The imaginary unit or unit imaginary number (i) is a solution to the
quadratic equation x2 + 1 = 0. Although there is no real number with
this property, i 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 i in a complex number is 2 + 3i.
Imaginary numbers are an important mathematical concept, which extend
the real number system ℝ to the complex number system ℂ, which in
turn provides at least one root for every nonconstant polynomial P(x).
(See
Contents 1 Definition 2 i and −i 2.1 Matrices 3 Proper use 4 Properties 4.1 Square roots
4.2 Cube roots
4.3
4.4.1 i raised to the power of i 4.5 Factorial 4.6 Other operations 5 Alternative notations 6 See also 7 Notes 8 References 9 Further reading 10 External links Definition[edit] The powers of i return cyclic values: ... (repeats the pattern from blue area) i−3 = i i−2 = −1 i−1 = −i i0 = 1 i1 = i i2 = −1 i3 = −i i4 = 1 i5 = i i6 = −1 ... (repeats the pattern from the blue area) The imaginary number i is defined solely by the property that its square is −1: i 2 = − 1. displaystyle i^ 2 =-1. With i defined this way, it follows directly from algebra that i and
−i 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.
i 3 = i 2 i = ( − 1 ) i = − i displaystyle i^ 3 =i^ 2 i=(-1)i=-i, i 4 = i 3 i = ( − i ) i = − ( i 2 ) = − ( − 1 ) = 1 displaystyle i^ 4 =i^ 3 i=(-i)i=-(i^ 2 )=-(-1)=1, i 5 = i 4 i = ( 1 ) i = i displaystyle i^ 5 =i^ 4 i=(1)i=i, Similarly, as with any non-zero real number: i 0 = i 1 − 1 = i 1 i − 1 = i 1 1 i = i 1 i = i i = 1 displaystyle i^ 0 =i^ 1-1 =i^ 1 i^ -1 =i^ 1 frac 1 i =i frac 1 i = frac i i =1, As a complex number, i is represented in rectangular form as
0 + i, having a unit imaginary component and no real
component (i.e., the real component is zero). In polar form, i is
represented as 1eiπ/2 (or just eiπ/2), having an absolute value (or
magnitude) of 1 and an argument (or angle) of π/2. In the complex
plane (also known as the Cartesian plane), i is the point located one
unit from the origin along the imaginary axis (which is at a right
angle to the real axis).
i and −i[edit]
Being a quadratic polynomial with no multiple root, the defining
equation x2 = −1 has two distinct solutions, which are equally valid
and which happen to be additive and multiplicative inverses of each
other. More precisely, once a solution i of the equation has been
fixed, the value −i, 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 results as long as one or other of the solutions is
chosen and labelled as "i", with the other one then being labelled as
−i. This is because, although −i and i are not quantitatively
equivalent (they are negatives of each other), there is no algebraic
difference between i and −i. Both imaginary numbers have equal claim
to being the number whose square is −1. If all mathematical
textbooks and published literature referring to imaginary or complex
numbers were rewritten with −i replacing every occurrence of +i (and
therefore every occurrence of −i replaced by −(−i) = +i), all
facts and theorems would continue to be equivalently valid. The
distinction between the two roots x of x2 + 1 = 0 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".[1]
The issue can be a subtle one. The most precise explanation is to say
that although the complex field, defined as ℝ[x]/(x2 + 1) (see
