Thus $$z \bar{z}=r^{2}=(|z|)^{2}$$. 1. 3 0 obj << The field is one of the key objects you will learn about in abstract algebra. We see that multiplying the exponential in Equation \ref{15.3} by a real constant corresponds to setting the radius of the complex number by the constant. That's complex numbers -- they allow an "extra dimension" of calculation. %PDF-1.3 A third set of numbers that forms a field is the set of complex numbers. In order to propely discuss the concept of vector spaces in linear algebra, it is necessary to develop the notion of a set of “scalars” by which we allow a vector to be multiplied. Adding and subtracting complex numbers expressed in Cartesian form is quite easy: You add (subtract) the real parts and imaginary parts separately. >> The general definition of a vector space allows scalars to be elements of any fixed field F. Exercise 4. Commutativity of S under $$*$$: For every $$x,y \in S$$, $$x*y=y*x$$. The real numbers also constitute a field, as do the complex numbers. The set of complex numbers See here for a complete list of set symbols. The system of complex numbers consists of all numbers of the form a + bi But there is … \end{align}\]. z_{1} z_{2} &=\left(a_{1}+j b_{1}\right)\left(a_{2}+j b_{2}\right) \nonumber \\ An imaginary number has the form $$j b=\sqrt{-b^{2}}$$. The complex conjugate of $$z$$, written as $$z^{*}$$, has the same real part as $$z$$ but an imaginary part of the opposite sign. }+\frac{x^{2}}{2 ! Complex Numbers and the Complex Exponential 1. These two cases are the ones used most often in engineering. The first of these is easily derived from the Taylor's series for the exponential. /Length 2139 h����:�^\����ï��~�nG���᎟�xI�#�᚞�^�w�B����c��_��w�@ ?���������v���������?#WJԖ��Z�����E�5*5�q� �7�����|7����1R�O,��ӈ!���(�a2kV8�Vk��dM(C� $Q0���G%�~��'2@2�^�7���#�xHR����3�Ĉ�ӌ�Y����n�˴�@O�T��=�aD���g-�ת��3��� �eN�edME|�,i$�4}a�X���V')� c��B��H��G�� ���T�&%2�{����k���:�Ef���f��;�2��Dx�Rh�'�@�F��W^ѐؕ��3*�W����{!��!t��0O~��z��X�L.=*(������������4� Division requires mathematical manipulation. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and p-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. &=a_{1} a_{2}-b_{1} b_{2}+j\left(a_{1} b_{2}+a_{2} b_{1}\right) [ "article:topic", "license:ccby", "imaginary number", "showtoc:no", "authorname:rbaraniuk", "complex conjugate", "complex number", "complex plane", "magnitude", "angle", "euler", "polar form" ], https://eng.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Feng.libretexts.org%2FBookshelves%2FElectrical_Engineering%2FSignal_Processing_and_Modeling%2FBook%253A_Signals_and_Systems_(Baraniuk_et_al. The system of complex numbers is a field, but it is not an ordered field. A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i represents the imaginary unit, satisfying the equation i = −1. Yes, m… Note that we are, in a sense, multiplying two vectors to obtain another vector. The Field of Complex Numbers. Definitions. (Yes, I know about phase shifts and Fourier transforms, but these are 8th graders, and for comprehensive testing, they're required to know a real world application of complex numbers, but not the details of how or why. Complex numbers are all the numbers that can be written in the form abi where a and b are real numbers, and i is the square root of -1. �̖�T� �ñAc�0ʕ��2���C���L�BI�R�LP�f< � From analytic geometry, we know that locations in the plane can be expressed as the sum of vectors, with the vectors corresponding to the $$x$$ and $$y$$ directions. The real numbers, R, and the complex numbers, C, are fields which have infinite dimension as Q-vector spaces, hence, they are not number fields. \end{align} \]. Let M_m,n (R) be the set of all mxn matrices over R. We denote by M_m,n (R) by M_n (R). An imaginary number can't be numerically added to a real number; rather, this notation for a complex number represents vector addition, but it provides a convenient notation when we perform arithmetic manipulations. There are three common forms of representing a complex number z: Cartesian: z = a + bi Dividing Complex Numbers Write the division of two complex numbers as a fraction. The set of complex numbers is denoted by either of the symbols ℂ or C. