complex numbers pdf notes

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1803 LECTURE NOTES SPRING 2014

7 1 Operations on complex numbers real part Re(x + yi) := x imaginary part Im(x + yi) := y (Note: It is y not yi so Im(x + yi) is real) complex conjugate x + yi := x yi (negate the imaginary component) One can add subtract multiply and divide complex numbers (except for division by 0)

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Lecture Notes: Mathematical Methods

1Complex numbers form an algebraically closed field where any polynomial equation has a root Cryptography (e g finding prime numbers) Machine learning: Using a pair of uniformly distributed random numbers (x;y) we can generate random numbers in polar form (r cos( );r sin( ))

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Notes on Complex Numbers

complex number: = a + bi; (2) where a; b are real is the sum of a real and an imaginary number The real part of z=a+bi: Refzg = a is a real number The imaginary part of z=a+bi: Imfzg = b is a also a real number A complex number z=a+bi represents a point (a; b) in a 2D space called the complex plane Im{z} z=a+bi b a −b z=a−bi Re{z}

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COMPLEX NUMBERS COURSE NOTES

Complex numbers are often denoted by z Complex numbers are built on the concept of being able to define the square root of negative one Let ????2=−බ ∴????=√−බ Just like how ℝ denotes the real number system (the set of all real numbers) we use ℂ to denote the set of complex numbers = +???? ∈ℂ for some ∈ℝ

PDF	101 Introduction to complex numbers The number belongs to the set of complex numbers ℂ i e ∈ℂ It comprises a real part ???? ( )=???? ( +???? )= and an imaginary part ???? ( )=???? ( +???? )= Complex numbers have some analogous properties to vectors as we will see during the unit

What is a set of complex numbers?
Just like how R denotes the real number system, (the set of all real numbers) we use C to denote the set of complex numbers. which is an element of the set of complex numbers where x and y are real numbers. So a number like 5 + 3 is a complex number. + 0 , so set of real numbers is a subset of the set of complex numbers. So what....
What is the difference between a real number and a complex number?
b = Im z. Note that real numbers are complex – a real number is simply a complex number with zero imaginary part. Remark 3 Note that two complex numbers are equal precisely when their real and imaginary parts are equal – that is a+bi = c+di if and only if a = c and b = d. This is called ‘comparing real and imaginary parts’.
What is a complex number 3?
Definition 2 A complex number 3 is a number of the form a + bi where a and b are real numbers. If z = a + bi then a is known as the real part of z and b as the imaginary part. We write a = Re z and b = Im z. Note that real numbers are complex – a real number is simply a complex number with zero imaginary part.
What is a complex number Z=a+bi?
Notes on Complex Numbers where a; b are real, is the sum of a real and an imaginary number. The real part of z=a+bi: Refzg = a is a real number. The imaginary part of z=a+bi: Imfzg = b is a also a real number. A complex number z=a+bi represents a point (a; b) in a 2D space, called the complex plane.

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