Definition:Imaginary Number

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Informal Definition

The quadratic equation $a x^2 + b x + c = 0$ 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:

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

and so on.

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.