Exponentials and Logarithms Properties in an Extended Complex Number Field

It is well established the complex exponential and logarithm are multivalued functions, both failing to maintain most identities originally valid over the positive integers domain. Moreover the general case of complex logarithm, with a complex base, is hardly mentionned in mathematic litterature. We study the exponentiation and logarithm as binary operations where all operands are complex. In a redeﬁned complex number system using an extension of the C ﬁeld, hereafter named E , we prove both operations always produce single value results and maintain the validity of identities such as log u ( wv ) = log u ( w ) + log u ( v ) where u, v, w ∈ E . There is a cost as some algebraic properties of the addition and subtraction will be diminished, though remaining valid to a certain extent. In order to handle formulas in a C and E dual number system, we introduce the notion of set precision and set truncation. We show the complex numbers as deﬁned in C are insuﬃciently precise to grasp all subtleties of some complex operations, resulting in multivaluation, identity failures and, in speciﬁc cases, wrong results. A geometric representation of the new complex number system is proposed, in which the complex plane appears as an orthogonal projection, and where the complex logarithm an exponentiation can be simply represented. Finally we attempt an algebraic formalization of E .


Introduction
In 1749 L. Euler [1] solved a decades old controversy between G.W. Leibniz and J. Bernoulli over the appropriate definition for logarithms of negative and imaginary values, by producing the formula ln(z) = ln(a + bi) = ln |z| + arg(z)i = ln |z| + θi + 2kπi, where |z| = √ a 2 + b 2 , θ the principal value of arg(z), k ∈ Z.
The formula for complex exponentiation z w = (a + bi) m+ni = x + yi, where both z, w ∈ C, was also given the same year by L.Euler in another study [2].
The first complex logarithm formula log z w = x + yi, where both z, w ∈ C, was given by M. Ohm in 1829 [3].
Both general complex exponentiation and logarithm formulas are nevertheless used by complex number calculators, though usually only the principal value at k = k z = k w = 0 is returned. The multivaluation of formulas 1.1 and 1.4 can be attributed to the multivalued complex logarithm function z → ln(z), each k integer corresponding to a branch of the logarithm.
In the same volume M. Ohm [3] studies the validity of the exponential and logarithm identities in C. He concludes the set of values on both sides of the identity equation can differ. As an example the left side of (z w ) v = z wv will produce many more results than the right side, since exponentiation is performed twice. He differentiates "complete" identities producing the same set of results on both sides of the equation, versus "incomplete" identities in which the results differ.
The formulas of Euler and Ohm show that all results of exponentiation and logarithm can be expressed in the form x + yi. Thus both operations are algebraically closed in C and can be defined either as multivalued functions or, when considering a particular branch, as ordinary functions f : C × C → C. However the closure has come at a cost, firstly most identities equations valid in R + can fail in C, secondly the multivaluation forces an arbitrary selection of a branch prior to any result evaluation. Furthermore one could consider the uselessness and geometric meaningless of the general complex logarithm as an abnormality.
In tables 1 and 2 we summarize the validity in R + and C of the exponentiation and logarithm main properties and identities. Logarithm inverse of exponentiation log z 1 (z 1 z 2 ) = z 2 yes yes, subset 1 1 The left side of the equation produces many more results, of which only a subset is equal to the right side. The equation always holds at principal value yes yes, if z 3 ∈ R + In this article we study the complex exponentiation and logarithm as binary operations, only the general case where all operands are complex is considered. The aim is to propose a redefinition of the complex number set in which the issues described above resolve. The idea is to introduce a new form of complex number, derived from the exponential form z = |z|e arg(z)i = |z|e θi+2kπi , that extends the possibilities of the algebraic form z = x + yi. This new form is hereafter named the complete form. It will become clear the complete form is necessary to grasp all the subtleties of the exponentiation and logarithm operations, and that a strict equality cannot be maintained between the complete form and algebraic form. The properties of the basic operations (+, −, ×, ÷) will be impacted by the redefinition, though most properties such as commutativity and associativity remain valid.
The sections 2 and 3 are dedicated to the definition of the set of complex numbers in complete form, hereafter named E, the equivalences between C and E, and to the definition of complex operations (+, −, ×, ÷, exp, log) in E. The exponentiation is no longer defined by the logarithm, instead the complex logarithm formula can be deduced from the exponentiation. Moreover all operations produce a single value result. In order to handle formulas in a C and E dual number system, we introduce here the notion of set precision and set truncation.
The section 4 includes all proofs and some examples over the validity of the exponential and logarithm identities in E. All the trivial identity failure cases given above resolve.
In the section 5 we show how to obtain explicit formulas linking the real and imaginary parts of some transcendental equations solutions.
The section 6 proposes a geometric representation of E, of which the complex plane appears as an orthogonal projection. The complex exponentiation z = z 1 z 2 and logarithm z = log z 1 (z 2 ), where z, z 1 , z 2 ∈ E, can be simply represented as a mapping of the two operands elements to the unique result element.
The section 7 lists all algebraic properties of E and compares them with the properties of the R and C fields.
In section 8 we argue why the exponentiation and logarithm multivalued results and identity failures in C are not induced by the operations, but are induced by an intrinsic limitation of the complex numbers algebraic form z = x + yi. The number set is hereafter named E. The real part (or real value) is defined as e a and the imaginary part e bi , where a is the real argument and b the imaginary argument. The element 0 is included for compatibility with C and R.

