The imaginary number i is defined solely by the property that its square is −1:
With i defined this way, it follows directly from
algebra that i and are both square roots of −1.
Although the construction is called "imaginary", and although the concept of an imaginary number may be intuitively more difficult to grasp than that of a real number, the construction is valid from a mathematical standpoint. Real number operations can be extended to imaginary and complex numbers, by treating i as an unknown quantity while manipulating an expression (and using the definition to replace any occurrence of with −1). Higher integral powers of i can also be replaced with , 1, i, or −1:
Similarly, as with any non-zero real number:
As a complex number, i can be represented in
rectangular form as , with a zero real component and a unit imaginary component. In
polar form, i can be represented as (or just ), with an
absolute value (or magnitude) of 1 and an
argument (or angle) of radians. (Adding any multiple of 2π to this angle works as well.) In the
complex plane (also known as the Argand plane), which is a special interpretation of a
Cartesian plane, i is the point located one unit from the origin along the
imaginary axis (which is orthogonal to the
i vs. −i
quadratic polynomial with no
multiple root, the defining equation has two distinct solutions, which are equally valid and which happen to be
multiplicative inverses of each other. Once a solution i of the equation has been fixed, the value , which is distinct from i, is also a solution. Since the equation is the only definition of i, it appears that the definition is ambiguous (more precisely, not
well-defined). However, no ambiguity will result as long as one or other of the solutions is chosen and labelled as "i", with the other one then being labelled as . After all, although and are not quantitatively equivalent (they are negatives of each other), there is no algebraic difference between and , as both imaginary numbers have equal claim to being the number whose square is −1.
In fact, if all mathematical textbooks and published literature referring to imaginary or complex numbers were to be rewritten with replacing every occurrence of (and, therefore, every occurrence of replaced by ), all facts and theorems would remain valid. The distinction between the two roots x of , with one of them labelled with a minus sign, is purely a notational relic; neither root can be said to be more primary or fundamental than the other, and neither of them is "positive" or "negative".
The issue can be a subtle one. One way of articulating the situation is that although the complex
unique (as an extension of the real numbers)
up toisomorphism, it is not unique up to a unique isomorphism. Indeed, there are two
field automorphisms of C that keep each real number fixed, namely the identity and
complex conjugation. For more on this general phenomenon, see
Using the concepts of
matrix multiplication, imaginary units can be represented in linear algebra. The value of 1 is represented by an
identity matrixI and the value of i is represented by any matrix J satisfying J2 = −I. A typical choice is
More generally, a real-valued 2 × 2 matrix J satisfies J2 = −I if and only if J has a
matrix trace of zero and a
matrix determinant of one, so J can be chosen to be
whenever −z2 − xy = 1. The product xy is negative because xy = −(1 + z2); thus, the points (x, y) lie on hyperbolas determined by z in quadrant II or IV.
Matrices larger than 2 × 2 can be used. For example, I could be chosen to be the 4 × 4 identity matrix with J chosen to be any of the three 4 × 4Dirac matrices for spatial dimensions, γ1, γ2, γ3.
Regardless of the choice of representation, the usual rules of complex number mathematics work with these matrices because I × I = I, I × J = J, J × I = J, and J × J = −I. For example,
The imaginary unit is sometimes written in advanced mathematics contexts (as well as in less advanced popular texts). However, great care needs to be taken when manipulating formulas involving
radicals. The radical sign notation is reserved either for the principal square root function, which is only defined for real , or for the principal branch of the complex square root function. Attempting to apply the calculation rules of the principal (real) square root function to manipulate the principal branch of the complex square root function can produce false results:
Generally, the calculation rules
are guaranteed to be valid for real, positive values of a and b only.
When a or b is real but negative, these problems can be avoided by writing and manipulating expressions like , rather than . For a more thorough discussion, see
square root and
Just like all nonzero complex numbers, i has two square roots: they are[a]
Many mathematical operations that can be carried out with real numbers can also be carried out with i, such as exponentiation, roots, logarithms, and trigonometric functions. The following functions are well-defined, single-valued functions when x is a positive real number.
which is a purely imaginary number when n is a real number.
In contrast, many functions involving i, including those that depend upon log i or the logarithm of another complex number, are
complexmulti-valued functions, with different values on different branches of the
Riemann surface the function is defined on. For example, if one chooses any branch where log i = πi/2 then one can write
when x is a positive real number. When x is not a positive real number in the above formulas then one must precisely specify the branch to get a single-valued function; see
^To find such a number, one can solve the equation where x and y are real parameters to be determined, or equivalently Because the real and imaginary parts are always separate, we regroup the terms, By
equating coefficients, separating the real part and imaginary part, we get a system of two equations:
Substituting into the first equation, we get Because x is a real number, this equation has two real solutions for x: and . Substituting either of these results into the equation in turn, we will get the corresponding result for y. Thus, the square roots of i are the numbers and .