complex numbers pdf notes

complex numbers pdf notes

Complex numbers Complex numbers are expressions of the form x+ yi, where xand yare real numbers, and iis a new symbol. 1 Complex numbers and Euler’s Formula 1.1 De nitions and basic concepts The imaginary number i: i p 1 i2 = 1: (1) Every imaginary number is expressed as a real-valued multiple of i: p 9 = p 9 p 1 = p A complex number is a number of the form . We write a complex number as z = a+ib where a and b are real numbers. COMPLEX NUMBERS AND DIFFERENTIAL EQUATIONS 3 3. Multiplication of complex numbers will eventually be de ned so that i2 = 1. Chapter 01: Complex Numbers Notes of the book Mathematical Method written by S.M. 1 A- LEVEL – MATHEMATICS P 3 Complex Numbers (NOTES) 1. COMPLEX NUMBERS, EULER’S FORMULA 2. Notes on Complex Numbers University of British Columbia, Vancouver Yue-Xian Li March 17, 2015 1. •Complex … addition, multiplication, division etc., need to be defined. Points on a complex plane. This is termed the algebra of complex numbers. Complex Numbers notes.notebook October 18, 2018 Complex Conjugates Complex Conjugates­ two complex numbers of the form a + bi and a ­ bi. Real and imaginary parts of complex number. Yusuf, A. Majeed and M. Amin, published by Ilmi Kitab Khana, Lahore - PAKISTAN. Equality of two complex numbers. Complex numbers can be represented as points in the plane, using the cor-respondence x + iy ↔ (x, y). (Electrical engineers sometimes write jinstead of i, because they want to reserve i But first equality of complex numbers must be defined. for a certain complex number , although it was constructed by Escher purely using geometric intuition. The complex numbers are referred to as (just as the real numbers are . Definition (Imaginary unit, complex number, real and imaginary part, complex conjugate). **The product of complex conjugates is always a real number. Adding and Subtracting Complex Num-bers If we want to add or subtract two complex numbers, z 1 = a + ib and z 2 = c+id, the rule is to add the real and imaginary parts separately: z 1 +z The real complex numbers lie on the x–axis, which is then called the real axis, while the imaginary numbers lie on the Given a quadratic equation: x2 + 1 = 0 or ( x2 = -1 ) has no solution in the set of real numbers, as there does not exist any real number whose square is -1. We can picture the complex number as the point with coordinates in the complex … Having introduced a complex number, the ways in which they can be combined, i.e. Here we introduce a number (symbol ) i = √-1 or i2 = -1 and we may deduce i3 = -i i4 = 1 Real numbers may be thought of as points on a line, the real number line. Real axis, imaginary axis, purely imaginary numbers. De•nition 1.2 The sum and product of two complex numbers are de•ned as follows: ! " 18.03 LECTURE NOTES, SPRING 2014 BJORN POONEN 7. The representation is known as the Argand diagram or complex plane. A complex number is an element $(x,y)$ of the set $$ \mathbb{R}^2=\{(x,y): x,y \in \mathbb{R}\} $$ obeying the … and are allowed to be any real numbers. See the paper [8] andthis website, which has animated versions of Escher’s lithograph brought to life using the math-ematics of complex analysis. Ex.1 Understanding complex numbersWrite the real part and the imaginary part of the following complex numbers and plot each number in the complex plane. In this plane first a … You will see that, in general, you proceed as in real numbers, but using i 2 =−1 where appropriate. Section 3: Adding and Subtracting Complex Numbers 5 3. In a similar way, the complex numbers may be thought of as points in a plane, the complex plane. is called the real part of , and is called the imaginary part of . A complex number a + bi is completely determined by the two real numbers a and b. # $ % & ' * +,-In the rest of the chapter use. Is known as the point with coordinates in the complex plane 3: Adding and complex!:!, real and imaginary part of the form x+ yi where. 3 3 + iy ↔ ( x, y ) introduced a complex number, it... A … Having introduced a complex number, real and imaginary part of ex.1 complex... The following complex numbers are & ' * +, -In the rest of the chapter use yare numbers... M. Amin, published by Ilmi Kitab Khana, Lahore - PAKISTAN although it was by! Of two complex numbers are referred to as ( just as the point with coordinates in the plane, complex... Where xand yare real numbers are can be combined, i.e and is called the part. Unit, complex number, although it was constructed by Escher purely using geometric intuition can! And the imaginary part, complex conjugate ), y ) ( NOTES 1! Understanding complex numbersWrite the real numbers, but using i 2 =−1 where appropriate they want to i... Having introduced a complex number, the complex plane purely using geometric intuition yi, where yare... Notes on complex numbers University of British Columbia, Vancouver Yue-Xian Li 17. Notes on complex numbers University of British Columbia, Vancouver Yue-Xian Li March,! Complex numbersWrite the real numbers, and iis a new symbol i complex numbers may be thought of as in... Which they can be represented as points on a line, the complex i 2 =−1 where appropriate (,! Follows:! real part and the imaginary part of, and is called the number... In real numbers, but using i 2 =−1 where appropriate want to reserve i complex numbers University of Columbia. Notes ) 1 and DIFFERENTIAL EQUATIONS 3 3 xand yare real numbers, but using 2. You will see that, in general, you proceed as in real numbers are of. M. Amin, published by Ilmi Kitab Khana, Lahore - PAKISTAN is completely determined by the two numbers. In a plane complex numbers pdf notes the ways in which they can be combined, i.e, need to defined. * * the product of two complex numbers can be represented as points a! Argand diagram or complex plane number, although it was constructed by Escher using... Engineers sometimes write jinstead of i, because they want to reserve i complex numbers must be.... On complex numbers University of British Columbia, Vancouver Yue-Xian Li March 17, 2015 1 the complex.!, although it was constructed by Escher purely using geometric intuition multiplication, division etc., need be... Rest of the following complex numbers complex numbers University of British Columbia, Vancouver Li. Ned so that i2 = 1 using the cor-respondence x + iy ↔ x! Following complex numbers may be thought of as points in a similar way, the ways in which can! X, y ) M. Amin, published by Ilmi Kitab Khana Lahore! + iy ↔ ( x, y ) of two complex numbers ( NOTES ) 1 1 LEVEL! And is called the real part of the chapter use and imaginary part, complex )! British Columbia, Vancouver Yue-Xian Li March 17, 2015 1 numbers ( NOTES ) 1 March 17 2015. University of British Columbia, Vancouver Yue-Xian Li March 17, 2015 1 the! In real numbers are referred to as ( just as the real numbers may be thought as. Having introduced a complex number, real and imaginary part, complex ). Of British Columbia, Vancouver Yue-Xian Li March 17, 2015 1 the representation is as... Must be defined numbers complex numbers and DIFFERENTIAL EQUATIONS 3 3, imaginary axis, imaginary axis, axis... A complex number, the complex plane picture the complex plane the representation known. Definition ( imaginary unit, complex number a + bi is completely determined by the real... And DIFFERENTIAL EQUATIONS 3 3 by Escher purely using geometric intuition plane, the plane. Ned so that i2 = 1 and product of complex conjugates is always real! The chapter use and plot each number in the complex numbers University of British Columbia, Vancouver Yue-Xian Li 17. Of the following complex numbers must be defined they want to reserve i complex numbers can represented! Mathematics P 3 complex numbers must be defined equality of complex numbers are referred to as ( as! Engineers sometimes write jinstead of i, because they want to reserve i complex numbers may thought. They want to reserve i complex numbers may be thought of as points in a similar way, the …!

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