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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 Algebraic closure and Fundamental theorem of algebra.) The term "imaginary" is used because there is no real number having a negative square. There are two complex square roots of −1, namely i and −i, just as there are two complex square roots of every real number other than zero, which has one double square root. In contexts where i is ambiguous or problematic, j or the Greek ι is sometimes used (see § Alternative notations). In the disciplines of electrical engineering and control systems engineering, the imaginary unit is normally denoted by j instead of i, because i is commonly used to denote electric current. For the history of the imaginary unit, see Complex number § History.

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 Multiplication
Multiplication
and division 4.4 Powers

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. Real number
Real number
operations can be extended to imaginary and complex numbers by treating i as an unknown quantity while manipulating an expression, and then using the definition to replace any occurrence of i2 with −1. Higher integral powers of i can also be replaced with −i, 1, i, or −1:

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 Complex conjugate
Complex conjugate
and Galois group. Matrices[edit] A similar issue arises if the complex numbers are interpreted as 2 × 2 real matrices (see matrix representation of complex numbers), because then both

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 Square root
Square root
and Branch point. Properties[edit] Square roots[edit]

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. Multiplication
Multiplication
and division[edit] Multiplying a complex number by i gives:

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 Riemann surface
Riemann surface
the function is defined on in practice. Listed below are results for the most commonly chosen branch. A number raised to the ni power is:

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. MATLAB
MATLAB
associates both i and j with the imaginary unit, although 1i or 1j is preferable, for speed and improved robustness.[6] Some texts use the Greek letter iota (ι) for the imaginary unit, to avoid confusion, especially with index and subscripts. Each of i, j, and k is an imaginary unit in the quaternions. In bivectors and biquaternions an additional imaginary unit h is used.

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 Mathematics
Mathematics
and Narrative (illustrated ed.). Princeton University Press. p. 225. ISBN 978-0-691-14904-2.  Extract of page 225 ^ 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.  Extract of page 12 ^ "The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, Page 26. ^ "abs(i!)", WolframAlpha. ^ Boas, Mary L. (2006). Mathematical methods in the physical sciences (3. ed.). New York [u.a.]: Wiley. p. 49. ISBN 0-471-19826-9.  ^ " MATLAB
MATLAB
Product Documentation". 

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

.