complex number), is unique up to isomorphism, it is not unique up to a
unique isomorphism — there are exactly two field automorphisms of
ℝ[x]/(x2 + 1) which keep each real number fixed: the identity and
the automorphism sending x to −x. See also
X = ( 0 − 1 1 0 ) displaystyle X= begin pmatrix 0&-1\1&;;0end pmatrix and X = ( 0 1 − 1 0 ) displaystyle X= begin pmatrix ;;0&1\-1&0end pmatrix are solutions to the matrix equation X 2 = − I = − ( 1 0 0 1 ) = ( − 1 0 0 − 1 ) . displaystyle X^ 2 =-I=- begin pmatrix 1&0\0&1end pmatrix = begin pmatrix -1&;;0\;;0&-1end pmatrix . 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 group SO(2, ℝ) has exactly two elements—the identity and the automorphism which exchanges "CW" (clockwise) and "CCW" (counter-clockwise) rotations. See orthogonal group. All these ambiguities can be solved by adopting a more rigorous definition of complex number, and 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. When the set of 2 × 2 real matrices M (2, ℝ) is used for a source, and the number one (1) is identified with the identity matrix, and minus one (−1) with the negative of the identity matrix, then there are many solutions to X 2 = −1. In fact, there are many solutions to X 2 = +1 and X 2 = 0 also. Any such X can be taken as a basis vector, along with 1, to form a planar subalgebra x I + y X : x , y ∈ R ⊂ M ( 2 , R ) . displaystyle xI+yX:x,yin mathbb R subset M(2,mathbb R ). Proper use[edit] The imaginary unit is sometimes written √−1 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 x ≥ 0, 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: − 1 = i ⋅ i = − 1 ⋅ − 1 = ( − 1 ) ⋅ ( − 1 ) = 1 = 1 displaystyle -1=icdot i= sqrt -1 cdot sqrt -1 = sqrt (-1)cdot (-1) = sqrt 1 =1 (incorrect). Similarly: 1 i = 1 − 1 = 1 − 1 = − 1 1 = − 1 = i displaystyle frac 1 i = frac sqrt 1 sqrt -1 = sqrt frac 1 -1 = sqrt frac -1 1 = sqrt -1 =i (incorrect). The calculation rules a ⋅ b = a ⋅ b displaystyle sqrt a cdot sqrt b = sqrt acdot b and a b = a b displaystyle frac sqrt a sqrt b = sqrt frac a b are only valid for real, non-negative values of a and b.[2]
These problems are avoided by writing and manipulating expressions
like i√7, rather than √−7. For a more thorough discussion, see
The two square roots of i in the complex plane The three cube roots of i in the complex plane i has two square roots, just like all complex numbers (except zero, which has a double root). These two roots can be expressed as the complex numbers:[nb 1] ± ( 2 2 + 2 2 i ) = ± 2 2 ( 1 + i ) . displaystyle pm left( frac sqrt 2 2 + frac sqrt 2 2 iright)=pm frac sqrt 2 2 (1+i). Indeed, squaring both expressions: ( ± 2 2 ( 1 + i ) ) 2
= ( ± 2 2 ) 2 ( 1 + i ) 2
= 1 2 ( 1 + 2 i + i 2 ) = 1 2 ( 1 + 2 i − 1 ) = i . displaystyle begin aligned left(pm frac sqrt 2 2 (1+i)right)^ 2 &=left(pm frac sqrt 2 2 right)^ 2 (1+i)^ 2 \&= frac 1 2 (1+2i+i^ 2 )\&= frac 1 2 (1+2i-1) \&=i. \end aligned Using the radical sign for the principal square root gives: i = 2 2 ( 1 + i ) . displaystyle sqrt i = frac sqrt 2 2 (1+i). Cube roots[edit] The three cube roots of i are: − i , displaystyle -i, 3 2 + i 2 , displaystyle frac sqrt 3 2 + frac i 2 , − 3 2 + i 2 . displaystyle - frac sqrt 3 2 + frac i 2 . Similar to all of the roots of 1, all of the roots of i are the
vertices of regular polygons inscribed within the unit circle in the
complex plane.
