Английская Википедия:Imaginary unit
Шаблон:Short description Шаблон:Redirect Шаблон:Use dmy dates
The imaginary unit or unit imaginary number (Шаблон:Mvar) is a solution to the quadratic equation Шаблон:Math Although there is no real number with this property, Шаблон:Mvar can be used to extend the real numbers to what are called complex numbers, using addition and multiplication. A simple example of the use of Шаблон:Mvar in a complex number is Шаблон:Math
Imaginary numbers are an important mathematical concept; they extend the real number system <math>\mathbb{R}</math> to the complex number system <math>\mathbb{C},</math> in which at least one root for every nonconstant polynomial exists (see Algebraic closure and Fundamental theorem of algebra). Here, the term "imaginary" is used because there is no real number having a negative square.
There are two complex square roots of Шаблон:Math Шаблон:Mvar and Шаблон:Math, just as there are two complex square roots of every real number other than zero (which has one double square root).
In contexts in which use of the letter Шаблон:Mvar is ambiguous or problematic, the letter Шаблон:Mvar is sometimes used instead. For example, in electrical engineering and control systems engineering, the imaginary unit is normally denoted by Шаблон:Mvar instead of Шаблон:Mvar, because Шаблон:Mvar is commonly used to denote electric current.[1]
Terminology
Шаблон:Further Square roots of negative numbers are called imaginary because in early-modern mathematics, only what are now called real numbers, obtainable by physical measurements or basic arithmetic, were considered to be numbers at all – even negative numbers were treated with skepticism – so the square root of a negative number was previously considered undefined or nonsensical. The name imaginary is generally credited to René Descartes, and Isaac Newton used the term as early as 1670.[2][3] The Шаблон:Mvar notation was introduced by Leonhard Euler.[4]
A unit is an undivided whole, and unity or the unit number is the number one (Шаблон:Math).
Definition
| The powers of Шаблон:Mvar are cyclic: |
|---|
| <math>\ \vdots</math> |
| <math>\ i^{-4} = \phantom-1\phantom{i}</math> |
| <math>\ i^{-3} = \phantom-i\phantom1</math> |
| <math>\ i^{-2} = -1\phantom{i}</math> |
| <math>\ i^{-1} = -i\phantom1</math> |
| <math>\ \ i^{0}\ = \phantom-1\phantom{i}</math> |
| <math>\ \ i^{1}\ = \phantom-i\phantom1</math> |
| <math>\ \ i^{2}\ = -1\phantom{i}</math> |
| <math>\ \ i^{3}\ = -i\phantom1</math> |
| <math>\ \ i^{4}\ = \phantom-1\phantom{i}</math> |
| <math>\ \ i^{5}\ = \phantom-i\phantom1</math> |
| <math>\ \ i^{6}\ = -1\phantom{i}</math> |
| <math>\ \ i^{7}\ = -i\phantom1</math> |
| <math>\ \vdots</math> |
The imaginary unit Шаблон:Mvar is defined solely by the property that its square is −1: <math display=block>i^2 = -1.</math>
With Шаблон:Mvar defined this way, it follows directly from algebra that Шаблон:Mvar and Шаблон:Math 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 Шаблон:Mvar as an unknown quantity while manipulating an expression (and using the definition to replace any occurrence of Шаблон:Math with Шаблон:Math). Higher integral powers of Шаблон:Mvar are thus <math display=block>\begin{alignat}{3} i^3 &= i^2 i &&= (-1) i &&= -i, \\[3mu] i^4 &= i^3 i &&= \;\!(-i) i &&= \ \,1, \\[3mu] i^5 &= i^4 i &&= \ \, (1) i &&= \ \ i, \end{alignat}</math> and so on, cycling through the four values Шаблон:Math, Шаблон:Mvar, Шаблон:Math, and Шаблон:Math. As with any non-zero real number, Шаблон:Math
As a complex number, Шаблон:Mvar can be represented in rectangular form as Шаблон:Math, with a zero real component and a unit imaginary component. In polar form, Шаблон:Mvar can be represented as Шаблон:Math (or just Шаблон:Math), with an absolute value (or magnitude) of 1 and an argument (or angle) of <math>\tfrac\pi2</math> radians. (Adding any integer multiple of Шаблон:Math to this angle works as well.) In the complex plane, which is a special interpretation of a Cartesian plane, Шаблон:Mvar is the point located one unit from the origin along the imaginary axis (which is orthogonal to the real axis).