Remark. Equivalence with the exponential form
The exponential form of complex numbers z = x + yi = |z|e arg(z)i = |z|e θi+2kπi has a similar but not identical definition. It remains explicitly linked to the algebraic form and must have a principal value θ of the argument arg(z) within the interval ] − π; π]. The purpose of the integer k is precisely to link all values of the exponential form to their unique corresponding algebraic form. Geometrically, the 2π periodicity of the imaginary argument is purposely maintaining the correlation with the complex plane.
In the complete form, the explicit link to the algebraic form and the constraint on the argument principal value are abolished. For example in E the numbers e 0 e 2πi and e 0 e 4πi are not equal, each having distinct properties as it will be demonstrated in further sections. Within C the symbolic and geometric representation of both numbers are equally represented by 1 and by the coordinates (x, y) = (1, 0) on the complex plane.
Replacing |z| by e a allows the establishment of more elegant and symmetrical formulas. We use the new denomination complete form to avoid any ambiguity.

Definition 2. Equivalence between C and E sets
Let the set E of complex numbers in complete form e a e bi be partitionned into C and E\C by restricting C to a 2π interval of the imaginary argument b, by convention the interval b ∈ ] − π; π]. Each number x + yi ∈ C converted into its unique corresponding complete form e a e bi forms then a distinct equivalence class together with numbers in the form e a e (b+2kπ)i ∈ E with k ∈ Z * .
The definition is equivalent as restricting C to the principal value of the exponential form of complex numbers. Even with this restriction, the algebraic definition of C and the complex plane definition are not altered.

Definition 3. Set precision and truncation
Let A be a set partitionned by an equivalence relation into two subsets A 1 and A 2 , and let each element a 1 ∈ A 1 form a distinct equivalence class with an arbitrary number of elements a 2 ∈ A 2 such as each element a is part of a unique given class. In such a set configuration, elements a 2 are defined as A precise, elements a 1 are defined as A 1 precise. Each element a 2 ∈ A 2 can be truncated to its unique corresponding a 1 ∈ A 1 element, thus at a lower precision level. The truncation is noted a 1 = | a 2 | A 1 .

Example 1. Z and N precision
Let the integer set Z be partitionned into N and Z <0 , an integer is Z precise if negative, and is N precise if positive or zero. The abs function is the truncation function from Z to N precision level.

