ... (previous) ... (next): Chapter $1$: Variables and Functions: The Set of Real Numbers