i ( a + b i ) = a i + b i 2 = − b + a i . displaystyle i,(a+bi)=ai+bi^ 2 =-b+ai. (This is equivalent to a 90° counter-clockwise rotation of a vector about the origin in the complex plane.) Dividing by i is equivalent to multiplying by the reciprocal of i: 1 i = 1 i ⋅ i i = i i 2 = i − 1 = − i . displaystyle frac 1 i = frac 1 i cdot frac i i = frac i i^ 2 = frac i -1 =-i. Using this identity to generalize division by i to all complex numbers gives: a + b i i = − i ( a + b i ) = − a i − b i 2 = b − a i . displaystyle frac a+bi i =-i,(a+bi)=-ai-bi^ 2 =b-ai. (This is equivalent to a 90° clockwise rotation of a vector about the origin in the complex plane.) Powers[edit] The powers of i repeat in a cycle expressible with the following pattern, where n is any integer: i 4 n = 1 displaystyle i^ 4n =1, i 4 n + 1 = i displaystyle i^ 4n+1 =i, i 4 n + 2 = − 1 displaystyle i^ 4n+2 =-1, i 4 n + 3 = − i . displaystyle i^ 4n+3 =-i., This leads to the conclusion that i n = i n mod 4 displaystyle i^ n =i^ n bmod 4 , where mod represents the modulo operation. Equivalently: i n = cos ( n π / 2 ) + i sin ( n π / 2 ) displaystyle i^ n =cos(npi /2)+isin(npi /2) i raised to the power of i[edit] Making use of Euler's formula, ii is i i = ( e i ( π / 2 + 2 k π ) ) i = e i 2 ( π / 2 + 2 k π ) = e − ( π / 2 + 2 k π ) displaystyle i^ i =left(e^ i(pi /2+2kpi ) right)^ i =e^ i^ 2 (pi /2+2kpi ) =e^ -(pi /2+2kpi ) where k ∈ Z displaystyle kin mathbb Z , the set of integers. The principal value (for k = 0) is e−π/2 or approximately 0.207879576...[3] Factorial[edit] The factorial of the imaginary unit i is most often given in terms of the gamma function evaluated at 1 + i: i ! = Γ ( 1 + i ) ≈ 0.4980 − 0.1549 i . displaystyle i!=Gamma (1+i)approx 0.4980-0.1549i. Also,
i !
= π sinh π displaystyle i!= sqrt pi over sinh pi [4] Other operations[edit]
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 complex multi-valued functions, and it should be clearly
stated which branch of the
x n i = cos ( n ln x ) + i sin ( n ln x ) . displaystyle x^ ni =cos(nln x)+isin(nln x). The nith root of a number is: x n i = cos ( ln x n ) − i sin ( ln x n ) . displaystyle sqrt[ ni ] x =cos left( frac ln x n right)-isin left( frac ln x n right). The imaginary-base logarithm of a number is: log i ( x ) = 2 ln x i π . displaystyle log _ i (x)= frac 2ln x ipi . As with any complex logarithm, the log base i is not uniquely defined. The cosine of i is a real number: cos i = cosh 1 = e + 1 / e 2 = e 2 + 1 2 e ≈ 1.54308064 … displaystyle cos i=cosh 1= frac e+1/e 2 = frac e^ 2 +1 2e approx 1.54308064ldots And the sine of i is purely imaginary: sin i = i sinh 1 = e − 1 / e 2 i = e 2 − 1 2 e i ≈ ( 1.17520119 … ) i . displaystyle sin i=isinh 1= frac e-1/e 2 i= frac e^ 2 -1 2e iapprox (1.17520119ldots )i. Alternative notations[edit] 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, traditionally denoted by i(t) or just i.[5] The
Python programming language also uses j to mark the imaginary part of
a complex number.
See also[edit] Multiplicity (mathematics) Root of unity Unit complex number Notes[edit] ^ To find such a number, one can solve the equation (x + iy)2 = i where x and y are real parameters to be determined, or equivalently x2 + 2ixy − y2 = i. Because the real and imaginary parts are always separate, we regroup the terms: x2 − y2 + 2ixy = 0 + i and by equating coefficients, real part and real coefficient of imaginary part separately, we get a system of two equations: x2 − y2 = 0 2xy = 1. Substituting y = 1/2x into the first equation, we get x2 − 1/4x2 = 0 x2 = 1/4x2 4x4 = 1 Because x is a real number, this equation has two real solutions for x: x = 1/√2 and x = −1/√2. Substituting either of these results into the equation 2xy = 1 in turn, we will get the corresponding result for y. Thus, the square roots of i are the numbers 1/√2 + i/√2 and −1/√2 − i/√2. (University of Toronto Mathematics Network: What is the square root of i? URL retrieved March 26, 2007.) References[edit] ^ Doxiadēs, Apostolos K.; Mazur, Barry (2012). Circles Disturbed: The
Interplay of
Further reading[edit] Nahin, Paul J. (1998). An Imaginary Tale: The Story of √−1. Chichester: Princeton University Press. ISBN 0-691-02795-1. External links[edit] Euler's work on Imaginary Roots of Polynomi |