Example 2. E and C precision
The Euler formula e bi = cos b + sin b i is de facto the truncation function from E to C precision. The truncation can be noted |z| C = |e a e bi | C = e a cos b + e a sin b i = e a e |b|ci , with the imaginary argument truncated such as : Equalities such as 1 = e 4πi or 1 = e 2kπi no longer hold whenever E precision is required, the notation |e 2kπi | C = e 0i = 1 or |e (2k+1)πi | C = e πi = −1 can be used to clearly indicate the truncation. Whenever the imaginary argument is inside the interval b ∈ ] − π; π], the complete or algebraic form can be used indifferently.

Remark.
The E set of complex numbers can be viewed as a "natural" extension of C. Within the set sequence N ⊂ Z ⊂ R ⊂ C ⊂ E each element in a given set is uniquely linked to a predecessor set element through an equivalence relation, therefore an element can always be truncated to the predecessor set precision level.
The Euler formula used for the conversion is not to be considered as an equality. From a E perspective an irreversible loss of information is induced when converting from complete to algebraic form if the imaginary argument is outside the interval ] − π; π].

Lemma 2. Converting from algebraic form to complete form
Using the definition of complex number modulus and argument. By definition z = 0 is equivalent in E and C.
z = x + yi = |z|e Arg(z)i = e ln |z| e θzi =⇒ e 1 2 ln (x 2 +y 2 ) e Atan ( y x )i = e a e bi (2.4) Remark. Usage of ln, Arg and Atan functions The natural logarithm function is applied to the domain R >0 , hence is single valued. In the formula 2.4 only the principal value of the arg function is considered to remain consistent with definition 2. The limits of the traditional arctan function, with the result in the interval ] − π 2 ; π 2 ], requires the use of the atan2 function with 2 arguments whose result is included in the interval ] − π; π] without singularities. In this study the notation Atan y x always refers to the atan2 function where both arguments remain as the fraction numerator and denominator. This notation adjustment will ease the readibility and handling of formulas, as obtained formulas always produce a fraction inside the Atan argument. The fraction can be simplified providing the numerator and denominator signum are preserved.

Binary operations in complete form
z z2 1 = e (e a 2 (a1 cos b2−b1 sin b2)) e (e a 2 (b1 cos b2+a1 sin b2))i Formulas are easier to handle when split between real and imaginary parts, in this study we mostly use the split notation. Let z = e a e bi :

Proof. Multiplication formula
Using the identity e w 1 · e w 2 = e w 1 +w 2 where w 1 ,

Proof. Division formula
Using the identity e w 1 / e w 2 = e w 1 −w 2 where w 1 , w 2 ∈ C [6]

Proof. Exponentiation formula
The formula u w = e w ln u with w, u ∈ C defines the complex exponentiation in C, the formula is necessary given the base cannot be exploited directly in algebraic form. The formula is equivalent as converting the base into an infinity of bases in the form u = e ln |u|+θi+2kπi . The exponent is then applied to the bases such as u w = (e ln |u|+θi+2kπi ) w = e w ln |u|+w(θ+2kπ)i . The result is then reconverted into algebraic form. When calculated separately for each integer k, the exponentiation can be defined as (e a e bi ) w = e aw e bwi with a single valued result, the base and result being in complete form and the exponent in algebraic form. Let z 1 = e a 1 e b 1 i and, using the conversion formula 2.1, let

Proof. Logarithm formula
The logarithm formula can be directly reversed from the exponentiation formula 3.3. Counter to the definition of the complex logarithm in C, both operands are here in E thus can be exploited directly in the formula without requiring any conversion. Let z 1 = e a 1 e b 1 i and z 2 = e a 2 e b 2 i .
Proof. Alternate proof of logarithm formula The result is in algebraic form and needs to be converted into complete form using conversion formula 2.4.

Proof. Addition and subtraction formulas
Both operands need to be converted into algebraic form using the formula 2.1, since no identity can be used directly in complete form. Let z 1 = e a 1 cos b 1 + e a 1 sin b 1 i and The result is in algebraic form and needs to be converted into complete form using conversion formula 2.4.
Theorem 1. Within a number system composed of the sets C ⊂ E, E precision is the highest possible precision level obtained as result of a multiplication, division or exponentiation operation From formulas 3.1, 3.2 and 3.3 we can easily deduce the result of the imaginary argument is not bounded by any limit and will be situated anywhere in b ∈ R.

