# complex numbers formulas pdf

<< /BBox [0 0 456 455] with complex numbers as well as the geometric representation of complex numbers in the euclidean plane. /x14 6 0 R Real numbers can be ordered, meaning that for any two real numbers aand b, one and (See Figure 5.1.) /ca 1 /BitsPerComponent 8 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: Real axis, imaginary axis, purely imaginary numbers. The complex numbers z= a+biand z= a biare called complex conjugate of each other. 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. To emphasize this, recall that forces, positions, momenta, potentials, electric and magnetic ﬁelds are all real quantities, and the equations describing them, These formulas, we can use in Excel 2013. /a0 /XObject Complex Number Formulas. << /XObject >> 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 This form, a+ bi, is called the standard form of a complex number. Exponentials 2. Using complex numbers and the roots formulas to prove trig. (See Figure 6.) stream The polar form of complex numbers gives insight into multiplication and division. T(�2P�01R0�4�3��Tе01Գ42R(JUW��*��)(�ԁ�@L=��\.�D��b� When graphing these, we can represent them on a coordinate plane called the complex plane. 1 0 obj and hyperbolic 4. @�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� �[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 ! >> + ...And he put i into it:eix = 1 + ix + (ix)22! /Subtype /Form /S /Transparency 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. Equality of two complex numbers. addition, multiplication, division etc., need to be defined. The set of all the complex numbers are generally represented by ‘C’. series 2. + (ix)33! Complex numbers can be de ned as pairs of real numbers (x;y) with special manipulation rules. stream /Interpolate true 5. complex numbers z = a+ib. << COMPLEX NUMBERS, EULER’S FORMULA 2. Logarithms 3. << 1 0 obj 9 0 obj and hyperbolic II. 2 0 obj The trigonometric form of a complex number provides a relatively quick and easy way to compute products of complex numbers. << 5.4 Polar representation of complex numbers For any complex number z= x+ iy(6= 0), its length and angle w.r.t. complex numbers. Complex Numbers and Euler’s Formula University of British Columbia, Vancouver Yue-Xian Li March 2017 1. << These are all multi-valued functions. When graphing these, we can represent them on a coordinate plane called the complex plane. Chapter 13: Complex Numbers >> Note that the formulas for addition and multiplication of complex numbers give the standard real number formulas as well. endobj endstream /Resources /Length 2187 COMPLEX NUMBERS AND DIFFERENTIAL EQUATIONS 3 3. >> /x19 9 0 R /Height 1894 There is built-in capability to work directly with complex numbers in Excel. endstream /Width 2480 Equality of complex numbers a + bi = c + di if and only if a = c and b = d Addition of complex numbers # \$ % & ' * +,-In the rest of the chapter use. /s13 7 0 R Complex Numbers and the Complex Exponential 1. Complex Number Formulas. endobj /ColorSpace /DeviceGray A complex number can be shown in polar form too that is associated with magnitude and direction like vectors in mathematics. *����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��\�. 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. �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?�/�. << Real numberslikez = 3.2areconsideredcomplexnumbers too. /Length 82 >> /BitsPerComponent 1 x�+� }w�^m���iHCn�O��,� ���׋[x��P#F�6�Di(2 ������L�!#W{,���,� T}I_��O�-hi��]V��,� T}��E�u /Length 457 /SMask 10 0 R /Type /XObject /Type /XObject /SMask 12 0 R /I true For any non zero complex number z = x + i y, there exists a complex number 1 z such that 1 1 z z⋅ = ⋅ =1 z z, i.e., multiplicative inverse of a + ib = 2 2 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. /ca 1 << 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 << 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. When the points of the plane are thought of as representing complex num­ bers in this way, the plane is called the complex plane. /G 13 0 R >> %PDF-1.4 2016 as well as 2019. l !"" − ... Now group all the i terms at the end:eix = ( 1 − x22! x�e�1 − ix33! We will therefore without further explanation view a complex number x+iy∈Cas representing a point or a vector (x,y) in R2, and according to our need we shall speak about a complex number or a point in the complex plane. endstream 5 0 obj 12 0 obj This form, a+ bi, is called the standard form of a complex number. >> stream '*G�Ջ^W�t�Ir4������t�/Q���HM���p��q��OVq���`�濜���ל�5��sjTy� V ��C�ڛ�h!���3�/"!�m���zRH+�3�iG��1��Ԧp� �vo{�"�HL\$���?���]�n�\��g�jn�_ɡ�� 䨬�§�X�q�(^E��~����rSG�R�KY|j���:.`3L3�(����Q���*�L��Pv�͸�c�v�yC�f�QBjS����q)}.