i vs. −i
Шаблон:Anchor Being a quadratic polynomial with no multiple root, the defining equation Шаблон:Math has two distinct solutions, which are equally valid and which happen to be additive and multiplicative 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 Шаблон:Math (or simply Шаблон:Mvar) and the other is labelled Шаблон:Math, though which is which is inherently ambiguous.
The only differences between Шаблон:Math and Шаблон:Math arise from this labelling. For example, by convention Шаблон:Math is said to have an argument of <math>+\tfrac\pi2</math> and Шаблон:Math is said to have an argument of <math>-\tfrac\pi2,</math> related to the convention of labelling orientations in the Cartesian plane relative to the positive Шаблон:Mvar-axis with positive angles turning anticlockwise in the direction of the positive Шаблон:Mvar-axis. Despite the signs written with them, neither Шаблон:Math nor Шаблон:Math is inherently positive or negative in the sense that real numbers are.[5]
A more formal expression of this indistinguishability of Шаблон:Math and Шаблон:Math is that, although the complex field is unique (as an extension of the real numbers) up to isomorphism, it is not unique up to a unique isomorphism. That is, there are two field automorphisms of the complex numbers <math>\C</math> that keep each real number fixed, namely the identity and complex conjugation. For more on this general phenomenon, see Galois group.
Matrices
Using the concepts of matrices and matrix multiplication, complex numbers can be represented in linear algebra. The real unit Шаблон:Math and imaginary unit Шаблон:Mvar can be represented by any pair of matrices Шаблон:Mvar and Шаблон:Mvar satisfying Шаблон:Math Шаблон:Math and Шаблон:Math Then a complex number Шаблон:Math can be represented by the matrix Шаблон:Math and all of the ordinary rules of complex arithmetic can be derived from the rules of matrix arithmetic.
The most common choice is to represent Шаблон:Math and Шаблон:Mvar by the Шаблон:Math identity matrix Шаблон:Mvar and the matrix Шаблон:Mvar,
<math display=block> I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad J = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}.</math>
Then an arbitrary complex number Шаблон:Math can be represented by:
<math display=block>aI + bJ = \begin{pmatrix} a & -b \\ b & a \end{pmatrix}.</math>
More generally, any real-valued Шаблон:Math matrix with a trace of zero and a determinant of one squares to Шаблон:Math, so could be chosen for Шаблон:Mvar. Larger matrices could also be used, for example Шаблон:Math could be represented by the Шаблон:Math identity matrix and Шаблон:Mvar could be represented by any of the Dirac matrices for spatial dimensions.
Root of Шаблон:Math
Polynomials (weighted sums of the powers of a variable) are a basic tool in algebra. Polynomials whose coefficients are real numbers form a ring, denoted <math>\R[x],</math> an algebraic structure with addition and multiplication and sharing many properties with the ring of integers.
The polynomial <math>x^2 + 1</math> has no real-number roots, but the set of all real-coefficient polynomials divisible by <math>x^2 + 1</math> forms an ideal, and so there is a quotient ring <math>\reals[x] / \langle x^2 + 1\rangle.</math> This quotient ring is isomorphic to the complex numbers, and the variable <math>x</math> expresses the imaginary unit.
