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 perfectly 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:
Being a quadratic polynomial with no multiple root, the defining equation has two distinct solutions, which are equally valid and which happen to be additive and 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 field is 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 Galois group.
( x, y ) is confined by hyperbola xy = –1 for an imaginary unit matrix.
In this case, the ambiguity results from the geometric choice of which "direction" around the unit circle is "positive" rotation. A more precise explanation is to say that the automorphism group of the special orthogonal groupSO(2, R) has exactly two elements: The identity and the automorphism which exchanges "CW" (clockwise) and "CCW" (counter-clockwise) rotations. For more, see orthogonal group.
All these ambiguities can be solved by adopting a more rigorous definition of complex number, and by explicitly choosing one of the solutions to the equation to be the imaginary unit. For example, the ordered pair (0, 1), in the usual construction of the complex numbers with two-dimensional vectors.
Consider the matrix equation
Here, , so the product xy is negative because ; thus, the point lies in quadrant II or IV. Furthermore,
so is bounded by the hyperbola .
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:
The calculation rules
are only valid for real, positive values of a and b.
These problems can be avoided by writing and manipulating expressions like , rather than . For a more thorough discussion, see square root and branch point.
The two square roots of i in the complex plane
The three cube roots of i in the complex plane
Just like all nonzero complex numbers, i has two square roots: they are[b]
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. All of the following functions are complexmulti-valued functions, and it should be clearly stated which branch of the Riemann surface the function is defined on in practice. Listed below are results for the most commonly chosen branch.
^Some texts[which?] use the Greek letter iota (ι) for the imaginary unit to avoid confusion, especially with indices and subscripts.
In electrical engineering and related fields, the imaginary unit is normally denoted by j to avoid confusion with electric current as a function of time, which is conventionally represented by or just i.
In the quaternions, each of i, j, and k is a distinct imaginary unit.
^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 .
^Boas, Mary L. (2006). Mathematical Methods in the Physical Sciences (3rd ed.). New York [u.a.]: Wiley. p. 49. ISBN 0-471-19826-9.
^ abWeisstein, Eric W. "Imaginary Unit". mathworld.wolfram.com. Retrieved 10 August 2020.
^Doxiadēs, Apostolos K.; Mazur, Barry (2012). Circles Disturbed: The interplay of mathematics and narrative (illustrated ed.). Princeton University Press. p. 225. ISBN 978-0-691-14904-2 – via Google Books.
^Bunch, Bryan (2012). Mathematical Fallacies and Paradoxes (illustrated ed.). Courier Corporation. p. 31-34. ISBN 978-0-486-13793-3 – via Google Books.
^Kramer, Arthur (2012). Math for Electricity & Electronics (4th ed.). Cengage Learning. p. 81. ISBN 978-1-133-70753-0 – via Google Books.
^Picciotto, Henri; Wah, Anita (1994). Algebra: Themes, tools, concepts (Teachers’ ed.). Henri Picciotto. p. 424. ISBN 978-1-56107-252-1 – via Google Books.
^Nahin, Paul J. (2010). An Imaginary Tale: The story of "i" [the square root of minus one]. Princeton University Press. p. 12. ISBN 978-1-4008-3029-9 – via Google Books.
^"What is the square root of i ?". University of Toronto Mathematics Network. Retrieved 26 March 2007.