By combining the real numbers with the imaginary unit usingaddition andmultiplication, a new number system known as thecomplex numbers is formed; it consists of all numbers of the forma +bi with real numbersa andb.
There are two complexsquare roots of−1: the imaginary uniti and itsadditive inverse−i. More generally, every nonzero complex number has two distinct complex-valued square roots, which are additive inverses of each other, whilezero has only zero as its (double) square root.
Historically, the imaginary unit was denoted by, though this is now rare. In contexts in which use of the letteri is ambiguous or problematic, the letterj is sometimes used instead. For example, inelectrical engineering the imaginary unit is normally denoted byj instead ofi, becausei is commonly used to denoteelectric current.[1]
Square roots of negative numbers are calledimaginary because inearly-modern mathematics, only what are now calledreal numbers, obtainable by physical measurements or basic arithmetic, were considered to be numbers at all—evennegative numbers were treated with skepticism—so the square root of a negative number was previously considered undefined or nonsensical. The nameimaginary is generally credited toRené Descartes, andIsaac Newton used the term as early as 1670.[2][3] Thei notation was introduced byLeonhard Euler.[4]
The imaginary uniti is defined solely by the property that its square is −1:
Withi defined this way, it follows directly fromalgebra thati and−i are both square roots of −1.
Although the construction is calledimaginary, 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 treatingi as an unknown quantity while manipulating an expression (and using the definition to replace any occurrence ofi2 with−1). Higher integral powers ofi are thusand so on, cycling through the four values1,i,−1, and−i. As with any non-zero real number,i0 = 1.
As a complex number,i can be represented inrectangular form as0 + 1i, with a zero real component and a unit imaginary component. Inpolar form,i can be represented as1 ×eπi /2 (or justeπi /2), with anabsolute value (or magnitude) of 1 and anargument (or angle) ofradians. (Adding any integer multiple of2π to this angle works as well.) In thecomplex plane, which is a special interpretation of aCartesian plane,i is the point located one unit from the origin along theimaginary axis (which isperpendicular to thereal axis).
Being aquadratic polynomial with nomultiple root, the defining equationx2 = −1 hastwo distinct solutions, which are equally valid and which happen to beadditive andmultiplicative inverses of each other. Although the two solutions are distinct numbers, their properties are indistinguishable; there is no property that one has that the other does not. One of these two solutions is labelled+i (or simplyi) and the other is labelled−i, though it is inherently ambiguous which is which.
The only differences between+i and−i arise from this labelling. For example, by convention+i is said to have anargument of and−i is said to have an argument of related to the convention of labelling orientations in theCartesian plane relative to the positivex-axis with positive angles turninganticlockwise in the direction of the positivey-axis. Also, despite the signs written with them, neither+i nor−i is inherently positive or negative in the sense that real numbers are.[5]
A more formal expression of this indistinguishability of+i and−i is that, although the complexfield isunique (as an extension of the real numbers)up toisomorphism, it isnot unique up to aunique isomorphism. That is, there are twofield automorphisms of the complex numbers that keep each real number fixed, namely the identity andcomplex conjugation. For more on this general phenomenon, seeGalois group.
Using the concepts ofmatrices andmatrix multiplication, complex numbers can be represented in linear algebra. The real unit1 and imaginary uniti can be represented by any pair of matricesI andJ satisfyingI2 =I,IJ =JI =J, andJ2 = −I. Then a complex numbera +bi can be represented by the matrixaI +bJ, and all of the ordinary rules of complex arithmetic can be derived from the rules of matrix arithmetic.
The most common choice is to represent1 andi by the2 × 2identity matrixI and the matrixJ,
Then an arbitrary complex numbera +bi can be represented by:
More generally, any real-valued2 × 2 matrix with atrace of zero and adeterminant of one squares to−I, so could be chosen forJ. Larger matrices could also be used; for example,1 could be represented by the4 × 4 identity matrix andi could be represented by any of theDirac matrices for spatial dimensions.
Polynomials (weighted sums of the powers of a variable) are a basic tool in algebra. Polynomials whosecoefficients are real numbers form aring, denoted an algebraic structure with addition and multiplication and sharing many properties with the ring ofintegers.
The polynomial has no real-numberroots, but the set of all real-coefficient polynomials divisible by forms anideal, and so there is aquotient ring This quotient ring isisomorphic to the complex numbers, and the variable expresses the imaginary unit.
The complex numbers can be represented graphically by drawing the realnumber line as the horizontal axis and the imaginary numbers as the vertical axis of aCartesian plane called thecomplex plane. In this representation, the numbers1 andi are at the same distance from0, with a right angle between them. Addition by a complex number corresponds totranslation in the plane, while multiplication by a unit-magnitude complex number corresponds to rotation about the origin. Everysimilarity transformation of the plane can be represented by a complex-linear function
In thegeometric algebra of theEuclidean plane, the geometric product or quotient of two arbitraryvectors is a sum of a scalar (real number) part and abivector part. (A scalar is a quantity with no orientation, a vector is a quantity oriented like a line, and a bivector is a quantity oriented like a plane.) The square of any vector is a positive scalar, representing its length squared, while the square of any bivector is a negative scalar.