Graphic representation
Шаблон:Main The complex numbers can be represented graphically by drawing the real number line as the horizontal axis and the imaginary numbers as the vertical axis of a Cartesian plane called the complex plane. In this representation, the numbers Шаблон:Math and Шаблон:Mvar are at the same distance from Шаблон:Math, with a right angle between them. Addition by a complex number corresponds to translation in the plane, while multiplication by a unit-magnitude complex number corresponds to rotation about the origin. Every similarity transformation of the plane can be represented by a complex-linear function <math>z \mapsto az + b.</math>
Geometric algebra
In the geometric algebra of the Euclidean plane, the geometric product or quotient of two arbitrary vectors is a sum of a scalar (real number) part and a bivector 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 scalar Шаблон:Math, and when multiplied by any vector leaves it unchanged (the identity transformation). The quotient of any two perpendicular vectors of the same magnitude, Шаблон:Math, which when multiplied rotates the divisor a quarter turn into the dividend, Шаблон:Math, is a unit bivector which squares to Шаблон:Math, 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 is isomorphic 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-dimensional Euclidean space, a unit bivector of any arbitrary planar orientation squares to Шаблон:Math, so can be taken to represent the imaginary unit Шаблон:Mvar.
Proper use
The imaginary unit was historically written <math display=inline>\sqrt{-1},</math> and still is in some modern works. However, great care needs to be taken when manipulating formulas involving radicals. The radical sign notation <math display=inline>\sqrt{x}</math> is reserved either for the principal square root function, which is defined for only real Шаблон:Math 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:[7] <math display=block>-1 = i \cdot i = \sqrt{-1} \cdot \sqrt{-1} \mathrel{\stackrel{\mathrm{fallacy}}{=}} {\textstyle \sqrt{(-1) \cdot (-1)}} = \sqrt{1} = 1 \qquad \text{(incorrect).}</math>
Generally, the calculation rules <math display=inline>\sqrt{x\vphantom{ty}} \cdot\! \sqrt{y\vphantom{ty}} = \sqrt{x \cdot y\vphantom{ty}}</math> and <math display=inline>\sqrt{x\vphantom{ty}}\big/\!\sqrt{y\vphantom{ty}} = \sqrt{x/y}</math> are guaranteed to be valid for real, positive values of Шаблон:Mvar and Шаблон:Mvar only.[8][9][10]
When Шаблон:Mvar or Шаблон:Mvar is real but negative, these problems can be avoided by writing and manipulating expressions like <math display=inline>i \sqrt{7}</math>, rather than <math display=inline>\sqrt{-7}</math>. For a more thorough discussion, see square root and branch point.
Properties
As a complex number, the imaginary unit follows all of the rules of complex arithmetic.
Imaginary integers and imaginary numbers
When the imaginary unit is repeatedly added or subtracted, the result is some integer times the imaginary unit, an imaginary integer; any such numbers can be added and the result is also an imaginary integer:
<math display=block>ai + bi = (a + b)i.</math>
Thus, the imaginary unit is the generator of a group under addition, specifically an infinite cyclic group.
The imaginary unit can also be multiplied by any arbitrary real number to form an imaginary number. These numbers can be pictured on a number line, the imaginary axis, which as part of the complex plane is typically drawn with a vertical orientation, perpendicular to the real axis which is drawn horizontally.
Gaussian integers
Integer sums of the real unit Шаблон:Math and the imaginary unit Шаблон:Mvar form a square lattice in the complex plane called the Gaussian integers. The sum, difference, or product of Gaussian integers is also a Gaussian integer:
<math display=block>\begin{align} (a + bi) + (c + di) &= (a + c) + (b + d)i, \\[5mu] (a + bi)(c + di) &= (ac - bd) + (ad + bc)i. \end{align}</math>
Quarter-turn rotation
When multiplied by the imaginary unit Шаблон:Mvar, any arbitrary complex number in the complex plane is rotated by a quarter turn (<math>\tfrac12\pi</math> radians or Шаблон:Math) anticlockwise. When multiplied by Шаблон:Math, any arbitrary complex number is rotated by a quarter turn clockwise. In polar form:
<math display=block>i \, re^{\varphi i} = re^{(\varphi + \pi/2)i}, \quad
-i \, re^{\varphi i} = re^{(\varphi - \pi/2)i}.</math>
In rectangular form,
<math display=block> i(a + bi) = -b + ai, \quad -i(a + bi) = b - ai.</math>
Integer powers
The powers of Шаблон:Mvar repeat in a cycle expressible with the following pattern, where Шаблон:Mvar is any integer:
<math display=block> i^{4n} = 1, \quad i^{4n+1} = i, \quad i^{4n+2} = -1, \quad i^{4n+3} = -i.</math>
Thus, under multiplication, Шаблон:Mvar is a generator of a cyclic group of order 4, a discrete subgroup of the continuous circle group of the unit complex numbers under multiplication.