Remark.
The operations can be defined as functions f : E × E → E, giving exactly four single variable continuous functions : z → w · z ; z → w/z ; power function z → z w ; exponential The complex exponentiation operation is more subtle since the exponent gets truncated to C precision by the cosine and sine functions used in the formula 3.3. On the other hand, the base and result require E precision.
Multiplication and division operands and results are at maximum E precise, no truncation is performed by the formulas 3.1 and 3.2. One can notice even with C precise operands, the result may be E precise.
Theorem 2. Within a number system composed of the sets C ⊂ E, C precision is the highest possible precision level obtained as result of a logarithm, addition or subtraction operation The formulas 3.4, 3.5 and 3.6 use the Atan function in the imaginary part, thus the result will always be situated inside the interval b ∈ ] − π; π], which is exactly the definition of the C precision. The domain of the corresponding functions is therefore f : E × E → C.

Remark.
Exactly four single variable continuous functions can be obtained: The singularities induced by the values 0 and e 0 e 0i = 1 are studied in a further section.
The complex logarithm operation requires mixed precision, both operands require the complete form which can therefore be at maximum E precise, but the result is always at maximum C precise.
The addition and subtraction are the only operations not requiring the complete form hence no E precision, operands exceeding the required precision are truncated to C precision by formulas 3.5 and 3.6.
Theorem 3. All binary complex operations defined in E are monovalued From the formulas 3.1 to 3.6, we can deduce that both the real and imaginary part will always give a single valued results, since no real multivalued function is used in the formulas.

Remark.
The Atan function as defined in this study is monovalued. An alternate definition with a multivalued result of periodicity 2π is possible and would imply the logarithm, addition and subtraction are multivalued in E. Though a matter of definition, the single valuation arctangent is far more consistent algebraically and also geometrically as it will be seen in further sections. The logarithm, addition and subtraction results are intrinsically limited to C precision, in the same way a function defined as f : Z × Z → N returns one positive integer, not all integers belonging to the same equivalence class.