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, and are fundamental in many aspects of the scientific description of the natural world. Polar form arises arises from the geometric interpretation of complex numbers. Existence of $$*$$ identity element: There is a $$e_* \in S$$ such that for every $$x \in S$$, $$e_*+x=x+e_*=x$$. (In fact, the real numbers are a subset of the complex numbers-any real number r can be written as r + 0 i, which is a complex representation.) The properties of the exponential make calculating the product and ratio of two complex numbers much simpler when the numbers are expressed in polar form. z=a+j b=r \angle \theta \\ (Note that there is no real number whose square is 1.) Both + and * are associative, which is obvious for addition. The angle velocity (ω) unit is radians per second. }-\frac{\theta^{2}}{2 ! We consider the real part as a function that works by selecting that component of a complex number not multiplied by $$j$$. $$z \bar{z}=(a+j b)(a-j b)=a^{2}+b^{2}$$. Surprisingly, the polar form of a complex number $$z$$ can be expressed mathematically as. L&�FJ����ATGyFxSx�h��,�H#I�G�c-y�ZS-z͇��ů��UrhrY�}�zlx�]�������)Z�y�����M#c�Llk The notion of the square root of $$-1$$ originated with the quadratic formula: the solution of certain quadratic equations mathematically exists only if the so-called imaginary quantity $$\sqrt{-1}$$ could be defined. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. Both + and * are commutative, i.e. Notice that if z = a + ib is a nonzero complex number, then a2 + b2 is a positive real number… A complex number is any number that includes i. That is, prove that if 2, w E C, then 2 +we C and 2.WE C. (Caution: Consider z. z. Prove the Closure property for the field of complex numbers. A complex number is any number that includes i. Complex numbers weren’t originally needed to solve quadratic equations, but higher order ones. Ampère used the symbol $$i$$ to denote current (intensité de current). Have questions or comments? Letz_1, z_2, z_3 \in \mathbb{C}$such that$z_1 = a_1 + b_1i$,$z_2 = a_2 + b_2i$, and$z_3 = a_3 + b_3i. There are other sets of numbers that form a field. If c is a positive real number, the symbol √ c will be used to denote the positive (real) square root of c. Also √ 0 = 0. Again, both the real and imaginary parts of a complex number are real-valued. The angle equals $$-\arctan \left(\frac{2}{3}\right)$$ or $$−0.588$$ radians ($$−33.7$$ degrees). Complex numbers are used insignal analysis and other fields for a convenient description for periodically varying signals. $z_{1} \pm z_{2}=\left(a_{1} \pm a_{2}\right)+j\left(b_{1} \pm b_{2}\right)$. To show this result, we use Euler's relations that express exponentials with imaginary arguments in terms of trigonometric functions. $$\operatorname{Re}(z)=\frac{z+z^{*}}{2}$$ and $$\operatorname{Im}(z)=\frac{z-z^{*}}{2 j}$$, $$z+\bar{z}=a+j b+a-j b=2 a=2 \operatorname{Re}(z)$$. Because no real number satisfies this equation, i is called an imaginary number. because $$j^2=-1$$, $$j^3=-j$$, and $$j^4=1$$. Consequently, multiplying a complex number by $$j$$. This video explores the various properties of addition and multiplication of complex numbers that allow us to call the algebraic structure (C,+,x) a field. We de–ne addition and multiplication for complex numbers in such a way that the rules of addition and multiplication are consistent with the rules for real numbers. The product of $$j$$ and a real number is an imaginary number: $$ja$$. A single complex number puts together two real quantities, making the numbers easier to work with. Complex Numbers and the Complex Exponential 1. Let us consider the order between i and 0. if i > 0 then i x i > 0, implies -1 > 0. not possible*. Thus, we would like a set with two associative, commutative operations (like standard addition and multiplication) and a notion of their inverse operations (like subtraction and division). x���r7�cw%�%>+�K\�a���r�s��H�-��r�q�> ��g�g4q9[.K�&o� H���O����:XYiD@\����ū��� The complex conjugate of the complex number z = a + ib is the complex number z = a − ib. z &=\operatorname{Re}(z)+j \operatorname{Im}(z) \nonumber \\ }+\ldots \nonumber\]. Hint: If the field of complex numbers were isomorphic to the field of real numbers, there would be no reason to define the notion of complex numbers when we already have the real numbers. Think of complex numbers as a collection of two real numbers. A field ($$S,+,*$$) is