Written as a special case of Euler's formula for an integer Шаблон:Mvar,
<math display=block> i^n = {\exp}\bigl(\tfrac12\pi i\bigr)^n
= {\exp}\bigl(\tfrac12 n \pi i\bigr)
= {\cos}\bigl(\tfrac12 n\pi \bigr) + {i \sin}\bigl(\tfrac12 n\pi \bigr).
</math>
With a careful choice of branch cuts and principal values, this last equation can also apply to arbitrary complex values of Шаблон:Mvar, including cases like Шаблон:Math.Шаблон:Cn
Roots
Just like all nonzero complex numbers, <math display=inline>i = e^{\pi i/ 2}</math> has two distinct square roots which are additive inverses. In polar form, they are <math display=block>\begin{alignat}{3} \sqrt{i} &= {\exp}\bigl(\tfrac12{\pi i}\bigr)^{1/2} &&{}= {\exp}\bigl(\tfrac14\pi i\bigr), \\ -\sqrt{i} &= {\exp}\bigl(\tfrac14{\pi i}-\pi i\bigr) &&{}= {\exp}\bigl({-\tfrac34\pi i}\bigr). \end{alignat}</math>
In rectangular form, they areШаблон:Efn</math> and <math>x=-\tfrac{1}{\sqrt{2}}</math>. Substituting either of these results into the equation Шаблон:Math in turn, we will get the corresponding result for Шаблон:Mvar. Thus, the square roots of Шаблон:Mvar are the numbers <math>\tfrac{1}{\sqrt{2}} + \tfrac{1}{\sqrt{2}}i</math> and <math>-\tfrac{1}{\sqrt{2}}-\tfrac{1}{\sqrt{2}}i</math>.[11]}}
<math display=block>\begin{alignat}{3} \sqrt{i} &= \ (1 + i)\big/ \sqrt2 &&{}= \phantom{-}\tfrac{\sqrt{2}}{2} + \tfrac{\sqrt{2}}{2}i, \\[5mu] -\sqrt{i} &= -(1 + i)\big/ \sqrt2 &&{}= - \tfrac{\sqrt{2}}{2} - \tfrac{\sqrt{2}}{2}i. \end{alignat}</math>
Squaring either expression yields <math display=block> \left( \pm \frac{1 + i}{\sqrt2} \right)^2 = \frac{1 + 2i - 1}{2} = \frac{2i}{2} = i. </math>
The three cube roots of Шаблон:Mvar are[12]
<math display=block> \sqrt[3]i = {\exp}\bigl(\tfrac16 \pi i\bigr) = \tfrac{\sqrt{3}}{2} + \tfrac12i, \quad
{\exp}\bigl(\tfrac56 \pi i\bigr) = -\tfrac{\sqrt{3}}{2} + \tfrac12i, \quad
{\exp}\bigl({-\tfrac12 \pi i}\bigr) = -i.
</math>
For a general positive integer Шаблон:Mvar, the [[nth root|Шаблон:Mvar-th roots]] of Шаблон:Mvar are, for Шаблон:Math <math display=block> \exp \left(2 \pi i \frac{k+\frac14}{n} \right) = \cos \left(\frac{4k+1}{2n}\pi \right) +
i \sin \left(\frac{4k+1}{2n}\pi \right).
</math> The value associated with Шаблон:Math is the principal Шаблон:Mvar-th root of Шаблон:Mvar. The set of roots equals the corresponding set of roots of unity rotated by the principal Шаблон:Mvar-th root of Шаблон:Mvar. These are the vertices of a regular polygon inscribed within the complex unit circle.
