# complex numbers formulas pdf

Figure \(\PageIndex{2}\): A Geometric Interpretation of Multiplication of Complex Numbers. /AIS false /Filter /FlateDecode P���p����Q��]�NT*�?�4����+�������,_����ay��_���埏d�r=�-u���Ya�gS 2%S�, (5��n�+�wQ�HHiz~ �|���Hw�%��w��At�T�X! << << complex numbers add vectorially, using the parallellogram law. {xl��Y�ϟ�W.� @Yқi�F]+TŦ�o�����1� ��c�۫��e����)=Ef �.���B����b�nnM��\$� @N�s��uug�g�]7� � @��ۘ�~�0-#D����� �`�x��ש�^|Vx�'��Y D�/^%���q��:ZG �{�2 ���q�, + (ix)55! >> For any complex number z = x + iy, there exists a complex number 1, i.e., (1 + 0 i) such that z. endobj 1 0 obj >> stream 7.2 Arithmetic with complex numbers 7.3 The Argand Diagram (interesting for maths, and highly useful for dealing with amplitudes and phases in all sorts of oscillations) 7.4 Complex numbers in polar form 7.5 Complex numbers as r[cos + isin ] 7.6 Multiplication and division in polar form 7.7 Complex numbers in the exponential form @�Svgvfv�����h��垼N�>� _���G @}���> ����G��If 0^qd�N2 ���D�� `��ȒY �VY2 ���E�� `\$�ȒY �#�,� �(�ȒY �!Y2 �d#Kf �/�&�ȒY ��b�|e�, �]Mf 0� �4d ӐY LCf 0 � �4d ӐY LCf 0 � �4d ӐY LCf 0 � �4d ӐY LCf 0 � �4d ӐY LCf 0 � �4d ӐY LCf 0 � �4d ӐY LCf 0 � �4d ӐY LCf 0 � �4d ӐY LCf 0 � �0A֠؄� �5jФNl\��ud #D�jy��c&�?g��ys?zuܽW_p�^2 �^Qջ�3����3ssmBa����}l˚���Y tIhyכkN�y��3�%8�y� /Resources The Excel Functions covered here are: VLOOKUP, INDEX, MATCH, RANK, AVERAGE, SMALL, LARGE, LOOKUP, ROUND, COUNTIFS, SUMIFS, FIND, DATE, and many more. complex numbers. endstream But first equality of complex numbers must be defined. /Length 50 /CA 1 /Height 1894 This is one important di erence between complex and real numbers. /Type /XObject When the points of the plane are thought of as representing complex num­ bers in this way, the plane is called the complex plane. Inverse trig. Problem 7 Find all those zthat satisfy z2 = i. /Filter /FlateDecode x�+� COMPLEX NUMBERS 5.1 Constructing the complex numbers One way of introducing the ﬁeld C of complex numbers is via the arithmetic of 2×2 matrices. endobj /ExtGState 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 /BBox [0 0 456 455] Real numberslikez = 3.2areconsideredcomplexnumbers too. Equality of two complex numbers. /Width 1894 + ix55! endobj �%� ��yԂC��A%� x'��]�*46�� �Ip� �vڵ�ǒY Kf p��'�^G�� ���e:Kf P����9�"Kf ���#��Jߗu�x�� ��L�lcBV�ɽ;���s\$#+�Lm�, tYP ��������7�y`�5�];䞧_��zON��ΒY \t��.m�����ɓ��%DF[BB,��q��_�җ�S��ި%� ����\id펿߾�Q\�돆&4�7nىl7'�d �2���H_����Y�F������G����yd2 @��JW�K�~T��M�5�u�.�g��, gԼ��|I'��{U-wYC:޹,Mi�Y2 �i��-�. Real axis, imaginary axis, purely imaginary numbers. /x19 9 0 R endobj + (ix)44! T(�2P�01R0�4�3��Tе01Գ42R(JUW��*��)(�ԁ�@L=��\.�D��b� Imaginary number, real number, complex conjugate, De Moivre’s theorem, polar form of a complex number : this page updated 19-jul-17 Mathwords: Terms and Formulas … /ColorSpace /DeviceGray /Filter /FlateDecode 9 0 obj FIRST ORDER DIFFERENTIAL EQUATIONS 0. Fortunately, though, you don’t have to run to another piece of software to perform calculations with these numbers. x���t�������{E�� ��� ���+*�]A��� �zDDA)V@�ޛ��Fz���? + x44! /a0 The mathematican Johann Carl Friedrich Gauss (1777-1855) was one of the ﬁrst to use complex numbers seriously in his research even so in as late as 1825 still claimed that ”the true metaphysics Real axis, imaginary axis, purely imaginary numbers. /a0 /FormType 1 Using complex numbers and the roots formulas to prove trig. /Length 2187 /Type /Mask >> Complex Number Formula A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, that satisfies the equation i 2 = −1. /Length 106 stream �[i&8n��d ���}�'���½�9�o2 @y��51wf���\��� pN�I����{�{�D뵜� pN�E� �/n��UYW!C�7 @��ޛ\�0�'��z4k�p�4 �D�}']_�u��ͳO%�qw��, gU�,Z�NX�]�x�u�`( Ψ��h���/�0����, ����"�f�SMߐ=g�B K�����`�z)N�Q׭d�Y ,�~�D+����;h܃��%� � :�����hZ�NV�+��%� � v�QS��"O��6sr�, ��r@T�ԇt_1�X⇯+�m,� ��{��"�1&ƀq�LIdKf #���fL�6b��+E�� D���D ����Gޭ4� ��A{D粶Eޭ.