��J�f�����M-q��K_,��(K�{Ԝ1��0( �6U���|^��)���G�/��2R��r�f��Q2e�hBZ�� �b��%}��kd��Mաk�����Ѱc�G! �%� ��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��-�. /Interpolate true Figure \(\PageIndex{2}\): A Geometric Interpretation of Multiplication of Complex Numbers. x�+�215�35S0 BS��H)\$�r�'(�+�WZ*��sr � complex numbers add vectorially, using the parallellogram law. /Subtype /Image /Height 3508 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; Real axis, imaginary axis, purely imaginary numbers. << /XObject Then Therefore, using the addition formulas for cosine and sine, we have This formula says that to multiply two complex numbers we multiply the moduli and add the arguments. /Length 56114 << complex numbers z = a+ib. /XObject << /Type /Group You will see that, in general, you proceed as in real numbers, but using i 2 =−1 where appropriate. endobj /ExtGState complex numbers, and to show that Euler’s formula will be satis ed for such an extension are given in the next two sections. /Type /Mask >> /AIS false He deﬁned the complex exponential, and proved the identity eiθ = cosθ +i sinθ. stream /CA 1 Suppose that z2 = iand z= a+bi,where aand bare real. 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. complex numbers. � >> /Type /XObject Excel Formulas PDF is a list of most useful or extensively used excel formulas in day to day working life with Excel. Deﬁnition (Imaginary unit, complex number, real and imaginary part, complex conjugate). Caspar Wessel (1745-1818), a Norwegian, was the ﬁrst one to obtain and publish a suitable presentation of 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. This means that if two complex numbers are equal, their real and imaginary parts must be equal. /Length 50 P���p����Q��]�NT*�?�4����+�������,_����ay��_���埏d�r=�-u���Ya�gS 2%S�, (5��n�+�wQ�HHiz~ �|���Hw�%��w��At�T�X! /Filter /FlateDecode /Filter /FlateDecode /ColorSpace /DeviceGray 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. 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. 7 0 obj /Type /XObject >> ), and he took this Taylor Series which was already known:ex = 1 + x + x22! EULER’S FORMULA FOR COMPLEX EXPONENTIALS According to Euler, we should regard the complex exponential eit as related to the trigonometric functions cos(t) and sin(t) via the following inspired deﬁnition:eit = cos t+i sin t where as usual in complex numbers i2 = ¡1: (1) The justiﬁcation of this notation is based on the formal derivative of both sides, Leonhard Euler was enjoying himself one day, playing with imaginary numbers (or so I imagine! Complex numbers of the form x 0 0 x are scalar matrices and are called /BBox [0 0 595.2 841.92] /CA 1 endobj Complex Number can be considered as the super-set of all the other different types of number. + x44! This is one important di erence between complex and real numbers. COMPLEX NUMBERS, EULER’S FORMULA 2. /Type /ExtGState However, they are not essential. /Type /XObject /BitsPerComponent 1 DEFINITION 5.1.1 A complex number is a matrix of the form x −y y x , where x and y are real numbers. T(�2�331T015�3� S��� /x6 2 0 R 10 0 obj Fortunately, though, you don’t have to run to another piece of software to perform calculations with these numbers. Above we noted that we can think of the real numbers as a subset of the complex numbers. /Subtype /Form /S /Alpha As a consequence, we will be able to quickly calculate powers of complex numbers, and even roots of complex numbers. This will leaf to the well-known Euler formula for complex numbers. /Matrix [1 0 0 1 0 0] + ... And because i2 = −1, it simplifies to:eix = 1 + ix − x22! Caspar Wessel (1745-1818), a Norwegian, was the ﬁrst one to obtain and publish a suitable presentation of complex numbers. De•nition 1.2 The sum and product of two complex numbers are de•ned as follows: ! " 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 + x55! endstream /ColorSpace /DeviceGray /Filter /FlateDecode {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�, >> /Length 1076 endobj Note that the formulas for addition and multiplication of complex numbers give the standard real number formulas as well. Real and imaginary parts of complex number. 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. /Length 1076 << /S /GoTo /D [2 0 R /Fit] >> /BitsPerComponent 1 /Height 1894 << << endobj /Group /Subtype /Image >> << << 3.1 e i as a solution of a di erential equation /Subtype /Form stream De•nition 1.2 The sum and product of two complex numbers are de•ned as follows: ! " x���1  �O�e� ��� 3 Complex Numbers and Vectors. << For any complex number z = x + iy, there exists a complex number 1, i.e., (1 + 0 i) such that z. 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 /CA 1 Algebra rules and formulas for complex numbers are listed below. COMPLEX NUMBERS, UNDETERMINED COEFFICIENTS, AND LAPLACE TRANSFORMS 3 1.3. Inverse trig. �0�{�~ �%���+k�R�6>�( /CA 1 11 0 obj Let be two complex numbers written in polar form. /Subtype /Image z2 = ihas two roots amongst the complex numbers. 