Exponential and logarithm
The complex exponential function relates complex addition in the domain to complex multiplication in the codomain. Real values in the domain represent scaling in the codomain (multiplication by a real scalar) with Шаблон:Math representing multiplication by Шаблон:Mvar, while imaginary values in the domain represent rotation in the codomain (multiplication by a unit complex number) with Шаблон:Mvar representing a rotation by Шаблон:Math radian. The complex exponential is thus a periodic function in the imaginary direction, with period Шаблон:Math and image Шаблон:Math at points Шаблон:Math for all integers Шаблон:Mvar, a real multiple of the lattice of imaginary integers.
The complex exponential can be broken into even and odd components, the hyperbolic functions Шаблон:Math and Шаблон:Math or the trigonometric functions Шаблон:Math and Шаблон:Math:
<math display=block>\exp z = \cosh z + \sinh z = \cos(-iz) + i\sin(-iz)</math>
Euler's formula decomposes the exponential of an imaginary number representing a rotation:
<math display=block>\exp i\varphi = \cos \varphi + i\sin \varphi.</math>
The quotient Шаблон:Math with appropriate scaling, can be represented as an infinite partial fraction decomposition as the sum of reciprocal functions translated by imaginary integers:[13] <math display=block> \pi \coth \pi z = \lim_{n\to\infty}\sum_{k=-n}^n \frac{1}{z + ki}. </math>
Other functions based on the complex exponential are well-defined with imaginary inputs. For example, a number raised to the Шаблон:Mvar power is: <math display=block>x^{n i} = \cos(n\ln x) + i \sin(n\ln x ).</math>
Because the exponential is periodic, its inverse the complex logarithm is a multi-valued function, with each complex number in the domain corresponding to multiple values in the codomain, separated from each-other by any integer multiple of Шаблон:Math One way of obtaining a single-valued function is to treat the codomain as a cylinder, with complex values separated by any integer multiple of Шаблон:Math treated as the same value; another is to take the domain to be a Riemann surface consisting of multiple copies of the complex plane stitched together along the negative real axis as a branch cut, with each branch in the domain corresponding to one infinite strip in the codomain.[14] 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 <math>\ln i = \tfrac12 \pi i</math> then when Шаблон:Mvar is a positive real number, <math display=block> \log_i x = -\frac{2i \ln x }{\pi}.</math>
Factorial
The factorial of the imaginary unit Шаблон:Mvar is most often given in terms of the gamma function evaluated at Шаблон:Math:[15]
<math display=block>i! = \Gamma(1+i) = i\Gamma(i) \approx 0.4980 - 0.1549\,i.</math>
The magnitude and argument of this number are:[16]
<math display=block> |\Gamma(1+i)| = \sqrt{\frac{\pi}{ \sinh \pi}} \approx 0.5216, \quad \arg{\Gamma(1+i)} \approx -0.3016. </math>
See also
Notes
References
Further reading
External links
- ↑ Шаблон:Cite book Шаблон:Pb Шаблон:Cite book
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite OED
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite book
- ↑ The interpretation of the imaginary unit as the ratio of two perpendicular vectors was proposed by Hermann Grassmann in the foreword to his Ausdehnungslehre of 1844; later William Clifford realized that this ratio could be interpreted as a bivector. Шаблон:Pb Шаблон:Cite book
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite book
- ↑ Euler expressed the partial fraction decomposition of the trigonometric cotangent as <math display=inline>\pi \cot \pi z = \frac1z + \frac1{z-1} + \frac1{z+1} + \frac1{z-2} + \frac1{z+2} + \cdots .</math> Шаблон:Pb Шаблон:Cite journal
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite journal Шаблон:Pb Sloane, N. J. A. (ed.). "Decimal expansion of the real part of i!", Sequence Шаблон:OEIS link; and "Decimal expansion of the negated imaginary part of i!", Sequence Шаблон:OEIS link. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ↑ Sloane, N. J. A. (ed.). "Decimal expansion of the absolute value of i!", Sequence Шаблон:OEIS link; and "Decimal expansion of the negated argument of i!", Sequence Шаблон:OEIS link. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