a set $$S$$ together with two binary operations $$+$$ and $$*$$ such that the following properties are satisfied. Similarly, $$z-\bar{z}=a+j b-(a-j b)=2 j b=2(j, \operatorname{Im}(z))$$, Complex numbers can also be expressed in an alternate form, polar form, which we will find quite useful. Note that $$a$$ and $$b$$ are real-valued numbers. When the scalar field is the complex numbers C, the vector space is called a complex vector space. The real numbers are isomorphic to constant polynomials, with addition and multiplication defined modulo p(X). Complex numbers are the building blocks of more intricate math, such as algebra. Here, $$a$$, the real part, is the $$x$$-coordinate and $$b$$, the imaginary part, is the $$y$$-coordinate. This post summarizes symbols used in complex number theory. Closure of S under $$*$$: For every $$x,y \in S$$, $$x*y \in S$$. z_{1} z_{2} &=r_{1} e^{j \theta_{1}} r_{2} e^{j \theta_{2}} \nonumber \\ Grouping separately the real-valued terms and the imaginary-valued ones, e^{j \theta}=1-\frac{\theta^{2}}{2 ! I want to know why these elements are complex. The real part of the complex number $$z=a+jb$$, written as $$\operatorname{Re}(z)$$, equals $$a$$. }-j \frac{\theta^{3}}{3 ! The imaginary number jb equals (0, b). \[\begin{array}{l} For the complex number a + bi, a is called the real part, and b is called the imaginary part. That is, the extension field C is the field of complex numbers. z^{*} &=\operatorname{Re}(z)-j \operatorname{Im}(z) This follows from the uncountability of R and C as sets, whereas every number field is necessarily countable. When any two numbers from this set are added, is the result always a number from this set? We thus obtain the polar form for complex numbers. For that reason and its importance to signal processing, it merits a brief explanation here. Therefore, the quotient ring is a field. When the original complex numbers are in Cartesian form, it's usually worth translating into polar form, then performing the multiplication or division (especially in the case of the latter). The quantity $$\theta$$ is the complex number's angle. While this definition is quite general, the two fields used most often in signal processing, at least within the scope of this course, are the real numbers and the complex numbers, each with their typical addition and multiplication operations. The quadratic formula solves ax2 + bx + c = 0 for the values of x. \[\begin{align} The imaginary numbers are polynomials of degree one and no constant term, with addition and multiplication defined modulo p(X). The imaginary part of $$z$$, $$\operatorname{Im}(z)$$, equals $$b$$: that part of a complex number that is multiplied by $$j$$. \frac{z_{1}}{z_{2}} &=\frac{a_{1}+j b_{1}}{a_{2}+j b_{2}} \nonumber \\ Imaginary numbers use the unit of 'i,' while real numbers use … }+\frac{x^{3}}{3 ! In using the arc-tangent formula to find the angle, we must take into account the quadrant in which the complex number lies. The real-valued terms correspond to the Taylor's series for $$\cos(\theta)$$, the imaginary ones to $$\sin(\theta)$$, and Euler's first relation results. Existence of $$+$$ identity element: There is a $$e_+ \in S$$ such that for every $$x \in S$$, $$e_+ + x = x+e_+=x$$. \[\begin{align} Because is irreducible in the polynomial ring, the ideal generated by is a maximal ideal. Deﬁnition. Notice that if z = a + ib is a nonzero complex number, then a2 + b2 is a positive real number… That is, there is no element y for which 2y = 1 in the integers. But there is … The set of non-negative even numbers is therefore closed under addition. Deﬁnition. In the travelling wave, the complex number can be used to simplify the calculations by convert trigonometric functions (sin(x) and cos(x)) to exponential functions (e x) and store the phase angle into a complex amplitude.. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. Fields are rather limited in number, the real R, the complex C are about the only ones you use in practice. Complex numbers can be used to solve quadratics for zeroes. Because the final result is so complicated, it's best to remember how to perform division—multiplying numerator and denominator by the complex conjugate of the denominator—than trying to remember the final result. &=r_{1} r_{2} e^{j\left(\theta_{1}+\theta_{2}\right)} A set of complex numbers forms a number field if and only if it contains more than one element and with any two elements \alpha and \beta their difference \alpha-\beta and quotient \alpha/\beta (\beta\neq0). A framework within which our concept of real numbers would fit is desireable. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. However, the field of complex numbers with the typical addition and multiplication operations may be unfamiliar to some. Using Cartesian notation, the following properties easily follow. The field of rational numbers is contained in every number field. This representation is known as the Cartesian form of $$\mathbf{z}$$. Hint: If the field of complex numbers were isomorphic to the field of real numbers, there would be no reason to define the notion of complex numbers when we already have the real numbers. When the scalar field F is the real numbers R, the vector space is called a real vector space. &=\frac{a_{1} a_{2}+b_{1} b_{2}+j\left(a_{2} b_{1}-a_{1} b_{2}\right)}{a_{2}^{2}+b_{2}^{2}} For example, consider this set of numbers: {0, 1, 2, 3}. There is no ordering of the complex numbers as there is for the field of real numbers and its subsets, so inequalities cannot be applied to complex numbers as they are to real numbers. Note that a and b are real-valued numbers. Is the set of even non-negative numbers also closed under multiplication? A complex number is a number that can be written in the form = +, where is the real component, is the imaginary component, and is a number satisfying = −. We will now verify that the set of complex numbers \mathbb{C} forms a field under the operations of addition and multiplication defined on complex numbers. a* (b+c)= (a*b)+ (a*c) After all, consider their definitions. (In fact, the real numbers are a subset of the complex numbers-any real number r can be written as r + 0i, which is a complex representation.) 1. r=|z|=\sqrt{a^{2}+b^{2}} \\ Associativity of S under $$*$$: For every $$x,y,z \in S$$, $$(x*y)*z=x*(y*z)$$. \begingroup you know I mean a real complex number such as (+/-)2.01(+/_)0.11 i. I have a matrix of complex numbers for electric field inside a medium. To divide, the radius equals the ratio of the radii and the angle the difference of the angles. It wasn't until the twentieth century that the importance of complex numbers to circuit theory became evident. The imaginary number $$jb$$ equals $$(0,b)$$. The distributive law holds, i.e. a+b=b+a and a*b=b*a \end{array} \nonumber. To convert $$3−2j$$ to polar form, we first locate the number in the complex plane in the fourth quadrant. Complex numbers satisfy many of the properties that real numbers have, such as commutativity and associativity. For given real functions representing actual physical quantities, often in terms of sines and cosines, corresponding complex functions are considered of which the … If a polynomial has no real roots, then it was interpreted that it didn’t have any roots (they had no need to fabricate a number field just to force solutions). so if you were to order i and 0, then -1 > 0 for the same order. Consequently, a complex number $$z$$ can be expressed as the (vector) sum $$z=a+jb$$ where $$j$$ indicates the $$y$$-coordinate. You may be surprised to find out that there is a relationship between complex numbers and vectors. Missed the LibreFest? }-\frac{\theta^{3}}{3 ! A complex number, z, consists of the ordered pair (a, b), a is the real component and b is the imaginary component (the j is suppressed because the imaginary component of the pair is always in the second position). The best way to explain the complex numbers is to introduce them as an extension of the field of real numbers. In mathematics, imaginary and complex numbers are two advanced mathematical concepts. For multiplication we nned to show that a* (b*c)=... 2. stream I don't understand this, but that's the way it is) To determine whether this set is a field, test to see if it satisfies each of the six field properties. Thus, 3 i, 2 + 5.4 i, and –π i are all complex numbers. The remaining relations are easily derived from the first. if I want to draw the quiver plot of these elements, it will be completely different if I … The product of $$j$$ and an imaginary number is a real number: $$j(jb)=−b$$ because $$j^2=-1$$. \theta=\arctan \left(\frac{b}{a}\right) The reader is undoubtedly already sufficiently familiar with the real numbers with the typical addition and multiplication operations. When you want … If c is a positive real number, the symbol √ c will be used to denote the positive (real) square root of c. Also √ 0 = 0. b=r \sin (\theta) \\ The mathematical algebraic construct that addresses this idea is the field.