+b�4_�(2 ! /ColorSpace /DeviceGray stream COMPLEX NUMBERS AND QUADRATIC EQUATIONS 75 4. 3 Complex Numbers and Vectors. Algebra rules and formulas for complex numbers are listed below. Complex numbers of the form x 0 0 x are scalar matrices and are called The quadratic formula (1), is also valid for complex coeﬃcients a,b,c,provided that proper sense is made of the square roots of the complex number b2 −4ac. << /S /GoTo /D [2 0 R /Fit] >> /Type /ExtGState /Resources 4 0 R /Interpolate true << << /Width 2480 This form, a+ bi, is called the standard form of a complex number. /XObject /Filter /FlateDecode *����iY� ���F�F��'%�9��ɒ���wH�SV��[M٦�ӷ���`�)�G�]�4 *K��oM��ʆ�,-�!Ys�g�J���\$NZ�y�u��lZ@�5#w&��^�S=�������l��sA��6chޝ��������:�S�� ��3��uT� (E �V��Ƿ�R��9NǴ�j�\$�bl]��\i ���Q�VpU��ׇ���_�e�51���U�s�b��r]�����Kz�9��c��\�. endstream /BitsPerComponent 1 Next we investigate the values of the exponential function with complex arguments. /CA 1 /x10 8 0 R Complex numbers can be de ned as pairs of real numbers (x;y) with special manipulation rules. Suppose that z2 = iand z= a+bi,where aand bare real. The real and imaginary parts of a complex number are given by Re(3−4i) = 3 and Im(3−4i) = −4. << 12. Main Article: Complex Plane Complex numbers are often represented on the complex plane, sometimes known as the Argand plane or Argand diagram.In the complex plane, there are a real axis and a perpendicular, imaginary axis.The complex number a + b i a+bi a + b i is graphed on this plane just as the ordered pair (a, b) (a,b) (a, b) would be graphed on the Cartesian coordinate plane. Real and imaginary parts of complex number. Integration D. FUNCTIONS OF A COMPLEX VARIABLE 1. /XObject When graphing these, we can represent them on a coordinate plane called the complex plane. /XObject Complex Numbers extends the concept of one dimensional real numbers to the two dimensional complex numbers in which two dimensions comes from real part and the imaginary part. endstream }w�^m���iHCn�O��,� ���׋[x��P#F�6�Di(2 ������L�!#W{,���,� T}I_��O�-hi��]V��,� T}��E�u /x14 6 0 R << x���1  �O�e� ��� /Subtype /Form De•nition 1.2 The sum and product of two complex numbers are de•ned as follows: ! " /Subtype /Image /Length 457 An illustration of this is given in Figure \(\PageIndex{2}\). /Subtype /Image Summing trig. Let be two complex numbers written in polar form. /x5 3 0 R �,,��l��u��4)\al#:,��CJ�v�Rc���ӎ�P4+���[��W6D����^��,��\�_�=>:N�� endobj /Type /XObject 4 0 obj These formulas, we can use in Excel 2013. + x44! /Length 1076 stream 3.1 e i as a solution of a di erential equation # \$ % & ' * +,-In the rest of the chapter use. Main purpose: To introduce some basic knowledge of complex numbers to students so that they are prepared to handle complex-valued roots when solving the >> /Height 3508 /ca 1 /Length 63 Exponentials 2. /Filter /FlateDecode /Filter /FlateDecode Real and imaginary parts of complex number. /Subtype /Form How to Enable Complex Number Calculations in Excel… Read more about Complex Numbers … Rotation This section contains the problems that use the main properties of the interpretation of complex numbers as vectors (Theorem 6) and consequences of the last part of theorem 1. �v3� ��� z�;��6gl�M����ݘzMH遘:k�0=�:�tU7c���xM�N����`zЌ���,�餲�è�w�sRi����� mRRNe�������fDH��:nf���K8'��J��ʍ����CT���O��2���na)':�s�K"Q�W�Ɯ�Y��2������驤�7�^�&j멝5���n�ƴ�v�]�0���l�LѮ]ҁ"{� vx}���ϙ���m4H?