Exponentials and logarithms identities in E
The result is strictly identical on both sides of the identity when z 1 , Theorem 5. The product and quotient logarithm identities valid in R * + are valid in E * The result is strictly identical on both sides of the identity when z 1 , z 2 , z 3 ∈ E * and z 1 = e 0 e 0i Theorem 6. The power and base substitution logarithm identities valid in R * + are valid in E * only at C precision level The result truncated to C precision is strictly identical on both sides of the identity when z 1 , z 2 , z 3 , z 4 ∈ E * and z 1 , z 4 = e 0 e 0i . The final operations on each side of the identity return different levels of precision, the identity cannot be a strict equality.
As demonstrated within the following proofs, the trivial cases of exponential and logarithm identity failures given in the introduction dissapear when both sides of the identity equation are calculated in E, thus when the formulas 3.1 to 3.6 are used at every calculation step.
Combining the multiplication and exponentiation formulas 3.1 and 3.3, let When the first expression is evaluated in algebraic form in C, the primary result is 1, the reason of the dissimilarity is because the result of the multiplication −1 · −1 was implicitly truncated to a C precision level. In E equating −1·−1 = 1 is an over simplification : e πi e πi = e 2πi = e 0i , though in algebraic form the 2 values are indistinctive. This imprecision, invisible at first glance, is revealed when the exponent ½ is applied on e 2πi or e 0i giving different values, respectively -1 and 1. Similarly, −i · −i = e − π 2 i e − π 2 i = e −πi = e πi and −1 · i = e πi e π 2 i = e 3π 2 i = e − π 2 i . On the other hand, i · i = −1 and i · −i = 1 are always valid.
is valid for all z 1 , z 2 , z 3 ∈ E * Combining the division and exponentiation formulas 3.2 and 3.3 Proof.
In the first expression the exponentiation base is taken as multivalued e 1 e 2πki , the exponent in algebraic form 1 + 2πki is also multivalued, with both k synchronised. Nothing wrong here. The result of the exponentiation will obviously be multivalued, the first formula given is correct assuming computation is done in E. In the second expression no exponentiation is performed, instead a double truncation from E to C precision. Equating e 1+2πki = e 1 e 2πki = e · 1 = e is imprecise, |e 1+2πki | C = e is correct. After the truncation only the value within the interval b ∈ ] − π; π] remains thus when k = 0.
The identity is similar to the identity e w 1 e w 2 = e w 1 +w 2 , with w 1 , w 2 ∈ C z z 1 +z 2 = z z 1 z z 2 (e a e bi ) z 1 +z 2 = (e a e bi ) z 1 (e a e bi ) z 2 (e a+bi ) z 1 +z 2 = (e a+bi ) z 1 (e a+bi ) z 2 e (a+bi)(z 1 +z 2 ) = e (a+bi)z 1 e (a+bi)z 2 (all exponents can be reduced into the form z = x + yi) The identity can be verified using the multiplication 3.
The identity is similar to the identity e w 1 /e w 2 = e w 1 −w 2 , with w 1 , w 2 ∈ C Combining the multiplication and logarithm formulas 3.1 and 3.4 Combining the exponentiation and logarithm formulas 3.3 and 3.4 Combining the logarithm and division formulas 3.4 and 3.2 Example 10. Identity failure at E precision level Proof. log z 1 (z 2 z 3 ) = log z 1 z 2 + log z 1 z 3 is valid for all z 1 , z 2 , z 3 ∈ E * Combining the multiplication and logarithm formulas 3.1 and 3.4 For simplicity, the algebraic form is used in the following equation, since neither the logarithm nor the addition require the complete form for the result representation The result in algebraic form needs to be converted into complete form using conversion formula 2.4 Example 11. ln(−1 · −1) = ln(−1) + ln(−1) ln(−1 · −1) = ln (e πi e πi ) = ln (e 2πi ) = 2πi ln(−1) + ln(−1) = πi + πi = 2πi Proof. log z 1 (z 2 /z 3 ) = log z 1 z 2 − log z 1 z 3 is valid for all z 1 , z 2 , z 3 ∈ E * Combining the division and logarithm formulas 3.2 and 3.4

Formulas for transcendental equations
The formulas 3.1 to 3.6 can be combined to obtain formulas linking the real and imaginary arguments of expressions using the complex operations.
Example 12. z 2 = z 1 w · w α where w, z 1 , z 2 ∈ E * , z 1 = e 0 e 0i , α ∈ R Explicit formulas linking the real and imaginary arguments a w , b w of w can be obtained.
From which the final formulas are obtained : Example 13. z 2 = log z 1 (w) · w α where w, z 1 , z 2 ∈ E * , w, z 1 = e 0 e 0i , α ∈ R Explicit formulas linking the real and imaginary arguments a w , b w of w can be obtained.
From which the final formulas are obtained : 6 Geometric representation of E

The complex helicoid Definition 5. Geometric representation of E : the complex helicoid
The complex plane is clearly insufficient to represent E precise numbers, one can notice only e a e bi with b ∈]−π; π] can be positioned in a unique way. The lack of "space" is solved by an additional axis, hereafter named the i axis, on which the imaginary argument b can translate rectilinearly without any boundaries. The rotation of the imaginary argument b is maintained with a 2π period, giving a unique perpendicular half straight line for each b argument on which the real part e a is positioned. Hereafter those half-lines are named "rays". Viewed in a three dimension euclidian space, with the origin situated at 0 on the i axis, every number w = e a e bi can be given a unique orthogonal coordinate (x, y, z) = (e a cos b, e a sin b, b). Thus the set E forms exactly an helicoid surface, hereafter named the complex helicoid. The i axis is a singularity itself, on which only the value 0 can be positioned, the value 0 was included into the E set only for algebraic purpose. The representation is similar as the Riemann surface of the complex logarithm function in C, but with a different meaning and purpose. From a E perspective the complex helicoid is the counterpart of the complex plane for C and the real axis for R, on which all numbers e a e bi are connected without any discontinuity. In this sense the complex helicoid is not to be considered as a layering of n complex plane sheets. In a similar way the complex plane is not usually considered as the gluing of n real axis.