� i�=�h�P4tM�xHѴl�rMÉ�N�c"�uj̦J:6�m�%�w��HhM����%�~�foj�r�ڡH��/ �#%;����d��\ Q��v�H������i2��޽%#lʸM��-m�4z�Ax ����9�2Ղ�y����u�l���^8��;��v��J�ྈ��O����O�i�t*�y4���fK|�s)�L�����}-�i�~o|��&;Y�3E�y�θ,���ke����A,zϙX�K�h�3���IoL�6��O��M/E�;�Ǘ,x^��(¦�_�zA��# wX��P�\$���8D�+��1�x�@�wi��iz���iB� A~䳪��H��6cy;�kP�. We call a the real part of the complex number, and we call bthe imaginary part of the complex number. We denote R and C the field of real numbers and the field of complex numbers respectively. Consider the set of non-negative even numbers: {0, 2, 4, 6, 8, 10, 12,…}. The complex number field is relevant in the mathematical formulation of quantum mechanics, where complex Hilbert spaces provide the context for one such formulation that is convenient and perhaps most standard. To multiply, the radius equals the product of the radii and the angle the sum of the angles. Closure. \begin{align} Exercise 3. )%2F15%253A_Appendix_B-_Hilbert_Spaces_Overview%2F15.01%253A_Fields_and_Complex_Numbers, Victor E. Cameron Professor (Electrical and Computer Engineering). Complex numbers The equation x2 + 1 = 0 has no solutions, because for any real number xthe square x 2is nonnegative, and so x + 1 can never be less than 1.In spite of this it turns out to be very useful to assume that there is a number ifor which one has Complex numbers are numbers that consist of two parts — a real number and an imaginary number. Euler first used $$i$$ for the imaginary unit but that notation did not take hold until roughly Ampère's time. The integers are not a field (no inverse). The Cartesian form of a complex number can be re-written as, \[a+j b=\sqrt{a^{2}+b^{2}}\left(\frac{a}{\sqrt{a^{2}+b^{2}}}+j \frac{b}{\sqrt{a^{2}+b^{2}}}\right) \nonumber. \end{align}\], $\frac{z_{1}}{z_{2}}=\frac{r_{1} e^{j \theta_{2}}}{r_{2} e^{j \theta_{2}}}=\frac{r_{1}}{r_{2}} e^{j\left(\theta_{1}-\theta_{2}\right)}$. Figure $$\PageIndex{1}$$ shows that we can locate a complex number in what we call the complex plane. Z, the integers, are not a field. Complex arithmetic provides a unique way of defining vector multiplication. Complex number … But either part can be 0, so all Real Numbers and Imaginary Numbers are also Complex Numbers. Fields generalize the real numbers and complex numbers. The system of complex numbers consists of all numbers of the form a + bi where a and b are real numbers. e^{x}=1+\frac{x}{1 ! A field consisting of complex (e.g., real) numbers. What is the product of a complex number and its conjugate? Another way to define the complex numbers comes from field theory. Closure of S under $$+$$: For every $$x$$, $$y \in S$$, $$x+y \in S$$. The distance from the origin to the complex number is the magnitude $$r$$, which equals $$\sqrt{13}=\sqrt{3^{2}+(-2)^{2}}$$. The importance of complex number in travelling waves. Addition and subtraction of polar forms amounts to converting to Cartesian form, performing the arithmetic operation, and converting back to polar form. Thus, 3i, 2 + 5.4i, and –πi are all complex numbers. xX}~��,�N%�AO6Ԫ�&����U뜢Й%�S�V4nD.���s���lRN���r��L���ETj�+׈_��-����A�R%�/�6��&_u0( ��^� V66��Xgr��ʶ�5�)v ms�h���)P�-�o;��@�kTű���0B{8�{�rc��YATW��fT��y�2oM�GI��^LVkd�/�SI�]�|�Ė�i[%���P&��v�R�6B���LT�T7P�c�n?�,o�iˍ�\r�+mرڈ�%#���f��繶y�s���s,��%\55@��it�D+W:E�ꠎY�� ���B�,�F*[�k����7ȶ< ;��WƦ�:�I0˼��n�3m�敯i;P��׽XF8P9���ڶ�JFO�.