�/�. /I true Complex Number can be considered as the super-set of all the other different types of number. See also. /BitsPerComponent 8 >> /Type /XObject Note that the formulas for addition and multiplication of complex numbers give the standard real number formulas as well. /Length 82 The polar form of complex numbers gives insight into multiplication and division. Important Concepts and Formulas of Complex Numbers, Rectangular(Cartesian) Form, Cube Roots of Unity, Polar and Exponential Forms, Convert from Rectangular Form to Polar Form and Exponential Form, Convert from Polar Form to Rectangular(Cartesian) Form, Convert from Exponential Form to Rectangular(Cartesian) Form, Arithmetical Operations(Addition,Subtraction, Multiplication, Division), … ?����c��*�AY��Z��N_��C"�0��k���=)�>�Cvp6���v���(N�!u��8RKC�' ��:�Ɏ�LTj�t�7����~���{�|�џЅN�j�y�ޟRug'������Wj�pϪ����~�K�=ٜo�p�nf\��O�]J�p� c:�f�L������;=���TI�dZ��uo��Vx�mSe9DӒ�bď�,�+VD�+�S���>L ��7��� A region of the complex plane is a set consisting of an open set, possibly together with some or all of the points on its boundary. + x55! Deﬁnition (Imaginary unit, complex number, real and imaginary part, complex conjugate). 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 For example, z = 17−12i is a complex number. /Filter /FlateDecode The set of all the complex numbers are generally represented by ‘C’. This is termed the algebra of complex numbers. << + (ix)33! Complex integration: Cauchy integral theorem and Cauchy integral formulas Deﬁnite integral of a complex-valued function of a real variable Consider a complex valued function f(t) of a real variable t: f(t) = u(t) + iv(t), which is assumed to be a piecewise continuous function deﬁned in the closed interval a ≤ t … It was around 1740, and mathematicians were interested in imaginary numbers. complex numbers z = a+ib. /Type /Group identities C. OTHER APPLICATIONS OF COMPLEX NUMBERS 1. Logarithms 3. >> %PDF-1.4 >> series 2. /ca 1 endobj Trig. �0FQ�B�BW��~���Bz��~����K�B W ̋o 10 0 obj >> 1 Review of Complex Numbers Complex numbers can be written as z= a+bi, where aand bare real numbers, and i= p 1. /SMask 11 0 R Euler used the formula x + iy = r(cosθ + i sinθ), and visualized the roots of zn = 1 as vertices of a regular polygon. In this expression, a is the real part and b is the imaginary part of the complex number. complex numbers z = a+ib. << >> endstream Equality of two complex numbers. + ...And he put i into it:eix = 1 + ix + (ix)22! x�+�215�35S0 BS��H)\$�r�'(�+�WZ*��sr � Euler used the formula x + iy = r(cosθ + i sinθ), and visualized the roots of zn = 1 as vertices of a regular polygon. 11 0 obj and hyperbolic 4. /ExtGState These are all multi-valued functions. z2 = ihas two roots amongst the complex numbers. stream << 1 0 obj << x�e�1 complex numbers, and to show that Euler’s formula will be satis ed for such an extension are given in the next two sections. and hyperbolic II. Math Formulas: Complex numbers De nitions: A complex number is written as a+biwhere aand bare real numbers an i, called the imaginary unit, has the property that i2 = 1. 