Constant functions representation on the complex helicoid
The constant function w = e a is the set of points situated at the position e a on each ray. The function appears as an infinite helix surrounding the i axis. The multiplication and division operations such as w = e a ± a translate the position of the point on each ray, thus bring closer or further the helix to the i axis.
The constant function w = e bi is the set of points on a ray pointing in the direction given by b, excluding the 0 situated on the i axis. The multiplication and division operations such as w = e (b ± b )i operate a rotation and a translation around and along the i axis. The constant functions underline the geometrical difference between real and imaginary values translations in E. When tied to a fixed imaginary part, the real part e a translates rectilinearly on a half-straight line. With a fixed real part, the imaginary part e bi spirals on an unbounded helix.

Representation of the complex logarithm operation
Let z 1 , z 2 ∈ E * with z 1 , z 2 = e 0 e 0i be 2 points on the complex helicoid. The representation of the point z = log z 1 (z 2 ) reveals, under a new perspective, a similar formula as the division on the complex plane.

Representation of the complex exponentiation operation
Let z 1 , z 2 ∈ E * with z 1 = e 0 e 0i be 2 points on the complex helicoid. The representation of the point z = z 1 z 2 is best visualised by 2 formulas. The exponent z 2 only being used at C precision, quite obviously the full b 2 distance on the i axis is not used in the formulas.

Complex helicoid projections on the plane
The orthogonal projection of the complex helicoid (x, y, z) to (x, y, 0) represents the complex plane, through a new perspective. The projection corresponds exactly to a C truncation of E and can be noted as P (w) = P (e a e bi ) = P (e a cos b, e a sin b, b) = (e a cos b, e a sin b, 0) or as a truncation |w| C = |e a e bi | C = e a cos b + e a sin b i. The singularity 0 is given the appearance of a normal point. The exponentials and logarithms identity failures in C represented on the complex plane are all due to a "careless" crossing of the Re-axis generating a C truncation. The projection should not be confused with the logarithmic representation of E which will be seen further, though both representations are graphicaly identical. Similarly, the orthogonal projections of the complex helicoid (x, y, z) to (x, 0, z) and (x, y, z) to (0, y, z), maps the constant helix into a cosine and sine curve.

Logarithmic representation
The complex numbers in complete form are identified in a unique way by their real and imaginary arguments. Positioning the arguments coordinates on a Wessel-Argand-Gauss diagram is therefore a logarithmic representation of E. One can notice (a; bi) and (a + bi) are equivalent notations for the coordinates, both are derived from the complete form e a e bi or e a+bi . Expressions at the exponent level only require C precision, thus all operations as defined in C can be used in an exponent. For example −1 · i = −i or (−2) 2 = 4, which both implicitly perfom a C truncation, can be used, the loss of precision will be without consequence. The number 0 is used in expressions as a normal number.

Representation of the addition and subtraction operations
The addition and subtraction do not require any E precision, representing them on the complex helicoid is basically useless, a projection on the complex plane is sufficient. Let z 1 , z 2 ∈ E * by 2 points on the complex helicoid, with their corresponding projections |z 1 | C = x 1 + y 1 i and |z 2 | C = x 2 + y 2 i on the complex plane.
x 1 = e a 1 cos b 1 y 1 = e a 1 sin b 1 x 2 = e a 2 cos b 2 y 2 = e a 2 sin b 2 z = z 1 ± z 2 = e  In the first line the E precision is preserved because the final operation is a multiplication, in the second line the addition operates a C truncation, hence the results can be different.

