Why are imaginary numbers called imaginary numbers

Question

Why do we call imaginary numbers "imaginary numbers"? As far as I can tell, there's nothing really imaginary about them. They exist. They're used all the time. What makes them so "imaginary"?

Answer 1

Those numbers were once poorly understood and regarded by some as fictitious. The use of imaginary numbers was not widely accepted by many. It was left to Euler and others to use them. Euler, in his Introduction to Algebra, wrote the following:

Because all conceivable numbers are either greater than zero or less than 0 or equal to 0, then it is clear that the square roots of negative numbers cannot be included among the possible numbers [real numbers]. Consequently we must say that these are impossible numbers. And this circumstance leads us to the concept of such number, which by their nature are impossible, and ordinarily are called imaginary or fancied numbers, because they exist only in imagination.

In short, it is a matter of acceptance of or the lack of it which gave them the name.

Answer 2

From the "Earliest Known Uses of Some of the Words of Mathematics" pages,

The terms IMAGINARY and REAL were introduced in French by Rene Descartes (1596-1650) in "La Geometrie" (1637):

...tant les vrayes racines que les fausses ne sont pas tousiours réelles; mais quelquefois seulement imaginaires; c'est à dire qu'on peut bien toujiours en imaginer autant que aiy dit en chàsque Equation; mais qu'il n'y a quelquefois aucune quantité, qui corresponde à celles qu'on imagine. comme encore qu'on en puisse imaginer trois en celle cy, $x^3−6xx+13x−10=0$, il n'y en a toutefois qu'une réelle, qui est 2, & pour les deux autres, quois qu'on les augmente, ou diminué, ou multiplié en la façon que ie viens d'éxpliquer, on ne sçauroit les rendre autres qu'imaginaires.

[...neither the true roots nor the false are always real; sometimes they are, however, imaginary; namely, whereas we can always imagine as many roots for each equation as I have predicted, there is still not always a quantity which corresponds to each root so imagined. Thus, while we may think of the equation$x^3−6xx+13x−10=0$ as having three roots, yet there is just one real root, which is 2, and the other two, however, increased, diminished, or multiplied them as we just laid down, remain always imaginary.] (page 380)

Continuing:

Imaginary and real are found in English in 1668 in Philosophical Transactions. The words are found in a review of the book Geometriæ Universalis:

And for the like reason a Cubick Æquation, having three reals roots, can never be reduced to a pure Æquation, which hath but one onely root, for in these Æquations, Reduction shall no wise profit, forasmuch as 'tis impossible, by aid thereof to change an Imaginary root into a real one, and the Converse. [Google print search by James A. Landau]

Further:

As a way of removing the stigma of the name, the American mathematician Arnold Dresden (1882-1954) suggested that imaginary numbers be called normal numbers, because the term "normal" is synonymous with perpendicular, and the $y$-axis is perpendicular to the $x$-axis (Kramer, p. 73). The suggestion appears in 1936 in his An Invitation to Mathematics.

Some other terms that have been used to refer to imaginary numbers include "sophistic" (Cardan), "nonsense" (Napier), "inexplicable" (Girard), "incomprehensible" (Huygens), and "impossible" (many authors).

The first edition of the Encyclopaedia Britannica (1768-1771) has:

Thus the square root of $-a^2$ cannot be assigned, and is what we call an impossible or imaginary quantity.

There are two modern meanings of the term imaginary number. In Merriam-Webster's Collegiate Dictionary, 10th ed., an imaginary number is a number of the form $a + bi$ where $b$ is not equal to $0$. In Calculus and Analytic Geometry (1992) by Stein and Barcellos, "a complex number that lies on the $y$ axis is called imaginary."

From a later entry:

The term IMAGINARY PART appears in 1748 in A Treatise of Algebra: In Three Parts by Colin Maclaurin. [Google print search by James A. Landau]

Also:

The term IMAGINARY UNIT was used (and apparently introduced) by Sir William Rowan Hamilton in "On a new Species of Imaginary Quantities connected with a theory of Quaternions," Proceedings of the Royal Irish Academy, Nov. 13, 1843:

...the extended expression...which may be called an imaginary unit, because its modulus is = 1, and its square is negative unity.

Answer 3

Mathematicians thought up an answer to the square root of -1, and they simply "imagined" a solution, hence imaginary numbers. They found out it could be very useful, and kept it that way. The name, however, has stuck.