�l�&��j������ � ��c���&�fGD�斊���u�4(�p��ӯ������S�z߸�E� \end{align}. An introduction to fields and complex numbers. &=\frac{a_{1}+j b_{1}}{a_{2}+j b_{2}} \frac{a_{2}-j b_{2}}{a_{2}-j b_{2}} \nonumber \\ Associativity of S under $$+$$: For every $$x,y,z \in S$$, $$(x+y)+z=x+(y+z)$$. Abstractly speaking, a vector is something that has both a direction and a len… We can choose the polynomials of degree at most 1 as the representatives for the equivalence classes in this quotient ring. For given real functions representing actual physical quantities, often in terms of sines and cosines, corresponding complex functions are considered of which the … To multiply two complex numbers in Cartesian form is not quite as easy, but follows directly from following the usual rules of arithmetic. $a_{1}+j b_{1}+a_{2}+j b_{2}=a_{1}+a_{2}+j\left(b_{1}+b_{2}\right) \nonumber$, Use the definition of addition to show that the real and imaginary parts can be expressed as a sum/difference of a complex number and its conjugate. /Filter /FlateDecode }+\ldots\right) \nonumber\]. So, a Complex Number has a real part and an imaginary part. Legal. Complex numbers The equation x2 + 1 = 0 has no solutions, because for any real number xthe square x 2is nonnegative, and so x + 1 can never be less than 1.In spite of this it turns out to be very useful to assume that there is a number ifor which one has The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. A complex number is a number of the form a + bi, where a and b are real numbers, and i is the imaginary number √(-1). The notion of the square root of $$-1$$ originated with the quadratic formula: the solution of certain quadratic equations mathematically exists only if the so-called imaginary quantity $$\sqrt{-1}$$ could be defined. Quaternions are non commuting and complicated to use. The quantity $$r$$ is known as the magnitude of the complex number $$z$$, and is frequently written as $$|z|$$. There is no multiplicative inverse for any elements other than ±1. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. Our first step must therefore be to explain what a field is. By forming a right triangle having sides $$a$$ and $$b$$, we see that the real and imaginary parts correspond to the cosine and sine of the triangle's base angle. a=r \cos (\theta) \\ Yes, adding two non-negative even numbers will always result in a non-negative even number. Watch the recordings here on Youtube! By then, using $$i$$ for current was entrenched and electrical engineers now choose $$j$$ for writing complex numbers. Commutativity of S under $$+$$: For every $$x,y \in S$$, $$x+y=y+x$$. A complex number, $$z$$, consists of the ordered pair $$(a,b)$$, $$a$$ is the real component and $$b$$ is the imaginary component (the $$j$$ is suppressed because the imaginary component of the pair is always in the second position). A complex number can be written in this form: Where x and y is the real number, and In complex number x is called real part and y is called the imaginary part. if i < 0 then -i > 0 then (-i)x(-i) > 0, implies -1 > 0. not possible*. Because complex numbers are defined such that they consist of two components, it … Distributivity of $$*$$ over $$+$$: For every $$x,y,z \in S$$, $$x*(y+z)=xy+xz$$. 2. Complex numbers are used insignal analysis and other fields for a convenient description for periodically varying signals. This property follows from the laws of vector addition. }+\cdots+j\left(\frac{\theta}{1 ! The final answer is $$\sqrt{13} \angle (-33.7)$$ degrees. If we add two complex numbers, the real part of the result equals the sum of the real parts and the imaginary part equals the sum of the imaginary parts. &=\frac{\left(a_{1}+j b_{1}\right)\left(a_{2}-j b_{2}\right)}{a_{2}^{2}+b_{2}^{2}} \nonumber \\ We convert the division problem into a multiplication problem by multiplying both the numerator and denominator by the conjugate of the denominator. The Field of Complex Numbers S. F. Ellermeyer The construction of the system of complex numbers begins by appending to the system of real numbers a number which we call i with the property that i2 = 1. The complex conjugate of the complex number z = a + ib is the complex number z = a − ib. If the formula provides a negative in the square root, complex numbers can be used to simplify the zero.Complex numbers are used in electronics and electromagnetism. Every number field contains infinitely many elements. Existence of $$+$$ inverse elements: For every $$x \in S$$ there is a $$y \in S$$ such that $$x+y=y+x=e_+$$. Existence of $$*$$ inverse elements: For every $$x \in S$$ with $$x \neq e_{+}$$ there is a $$y \in S$$ such that $$x*y=y*x=e_*$$. \begin{align} }+\ldots \nonumber, Substituting $$j \theta$$ for $$x$$, we find that, $e^{j \theta}=1+j \frac{\theta}{1 ! \[e^{j \theta}=\cos (\theta)+j \sin (\theta) \label{15.3}$, $\cos (\theta)=\frac{e^{j \theta}+e^{-(j \theta)}}{2} \label{15.4}$, $\sin (\theta)=\frac{e^{j \theta}-e^{-(j \theta)}}{2 j}$.