FIRST ORDER DIFFERENTIAL EQUATIONS 0. /ExtGState 1 Complex Numbers De•nitions De•nition 1.1 Complex numbers are de•ned as ordered pairs Points on a complex plane. >> /Subtype /Form Complex Number Formulas. /CA 1 endstream endobj 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. /ColorSpace /DeviceGray /Height 3508 C�|�@ ��� 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. /Group Equality of complex numbers a + bi = c + di if and only if a = c and b = d Addition of complex numbers Suppose that z2 = iand z= a+bi,where aand bare real. 12. �v3� ���
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C� @�t������0O��٥Cf��#YC�&. 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. << When graphing these, we can represent them on a coordinate plane called the complex plane. /Matrix [1 0 0 1 0 0] the horizontal axis are both uniquely de ned. /Type /XObject Summing trig. These formulas, we can use in Excel 2013. /a0 /ExtGState 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, '*G�Ջ^W�t�Ir4������t�/Q���HM���p��q��OVq���`�濜���ל�5��sjTy� V
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" /XObject Deﬁnition (Imaginary unit, complex number, real and imaginary part, complex conjugate). 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, 1 = 1 .z = z, known as identity element for multiplication. Complex Numbers and Euler’s Formula University of British Columbia, Vancouver Yue-Xian Li March 2017 1. %���� /Interpolate true Using complex numbers and the roots formulas to prove trig. + x44! /CA 1 and hyperbolic 4. endobj /Subtype /Form /ExtGState endobj In this expression, a is the real part and b is the imaginary part of the complex number. Above we noted that we can think of the real numbers as a subset of the complex numbers. /XObject + (ix)55! << /x6 2 0 R 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 Equality of two complex numbers. 2016 as well as 2019. /XObject 3 Complex Numbers and Vectors. >> {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�, 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 Logarithms 3. endobj << 4. /a0 /Subtype /Image /AIS false >> /Filter /FlateDecode /CA 1 x�e�1 Real numberslikez = 3.2areconsideredcomplexnumbers too. 1 Complex Numbers in Quantum Mechanics Complex numbers and variables can be useful in classical physics. /BitsPerComponent 8 Inverse trig. identities C. OTHER APPLICATIONS OF COMPLEX NUMBERS 1. /Length 63 stream 5.4 Polar representation of complex numbers For any complex number z= x+ iy(6= 0), its length and angle w.r.t. He deﬁned the complex exponential, and proved the identity eiθ = cosθ +i sinθ. >> /Type /XObject 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 >> /S /Transparency /Type /XObject The set of all the complex numbers are generally represented by ‘C’. >> >> << /XObject stream /Filter /FlateDecode 3. 12. << He deﬁned the complex exponential, and proved the identity eiθ = cosθ +i sinθ. Euler’s Formula, Polar Representation 1. P���p����Q��]�NT*�?�4����+�������,_����ay��_���埏d�r=�-u���Ya�gS 2%S�, (5��n�+�wQ�HHiz~ �|���Hw�%��w��At�T�X! /Length 2187 /BBox [0 0 456 455] /Filter /FlateDecode COMPLEX NUMBERS AND DIFFERENTIAL EQUATIONS 3 3. /Type /Mask /Filter /FlateDecode Equality of two complex numbers. /SMask 12 0 R stream stream Note that the formulas for addition and multiplication of complex numbers give the standard real number formulas as well. x���1 �O�e� ��� − ix33! << << Figure \(\PageIndex{2}\): A Geometric Interpretation of Multiplication of Complex Numbers. %PDF-1.4 /Subtype /Image This is termed the algebra of complex numbers. with complex numbers as well as the geometric representation of complex numbers in the euclidean plane. endstream /Resources 2 0 obj >> This is one important di erence between complex and real numbers. >> Integration D. FUNCTIONS OF A COMPLEX VARIABLE 1. Dividing complex numbers. Problem 7 Find all those zthat satisfy z2 = i. /Subtype /Image endobj /Width 2480 << x���1 �O�e� ��� }w�^m���iHCn�O��,� ���[x��P#F�6�Di(2 ������L�!