Identity element
The identity element of addition and multiplication : z 1 × e 0 e 0i = z 1 (7.9) z 1 + 0 = z 1 (7.10) The right identity element of division and subtraction, exponentiation having an infinite set of right identities :

Inverse
Multiplication, division and exponentiation are the exact reciprocal of their inverse operation : Logarithm, addition and subtraction are only the C precise reciprocal of their inverse operation: Proof. Exponentiation is the exact inverse of logarithm Using the logarithm formula 3.4 converted into algebraic form, let z 1 = e a 1 e b 1 i , z 2 = e a 2 e b 2 i and log z 1 (z 2 ) = a 1 a 2 +b 1 b 2

Symmetry
e a e bi · e −a e −bi = e 0 e 0i (7.20) e a e bi e a e bi = e 0 e 0i (7.21) e a e bi + e a e bi+(2k+1)πi = 0 (with k ∈ Z) e a e bi − e a e bi+2kπi = 0 (7.23)

Singularities
At first we consider the singularities of operations where both operands are in E\{0}. log (e 0 e 0i ) (z 2 ) = ∞ (7.24) log z 1 (e 0 e 0i ) = 0 (7.25) log (e 0 e 0i ) (e 0 e 0i ) = undefined (7.26) From the formulas 3.5 and 3.6, it is possible to deduce both addition and subtraction have singularities caused by the ln with operand 0 and the Atan with 0/0 argument.
e a e bi + e a e bi+(2k+1)πi = 0 (with k ∈ Z) (7.27) e a e bi − e a e bi+2kπi = 0 (7.28) The introduction of the element 0 allows to reduce some of the above singularities, but also adds new ones.

Algebraic structure of E
Conclusions can be made from formulas 3.1 to 3.6 and from the properties listed above : For each of the 6 complex operations, E has a closed algebraic structure, except for the singularities all results can be represented The multiplication and division maintain all their intrinsic properties such as in C The addition and subtraction maintain all their intrinsic properties but only at C precision, since both operations do not require nor can provide any E precision The distributivity property generally only holds when the left side is truncated to C precision, thus distributivity is only C precise The multiplication is clearly the defining operation and possess all the properties to constitute a multiplicative group (E * , ·) The field axioms are not all verified, since the addition/subtraction reciprocity and the distributivity do not hold exactly in E It would be a mistake to limit E to a multiplicative group, as many properties over the exponentiation and logarithm operations are added. All properties and identities hold to a certain extent, only limited by the operations maximum precision level. E is more to be considered as a complete number system.
7.9 Properties comparison between R, C and E   and cosine functions convert the result into algebraic form. However, during the conversion, precision is lost and the result principal value may be shifted. For example (−1) 3 = (e ln(−1) ) 3 = (e πi+2kπi ) 3 = e 3πi+6kπi . Converting those values into algebraic form returns −1 = e πi+2kπi , thus the principal value is reset to e πi . Moreover there is no possibility to convert expressions such as e 6kπi without loss of information. In the example (i −5 ) i , when the result of i −5 is reconverted into algebraic form the principal value result is shifted from e − 5πi 2 to e − πi 2 , the result of (i −5 ) i becoming e π 2 instead of e 5π 2 .
The multivaluation of the complex exponentiation is not induced by the logarithm, but by the algebraic form of the base. Since no identity is available to exploit the base as such, the formula 1.1 is equivalent as substituting the base by an infinity of bases, the so-called exponential form, using the formula z = |z|e arg(z)i = |z|e θi+2kπi . In general the multivaluation is assumed, unless explicitly restricting to real positives with notations such as |z| α or √ a 2 + b 2 which both assume a single valued real positive base.
The complex logarithm as defined by L. Euler [1] is restricted to the base e or at least to real positive values. Euler himself does not mention a multivalued logarithm function, rather he speaks of each real or complex number having an infinite number of logarithms. Indeed, as for the exponentiation base, the logarithm operand cannot be exploited directly in algebraic form, thus has to be converted into exponential form, ln(z) = ln(|z|e arg(z)i ) = ln(|z|e θi+2kπi ) = ln |z| + θi + 2kπi. The primary result being in algebraic form, no conversion is required nor any loss of precision is induced. The multivaluation is solely induced by the operand substitution, for example ln(1) = 2kπi and ln(−1) = πi + 2kπi. On the other hand ln |z| is assumed single valued as the operand is implicitly substituted by xe 0i .
Notations such as ln, log 2 or log 10 assume the logarithm base is in the form xe 0i . For bases the same logic applies as for the exponentiation, a base in algebraic form can be substituted by the equivalent exponential form, or by any particular value in complete form. As an example, for log −1 the base can be assumed as monovalued e πi or multivalued e πi+2kπi .
It is clear there is only one unique exponentiation and one unique logarithm complex operation. The different notations conventions and different assumptions regarding the operands substitutions are creating some confusion, which can be blamed on the lack of precision of the algebraic form. In complete form, real positive numbers are not fundamentally different, all operands are in the form e a e bi , moreover the concepts of principal value and branches are no longer necessary. Expressions such as (e πi ) 1 3 or ln (e πi ) return a single valued result in E. The same expressions in C are multivalued because e πi is converted into algebraic form, assuming e πi = −1 = e πi+2kπi In general, when dealing with exponentiation and logarithm in C the equality e αi = e αi+2kπi is automatically assumed, by an analogy with the trigonometric circle where an angle of α is equal to α + 2kπ. It turns out this assumption is responsible for the exponentiation and logarithm multivaluation. From a E perspective only |e αi | C = |e αi+2kπi | C is valid. The formulas cos(α) + i sin(α) = cos(α + 2kπ) + i sin(α + 2kπ) = ∞ n=0 (αi) n n! = ∞ n=0 (αi+2kπi) n n! are strictly equal, but are also equal in deconstructing the complete form and reconstructing a result in algebraic form, as such they literally truncate the precision of the complete form.

