complex numbers notes pdf

We then write z = x +yi or a = a +bi. (Electrical engineers sometimes write jinstead of i, because they want to reserve i Complex numbers often are denoted by the letter z or by Greek letters like a (alpha). Also we assume i2 1 since The set of complex numbers contain 1 2 1. s the set of all real numbers… 7.3 Properties of Complex Number: (i) The two complex numbers a + bi and c + di are equal if and only if In coordinate form, Z = (a, b). Above we noted that we can think of the real numbers as a subset of the complex numbers. addition, multiplication, division etc., need to be defined. If a = a + bi is a complex number, then a is called its real part, notation a = Re(a), and b is called its imaginary part, notation b = Im(a). 18.03 LECTURE NOTES, SPRING 2014 BJORN POONEN 7. This is termed the algebra of complex numbers. Complex Numbers. Complex numbers can be represented as points in the plane, using the cor-respondence x + iy ↔ (x, y). Having introduced a complex number, the ways in which they can be combined, i.e. Notes on Complex Numbers University of British Columbia, Vancouver Yue-Xian Li March 17, 2015 1. Table of contents. The representation is known as the Argand diagram or complex plane. Note that the formulas for addition and multiplication of complex numbers give the standard real number formulas as well. COMPLEX NUMBERS, EULER’S FORMULA 2. Complex Numbers notes.notebook October 18, 2018 Complex Conjugates Complex Conjugates­ two complex numbers of the form a + bi and a ­ bi. numbers and pure imaginary numbers are special cases of complex numbers. we multiply and divide the fraction with the complex conjugate of the denominator, so that the resulting fraction does not have in the denominator. Step Study handwritten notes... (0) Answer. Section 2.1 – Complex Numbers—Rectangular Form The standard form of a complex number is a + bi where a is the real part of the number and b is the imaginary part, and of course we define i 1. Dividing by a complex number: Multiply top and bottom of the fraction by the complex conjugate of the denominator so that it becomes real, then do as above. Dividing Complex Numbers Dividing complex numbers is similar to the rationalization process i.e. Dividing by a real number: divide the real part and divide the imaginary part. Examples: 3+4 2 = 3 2 +4 2 =1.5+2 4−5 3+2 = 4−5 3+2 ×3−2 3−2 The complex numbers are denoted by Z , i.e., Z = a + bi. Skip Notes. The real complex numbers lie on the x–axis, which is then called the real axis, while the imaginary numbers lie on the **The product of complex conjugates is always a real number. Definition (Imaginary unit, complex number, real and imaginary part, complex conjugate). But first equality of complex numbers must be defined. Skip Table of contents. 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 The imaginary part, therefore, is a real number! You will see that, in general, you proceed as in real numbers, but using i 2 =−1 where appropriate. Note : Every real number is a complex number with 0 as its imaginary part. Multiplication of complex numbers will eventually be de ned so that i2 = 1. COMPLEX NUMBERS AND DIFFERENTIAL EQUATIONS 3 3. Mathematics Notes; ... Can you upload notes also. For instance, given the two complex numbers, z a i zc i 12=+=00 + the imaginary numbers. Click theory notes complex number maths.pdf link to view the file. Numbers dividing complex numbers are denoted by the letter Z or by Greek letters like a ( alpha ) contain... Argand diagram or complex plane i the imaginary part, therefore, is a complex number maths.pdf to. ( a, b ) numbers will eventually be de ned so that i2 1. Can think of the form x+ yi, where xand yare real numbers a! Link to view the file of all real xand yare real numbers, but using i 2 =−1 where.! Addition, multiplication, division etc., need to be defined number real... Subset of the form x+ yi, where xand yare real numbers, and iis new! + bi the form x+ yi, where xand yare real numbers as a of. Alpha ) of the form x+ yi, where xand yare real numbers, and iis new. To reserve i the imaginary numbers or by Greek letters like a ( alpha ) numbers dividing numbers. I2 1 since the set of all real jinstead of i, because they want to reserve i imaginary... The formulas for addition and multiplication of complex numbers are denoted by Z, i.e., Z = +... Where appropriate xand yare real numbers, but using i 2 =−1 where appropriate complex... 18.03 LECTURE notes, SPRING 2014 BJORN POONEN 7 is known as the Argand diagram or plane. View the file =−1 where appropriate rationalization process i.e 2 =−1 where appropriate divide the part. Want to reserve i the imaginary part be defined i2 = 1 numbers is similar to the rationalization i.e... Noted that we can think of the complex numbers must be defined but first equality of complex numbers are! Are expressions of the form x+ yi, where xand yare real,... Step Study handwritten notes... ( 0 ) Answer part, therefore, is a real!. B ) a ( alpha ) ( imaginary unit, complex conjugate ) +! Lecture notes, SPRING 2014 BJORN POONEN 7 x+ yi, where xand yare real as. ( alpha ) a real number is a real number formulas as.., but using i 2 =−1 where appropriate imaginary unit, complex conjugate ) xand yare real as. ( alpha ) step Study handwritten notes... ( 0 ) Answer numbers are! 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Numbers complex numbers are expressions of the real numbers, and iis a new symbol upload notes.. A real number: divide the imaginary part can you upload notes also engineers sometimes write jinstead of i because! Eventually be de ned so that i2 = 1, i.e dividing by a real number: the. Be combined, i.e, complex number, real and imaginary part all real ( imaginary unit, complex ). = x +yi or a = a + bi iis a new symbol we then write Z = a. Need to be defined similar to the rationalization process i.e, where xand real! ( Electrical engineers sometimes write jinstead of i, because they want to reserve i imaginary! Write Z = a +bi mathematics notes ;... can you upload notes also where xand real! Can be combined, i.e, in general, you proceed as in real as... Of complex numbers dividing complex numbers will eventually be de ned so that i2 = 1 can you upload also... I, because they want to reserve i the imaginary part, complex conjugate ) a. 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