The quotient of a vector with itself is the scalar1 =u/u, and when multiplied by any vector leaves it unchanged (theidentity transformation). The quotient of any two perpendicular vectors of the same magnitude,J =u/v, which when multiplied rotates the divisor a quarter turn into the dividend,Jv =u, is a unit bivector which squares to−1, and can thus be taken as a representative of the imaginary unit. Any sum of a scalar and bivector can be multiplied by a vector to scale and rotate it, and the algebra of such sums isisomorphic to the algebra of complex numbers. In this interpretation points, vectors, and sums of scalars and bivectors are all distinct types of geometric objects.[6]
More generally, in the geometric algebra of any higher-dimensionalEuclidean space, a unit bivector of any arbitrary planar orientation squares to−1, so can be taken to represent the imaginary uniti.
The imaginary unit was historically written and still is in some modern works. However, great care needs to be taken when manipulating formulas involvingradicals. The radical sign notation is reserved either for the principal (positive) square root of a positive real number or for theprincipal square root of a complex number. Attempting to apply the calculation rules of square roots of positive real numbers to manipulate square roots of complex numbers can produce false results:[7]
Generally, the calculation rulesandare guaranteed to be valid only whenx andy are both positive real numbers.[8][9][10]
Whenx ory is real but negative, these problems can be avoided by writing and manipulating expressions like, rather than. For a more thorough discussion, see the articlesSquare root andBranch point.
When the imaginary unit is repeatedly added or subtracted, the result is someinteger times the imaginary unit, animaginary integer; any such numbers can be added and the result is also an imaginary integer:
Thus, the imaginary unit is the generator of agroup under addition, specifically an infinitecyclic group.
The imaginary unit can also be multiplied by any arbitraryreal number to form animaginary number. These numbers can be pictured on anumber line, theimaginary axis, which as part of the complex plane is typically drawn with a vertical orientation, perpendicular to the real axis which is drawn horizontally.
Integer sums of the real unit1 and the imaginary uniti form asquare lattice in the complex plane called theGaussian integers. The sum, difference, or product of Gaussian integers is also a Gaussian integer:
When multiplied by the imaginary uniti, any arbitrary complex number in the complex plane is rotated by a quarter turn( radians or90°)anticlockwise. When multiplied by−i, any arbitrary complex number is rotated by a quarter turn clockwise. In polar form:
The powers ofi repeat in a cycle expressible with the following pattern, wheren is any integer:
Thus, under multiplication,i is a generator of acyclic group of order 4, a discrete subgroup of the continuouscircle group of the unit complex numbers under multiplication.
Written as a special case ofEuler's formula for an integern,
For a general positive integern, then-th roots ofi are, fork = 0, 1, ...,n − 1,The value associated withk = 0 is theprincipaln-th root ofi. The set of roots equals the corresponding set ofroots of unity rotated by the principaln-th root ofi. These are the vertices of aregular polygon inscribed within the complexunit circle.
Thecomplex exponential function relates complex addition in the domain tocomplex multiplication in the codomain. Real values in the domain represent scaling in the codomain (multiplication by a real scalar) with1 representing multiplication bye, while imaginary values in the domain represent rotation in the codomain (multiplication by a unit complex number) withi representing a rotation by1 radian. The complex exponential is thus aperiodic function in the imaginary direction, with period2πi and image1 at points2kπi for all integersk, a real multiple of the lattice of imaginary integers.
Other functions based on the complex exponential are well-defined with imaginary inputs. For example, a number raised to theni power is:
Because the exponential is periodic, its inverse thecomplex logarithm is amulti-valued function, with each complex number in the domain corresponding to multiple values in the codomain, separated from each-other by any integer multiple of2πi. One way of obtaining a single-valued function is to treat the codomain as acylinder, with complex values separated by any integer multiple of2πi treated as the same value; another is to take the domain to be aRiemann surface consisting of multiple copies of the complex plane stitched together along the negative real axis as abranch cut, with each branch in the domain corresponding to one infinite strip in the codomain.[15] Functions depending on the complex logarithm therefore depend on careful choice of branch to define and evaluate clearly.
For example, if one chooses any branch where then whenx is a positive real number,
^To find such a number, one can solve the equation(x +iy)2 =i wherex andy are real parameters to be determined, or equivalentlyx2 + 2ixy −y2 =i. Because the real and imaginary parts are always separate, we regroup the terms,x2 −y2 + 2ixy = 0 +i. Byequating coefficients, separating the real part and imaginary part, we have a system of two equations:Substituting into the first equation, we get Becausex is a real number, this equation has two real solutions forx and. Substituting either of these results into the equation2xy = 1 in turn, we will get the corresponding result fory. Thus, the square roots ofi are the numbers and.[11]
^The interpretation of the imaginary unit as the ratio of two perpendicular vectors was proposed byHermann Grassmann in the foreword to hisAusdehnungslehre of 1844; laterWilliam Clifford realized that this ratio could be interpreted as a bivector.Hestenes, David (1996)."Grassmann's Vision"(PDF). In Schubring, G. (ed.).Hermann Günther Graßmann (1809–1877). Boston Studies in the Philosophy of Science. Vol. 187. Springer. pp. 243–254.doi:10.1007/978-94-015-8753-2_20.ISBN978-90-481-4758-8.
^Zill, Dennis G.; Shanahan, Patrick D. (2003).A first course in complex analysis with applications. Boston: Jones and Bartlett. pp. 24–25.ISBN0-7637-1437-2.OCLC50495529.