Conclusion
As demonstrated in this article, the complex exponentiation base and result, the complex logarithm base and operand cannot be represented precisely in algebraic form. The same observation holds for the multiplication and division results when used in combination with an exponentiation or logarithm. For this reason alone, multivalued results, identity failures and even wrong results are obtained when computing exclusively in C.
The establishment of the complete form is an attempt to restore the properties of exponentiation and logarithm, and to ease the conceptualization and handling of both operations when all operands are complex. Moreover the E set of complex numbers in complete form can be viewed as a "natural" extension of C. Within the sequence N ⊂ Z ⊂ R ⊂ C ⊂ E each set extends the capacity of the predecessor set by providing new elements, thus new symbolic representations of numbers. Each element in a given set is uniquely linked to a predecessor set element through an equivalence relation, therefore an element can always be truncated to the predecessor set precision level. Similarly the geometric representations are extended while preserving the predecessor sets representations.
Labelling expressions such as e a e bi as numbers might seem strange, though we believe it is totally justified by the extra precision and possibilities they introduce, as they overcome some limitations encountered in C with the algebraic form. As we have frequently illustrated with examples, it remains possible to combine the algebraic and complete form inside expressions and formulas. Within the ] − π; π] boundary of the imaginary argument both forms can be used indifferently, but outside that interval, the complete form and formulas 3.1 to 3.6 should be used in complex number calculators.
We do not consider the C precision limitation on some complex operations properties and identities as an insurmountable issue. Expressions involving additions and subtractions, such as polynomials, do not require E precision, the algebraic form and the basic operations (+, −, ×, ÷) as defined in C are always sufficient. The required precision is more to be considered as a matter of choice depending on the context where the complex operations are used.

Daniel Tischhauser
Independent researcher, Geneva, Switzerland Email for correspondence: dtischhauser.math@gmail.com Declarations of interest: none