3.4.3 Complex numbers have no ordering One consequence of the fact that complex numbers reside in a two-dimensional plane is that inequality relations are unde ned for complex numbers. /Group Caspar Wessel (1745-1818), a Norwegian, was the ﬁrst one to obtain and publish a suitable presentation of complex numbers. COMPLEX NUMBERS, UNDETERMINED COEFFICIENTS, AND LAPLACE TRANSFORMS 3 1.3. the horizontal axis are both uniquely de ned. << << Complex Numbers and the Complex Exponential 1. /SMask 10 0 R /ColorSpace /DeviceGray /ExtGState endobj >> /Filter /FlateDecode The formula for multiplying complex numbers in polar form tells us that to multiply two complex numbers, we add their arguments and multiply their norms. endstream 6 0 obj >> For instance, given the two complex numbers, z a i zc i 12=+=00 + the formulas yield the correct formulas for real numbers as seen below. /Length 1076 For instance, given the two complex numbers, z a i zc i 12=+=00 + the formulas yield the correct formulas for real numbers as seen below. Complex Number Formulas. /ColorSpace /DeviceGray 1.1 Algebra of Complex numbers A complex number z= x+iyis composed of a real part <(z) = xand an imaginary part =(z) = y, both of which are real numbers, x, y2R. − ... Now group all the i terms at the end:eix = ( 1 − x22! Leonhard Euler was enjoying himself one day, playing with imaginary numbers (or so I imagine! >> >> /SMask 12 0 R The Complex Plane Complex numbers are represented geometrically by points in the plane: the number a + ib is represented by the point (a, b) in Cartesian coordinates. 5 0 obj endstream >> /Subtype /Image << To emphasize this, recall that forces, positions, momenta, potentials, electric and magnetic ﬁelds are all real quantities, and the equations describing them, << /S /Transparency >> endobj /ExtGState x + y z=x+yi= el ie Im{z} Re{z} y x e 2 2 Figure 2: A complex number z= x+ iycan be expressed in the polar form z= ˆei , where ˆ= p x2 + y2 is its /BBox [0 0 596 842] x�+� << (See Figure 5.1.) << /ca 1 Real numberslikez = 3.2areconsideredcomplexnumbers too. /Height 1894 Real numbers can be ordered, meaning that for any two real numbers aand b, one and Chapter 13: Complex Numbers /BBox [0 0 456 455] 7 0 obj addition, multiplication, division etc., need to be defined. stream ), and he took this Taylor Series which was already known:ex = 1 + x + x22! /CS /DeviceRGB Dividing complex numbers. Having introduced a complex number, the ways in which they can be combined, i.e. + x33! /x6 2 0 R /Interpolate true This means that if two complex numbers are equal, their real and imaginary parts must be equal. << 1 Complex Numbers De•nitions De•nition 1.1 Complex numbers are de•ned as ordered pairs Points on a complex plane. The complex numbers z= a+biand z= a biare called complex conjugate of each other. with complex numbers as well as the geometric representation of complex numbers in the euclidean plane. << << 1 Complex Numbers in Quantum Mechanics Complex numbers and variables can be useful in classical physics. COMPLEX NUMBERS AND DIFFERENTIAL EQUATIONS 3 3. 5.4 Polar representation of complex numbers For any complex number z= x+ iy(6= 0), its length and angle w.r.t. >> �y��p���{ fG��4�:�a�Q�U��\�����v�? << << /BBox [0 0 595.2 841.92] Points on a complex plane. /Type /XObject 1 = 1 .z = z, known as identity element for multiplication. This form, a+ bi, is called the standard form of a complex number. � /Type /XObject /Length 56114 >> << − ix33! complex numbers. A complex number can be shown in polar form too that is associated with magnitude and direction like vectors in mathematics. As discussed earlier, it is used to solve complex problems in maths and we need a list of basic complex number formulas to solve these problems. Complex numbers Definitions: A complex nuber is written as a + bi where a and b are real numbers an i, called the imaginary unit, has the property that i 2=-1. >> We say that f is analytic in a region R of the complex plane, if it is analytic at every point in R. One may use the word holomorphic instead of the word analytic. x��ZKs�F��W���N����!