#W{,���,� T}I_��O�-hi��]V��,� T}��E�u These are all multi-valued functions. 11 0 obj + ...And he put i into it:eix = 1 + ix + (ix)22! complex numbers, and to show that Euler’s formula will be satis ed for such an extension are given in the next two sections. + (ix)44! /Height 3508 Real numbers can be ordered, meaning that for any two real numbers aand b, one and stream x�+� The Excel Functions covered here are: VLOOKUP, INDEX, MATCH, RANK, AVERAGE, SMALL, LARGE, LOOKUP, ROUND, COUNTIFS, SUMIFS, FIND, DATE, and many more. 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. 7 0 obj x�+�215�35S0 BS��H)$�r�'(�+�WZ*��sr � + ix55! When graphing these, we can represent them on a coordinate plane called the complex plane. /Width 1894 8 0 obj /Length 457 DEFINITION 5.1.1 A complex number is a matrix of the form x −y y x , where x and y are real numbers. /Subtype /Form endstream Real and imaginary parts of complex number. 1 Review of Complex Numbers Complex numbers can be written as z= a+bi, where aand bare real numbers, and i= p 1. /ColorSpace /DeviceGray /ColorSpace /DeviceGray stream x��ZKs�F��W���N����!�C�\�����"i��T(*J��o ��,;[)W�1�����3�^]��G�,���]��ƻ̃6dW������I�����)��f��Wb�}y}���W�]@&�$/K���fwo�e6��?e�S��S��.��2X���~���ŷQ�Ja-�( @�U�^�R�7$��T93��2h���R��q�?|}95RN���ݯ�k��CZ���'��C��`Z(m1��Z&dSmD0����� z��-7k"^���2�"��T��b �dv�/�'��?�S`�ؖ��傧�r�[���l��
�iG@\�cA��ϿdH���/ 9������z���v�]0��l{��B)x��s; Exponentials 2. Euler used the formula x + iy = r(cosθ + i sinθ), and visualized the roots of zn = 1 as vertices of a regular polygon. 3.1 e i as a solution of a di erential equation /SMask 11 0 R << /BitsPerComponent 1 >> >> /Height 1894 For any non zero complex number z = x + i y, there exists a complex number 1 z such that 1 1 z z⋅ = ⋅ =1 z z, i.e., multiplicative inverse of a + ib = 2 2 << endstream Caspar Wessel (1745-1818), a Norwegian, was the ﬁrst one to obtain and publish a suitable presentation of complex numbers. �0FQ�B�BW��~���Bz��~����K�B W ̋o Complex numbers can be de ned as pairs of real numbers (x;y) with special manipulation rules. Real axis, imaginary axis, purely imaginary numbers. endobj x���
�Om �i�� Note that the formulas for addition and multiplication of complex numbers give the standard real number formulas as well. /Subtype /Form Complex Number can be considered as the super-set of all the other different types of number. Real axis, imaginary axis, purely imaginary numbers. /a0 x�+� For example, z = 17−12i is a complex number. Above we noted that we can think of the real numbers as a subset of the complex numbers. ), and he took this Taylor Series which was already known:ex = 1 + x + x22! l !"" << /ca 1 The real and imaginary parts of a complex number are given by Re(3−4i) = 3 and Im(3−4i) = −4. �%� ��yԂC��A%� x'��]�*46�� �Ip�
�vڵ�ǒY Kf p��'�^G�� ���e:Kf P����9�"Kf ���#��Jߗu�x�� ��L�lcBV�ɽ;���s$#+�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��-�. 1 0 obj /Length 82 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. << /S /GoTo /D [2 0 R /Fit] >> This means that if two complex numbers are equal, their real and imaginary parts must be equal. >> 5 0 obj 9 0 obj /CS /DeviceRGB # $ % & ' * +,-In the rest of the chapter use. 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. 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 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 /Interpolate true ?����c��*�AY��Z��N_��C"�0��k���=)�>�Cvp6���v���(N�!u��8RKC�'
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