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 endobj x���1  �O�e� ��� For example, z = 17−12i is a complex number. /Interpolate true >> << /Filter /FlateDecode /Height 3508 The complex inverse trigonometric and hyperbolic functions In these notes, we examine the inverse trigonometric and hyperbolic functions, where the arguments of these functions can be complex numbers. 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 5.1 Constructing the complex numbers One way of introducing the ﬁeld C of complex numbers is via the arithmetic of 2×2 matrices. 5 0 obj << stream 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. 1 = 1 .z = z, known as identity element for multiplication. 6 0 obj �0FQ�B�BW��~���Bz��~����K�B W ̋o Next we investigate the values of the exponential function with complex arguments. An illustration of this is given in Figure \(\PageIndex{2}\). He deﬁned the complex exponential, and proved the identity eiθ = cosθ +i sinθ. stream >> /Resources 5 0 R Deﬁnition (Imaginary unit, complex number, real and imaginary part, complex conjugate). + (ix)44! ������, �� U]�M�G�s�4�1����|��%� ��-����ǟ���7f��sݟ̒Y @��x^��}Y�74d�С{=T�� ���I9��}�!��-=��Y�s�y�� ���:t��|B�� ��W�`�_ /cR C� @�t������0O��٥Cf��#YC�&. /Width 2480 x���t�������{E�� ��� ���+*�]A��� �zDDA)V@�ޛ��Fz���? << The Excel Functions covered here are: VLOOKUP, INDEX, MATCH, RANK, AVERAGE, SMALL, LARGE, LOOKUP, ROUND, COUNTIFS, SUMIFS, FIND, DATE, and many more. Above we noted that we can think of the real numbers as a subset of the complex numbers. endobj + ix55! Summing trig. /a0 �,,��l��u��4)\al#:,��CJ�v�Rc���ӎ�P4+���[��W6D����^��,��\�_�=>:N�� See also. Formulas: Equality of complex numbers 1. a+bi= c+di()a= c and b= d Addition of complex numbers 2. >> Euler used the formula x + iy = r(cosθ + i sinθ), and visualized the roots of zn = 1 as vertices of a regular polygon. It was around 1740, and mathematicians were interested in imaginary numbers. << >> /ca 1 >> /BBox [0 0 596 842] 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. COMPLEX NUMBERS AND DIFFERENTIAL EQUATIONS 3 3. 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 … Real numberslikez = 3.2areconsideredcomplexnumbers too. /Filter /FlateDecode >> 1 Review of Complex Numbers Complex numbers can be written as z= a+bi, where aand bare real numbers, and i= p 1. stream /ExtGState endobj %���� >> Equality of two complex numbers. 8 0 obj 1 Complex Numbers in Quantum Mechanics Complex numbers and variables can be useful in classical physics. 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 … /FormType 1 1 Complex Numbers De•nitions De•nition 1.1 Complex numbers are de•ned as ordered pairs Points on a complex plane. 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. /SMask 11 0 R stream Problem 7 Find all those zthat satisfy z2 = i. /Filter /FlateDecode /BBox [0 0 456 455] >> 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), … << << + (ix)55! Integration D. FUNCTIONS OF A COMPLEX VARIABLE 1. endstream Euler used the formula x + iy = r(cosθ + i sinθ), and visualized the roots of zn = 1 as vertices of a regular polygon. /Length 63 endobj But first equality of complex numbers must be defined. endobj 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. We also carefully deﬁne the … �y��p���{ fG��4�:�a�Q�U��\�����v�? C�|�@ ��� In this expression, a is the real part and b is the imaginary part of the complex number. Points on a complex plane. >> COMPLEX NUMBERS AND QUADRATIC EQUATIONS 75 4. << The real and imaginary parts of a complex number are given by Re(3−4i) = 3 and Im(3−4i) = −4. /Subtype /Image 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. Dividing complex numbers. /Subtype /Form identities C. OTHER APPLICATIONS OF COMPLEX NUMBERS 1. /ca 1 /Filter /FlateDecode >> For example, z = 17−12i is a complex number. ?����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��� >> /ExtGState 4. /Type /XObject << /ColorSpace /DeviceGray x�+� 3. 3 0 obj Trig. endstream + x44! Having introduced a complex number, the ways in which they can be combined, i.e. 1 Review of Complex Numbers Complex numbers can be written as z= a+bi, where aand bare real numbers, and i= p 1. 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. /Type /XObject %���� 12. endstream In some branches of engineering, it’s inevitable that you’re going to end up working with complex numbers. /x10 8 0 R >> >> /a0 >> << FIRST ORDER DIFFERENTIAL EQUATIONS 0. /x5 3 0 R /Interpolate true Complex Numbers and the Complex Exponential 1. + x33! /Type /XObject /Length 106 %PDF-1.4 12. /Filter /FlateDecode The complex numbers a+bi and a-bi are called complex conjugate of each other. Euler’s Formula, Polar Representation 1. /ExtGState /Filter /FlateDecode the horizontal axis are both uniquely de ned. /Width 1894 /CS /DeviceRGB x���  �Om �i�� 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. << /Width 1894 /Resources >> Real and imaginary parts of complex number. 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