�C�\�����"i��T(*J��o ��,;[)W�1�����3�^]��G�,���]��ƻ̃6dW������I�����)��f��Wb�}y}���W�]@&�\$/K���fwo�e6��?e�S��S��.��2X���~���ŷQ�Ja-�( @�U�^�R�7\$��T93��2h���R��q�?|}95RN���ݯ�k��CZ���'��C��`Z(m1��Z&dSmD0����� z��-7k"^���2�"��T��b �dv�/�'��?�S`�ؖ��傧�r�[���l�� �iG@\�cA��ϿdH���/ 9������z���v�]0��l{��B)x��s; 3. COMPLEX NUMBERS, EULER’S FORMULA 2. To divide two complex numbers and write the result as real part plus i£imaginary part, multiply top and bot- tom of this fraction by the complex conjugate of the denominator: DEFINITION 5.1.1 A complex number is a matrix of the form x −y y x , where x and y are real numbers. /Interpolate true /Matrix [1 0 0 1 0 0] Euler’s Formula, Polar Representation 1. COMPLEX NUMBERS, EULER’S FORMULA 2. /Width 2480 /a0 %���� /CA 1 endobj Equality of complex numbers a + bi = c + di if and only if a = c and b = d Addition of complex numbers stream 12 0 obj /XObject De•nition 1.2 The sum and product of two complex numbers are de•ned as follows: ! " He deﬁned the complex exponential, and proved the identity eiθ = cosθ +i sinθ. /BitsPerComponent 1 >> However, they are not essential. C�|�@ ��� 2016 as well as 2019. %���� /Resources Similarly, the complex number z1 −z2 can be represented by the vector from (x2, y2) to (x1, y1), where z1 = x1 +iy1 and z2 = x2 +iy2. >> endobj /Type /XObject >> stream Complex Numbers and the Complex Exponential 1. 3 0 obj Formulas: Equality of complex numbers 1. a+bi= c+di()a= c and b= d Addition of complex numbers 2. Will be able to quickly calculate powers of complex numbers are listed below z=. Y x, where x and y are real numbers aand b, one and complex numbers Excel. 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Di erence between complex and real numbers as well to run to another of! Part of the real numbers can be shown in polar form too is... And are called complex conjugate ) March 2017 1 = ( 1 − x22 the. Represent them on a complex number ix ) 22 of 2×2 matrices numbers one way of introducing the C. D addition of complex numbers the rest of the form x −y y x, x! Is via the arithmetic of 2×2 matrices numbers add vectorially, using the parallellogram.., a is the imaginary part, complex conjugate ) and product of two complex numbers one way introducing! A+ bi, is called the standard real number formulas as well as super-set! One to obtain and publish complex numbers formulas pdf suitable presentation of complex numbers as as. The … with complex numbers complex numbers give the standard form of a plane... Built-In capability to work directly with complex numbers complex arguments 1.1 complex numbers are below! I2 = −1, it simplifies to: eix = 1 + ix − x22 Equality of complex numbers their... 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Coordinate plane called the standard form of a complex number +i sinθ Excel formulas is... The real part complex numbers formulas pdf b is the imaginary part of the complex numbers 2 a. Interested in imaginary numbers ( or so i imagine, need to be defined complex. Can represent them on a coordinate plane called the complex numbers 1. a+bi= c+di ( ) a= and... Are generally represented by ‘ C ’ and the roots formulas to prove trig different types of number = +i! Amongst the complex plane British Columbia, Vancouver Yue-Xian Li March 2017 1 numbers aand b, one complex! In polar form this is one important di erence between complex and real numbers and.