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Text of the page (random words):
is the imaginary unit or unit imaginary number i is a solution to the quadratic equation x 2 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 3 i imaginary numbers are an important mathematical concept they extend the real number system r displaystyle mathbb r to the complex number system c displaystyle mathbb c 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 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 in which use of the letter i is ambiguous or problematic the letter j or the greek ι is sometimes used instead a for example in 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 vs i 2 1 matrices 3 proper use 4 properties 4 1 square roots 4 2 cube roots 4 3 multiplication and division 4 4 powers 4 4 1 i raised to the power of i 4 5 factorial 4 6 other operations 5 history 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 bold blue area i 3 i i 2 1 i 1 i i 0 1 i 1 i i 2 1 i 3 i i 4 1 i 5 i i 6 1 repeats the pattern from bold 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 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 i 2 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 or equivalently i 4 i 2 i 2 1 1 1 displaystyle i 4 i 2 i 2 1 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 1 i with a zero real component and a unit imaginary component in polar form i is represented as 1 e iπ 2 or just e iπ 2 with an absolute value or magnitude of 1 and an argument or angle of π 2 in the complex plane also known as the argand plane which is a special interpretation of a cartesian plane i is the point located one unit from the origin along the imaginary axis which is orthogonal to the real axis i vs i edit being a quadratic polynomial with no multiple root the defining equation x 2 1 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 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 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 i 3 after all although i and i are not quantitatively equivalent they are negatives of each other there is no algebraic difference between i and i 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 i replacing every occurrence of i and therefore every occurrence of i replaced by i i all facts and theorems would remain valid the distinction between the two roots x of x 2 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 4 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 to isomorphism it is not unique up to a unique isomorphism indeed there are two field automorphisms of c which keep each real number fixed namely the identity and complex conjugation for more on this general phenomenon see galois group matrices edit x y is confined by hyperbola xy 1 for an imaginary unit matrix 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 0 end pmatrix and x 0 1 1 0 displaystyle x begin pmatrix 0 1 1 0 end pmatrix would be 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 1 end pmatrix begin pmatrix 1 0 0 1 end 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 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 z x y z 2 1 0 0 1 displaystyle begin pmatrix z x y z end pmatrix 2 begin pmatrix 1 0 0 1 end pmatrix here z 2 xy 1 so the product xy is negative because xy 1 z 2 thus the point x y lies in quadrant ii or iv furthermore z 2 1 x y 0 x y 1 displaystyle z 2 1 xy geq 0 implies xy leq 1 so x y is bounded by the hyperbola xy 1 proper use edit the imaginary unit is sometimes written 1 displaystyle sqrt 1 in advanced mathematics contexts 3 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 5 1 i i 1 1 1 1 1 1 incorrect displaystyle 1 i cdot i sqrt 1 cdot sqrt 1 sqrt 1 cdot 1 sqrt 1 1 qquad text incorrect similarly 1 i 1 1 1 1 1 1 1 i incorrect displaystyle frac 1 i frac sqrt 1 sqrt 1 sqrt frac 1 1 sqrt frac 1 1 sqrt 1 i qquad text incorrect the calculation rules a b a b displaystyle sqrt a cdot sqrt b sqrt a cdot b and a b a b displaystyle frac sqrt a sqrt b sqrt frac a b are only valid for real positive values of a and b 6 7 8 these problems can be avoided by writing and manipulating expressions like i 7 displaystyle i sqrt 7 rather than 7 displaystyle sqrt 7 for a more thorough discussion see 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 just like all nonzero complex numbers i has two square roots they are b 2 2 2 2 i 2 2 1 i displaystyle pm left frac sqrt 2 2 frac sqrt 2 2 i right pm frac sqrt 2 2 1 i indeed squaring both expressions yields 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 we get 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 and 3 2 i 2 displaystyle frac sqrt 3 2 frac i 2 similar to all the roots of 1 all the roots of i are the vertices of regular polygons which are inscribed within the unit circle in the complex plane 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 n pi 2 i sin n pi 2 i raised to the power of i edit making use of euler s formula i i 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 2k pi right i e i 2 pi 2 2k pi e pi 2 2k pi where k z the set of integers the principal value for k 0 is e π 2 or approximately 0 207879576 10 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 frac pi sinh pi 11 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 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 n ln x i sin n ln x the ni th 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 i sin 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 2 ln x i pi 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 54308064 ldots 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 i sinh 1 frac e 1 e 2 i frac e 2 1 2e i approx 1 17520119 ldots i history edit further information complex number history see also edit euler s identity mathematical constant multiplicity mathematics root of unity unit complex number notes edit 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 i t or just i 1 the python programming language also uses j to mark the imaginary part of a complex number matlab associates both i and j with the imaginary unit although the input 1 i or 1 j is preferable for speed and more robust expression parsing 2 in the quaternions each of i j and k is a distinct imaginary unit in bivectors and biquaternions an additional imaginary unit h or ℓ is used 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 x 2 2 ixy y 2 i because the real and imaginary parts are always separate we regroup the terms x 2 y 2 2 ixy 0 i and by equating coefficients real part and real coefficient of imaginary part separately we get a system of two equations x 2 y 2 0 2 xy 1 substituting y ½ x into the first equation we get x 2 ¼ x 2 0 x 2 ¼ x 2 4 x 4 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 2 xy 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 9 references edit boas mary l 2006 mathematical methods in the physical sciences 3rd ed new york u a wiley p 49 isbn 0 471 19826 9 matlab product documentation a b weisstein 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 wells david 1997 1986 the penguin dictionary of curious and interesting numbers revised ed uk penguin books p 26 isbn 0 14 026149 4 abs i wolfram alpha further reading edit nahin paul j 1998 an imaginary tale the story of i the square root of minus one chichester princeton university press isbn 0 691 02795 1 via archive org external links edit euler leonhard imaginary roots of polynomials at convergence mathdl maa org mathematical association of america archived from the original on 13 july 2007 retrieved from https en wikipedia org w index php title imaginary_unit oldid 1099992485 categories complex numbers algebraic numbers quadratic irrational numbers mathematical constants hidden categories all articles with specifically marked weasel worded phrases articles with specifically marked weasel worded phrases from february 2020 articles with short desc...