Mathematics
Imaginary Unit and Polar Bijection
The imaginary unit is denoted by the symbol "i" and is defined as the square root of -1. It is a fundamental concept in complex numbers and is used to extend the real number system. A polar bijection is a mapping that converts complex numbers from the rectangular form to the polar form, providing a way to represent complex numbers using magnitude and angle.
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8 Key excerpts on "Imaginary Unit and Polar Bijection"
eBook - PDF- Cynthia Y. Young(Author)
- 2021(Publication Date)
- Wiley(Publisher)
In order to solve such equations, mathematicians created • Perform operations on complex numbers. • Express complex numbers in polar form. • Find products, quotients, powers, and roots of complex numbers using polar form. • Convert between rectangular and polar coordinates. • Use parametric equations to model paths: spirals and projectiles. LEARNING OBJECTIVES 8.1 Complex Numbers 407 Recall that for positive real numbers a and b, we defined the principal square root as b = √ _ a , which means b 2 = a Similarly, we define the principal square root of a negative number as √ _ −a = i√ _ a , since (i√ _ a ) 2 = i 2 a = −a, for a > 0. The imaginary unit is denoted by the letter i and is defined as i = √ _ −1 where i 2 = −1. The Imaginary Unit i If −a is a negative real number, then the principal square root of −a is √ _ −a = i√ _ a where i is the imaginary unit and i 2 = −1. Principal Square Root It is customary to write i√ _ a instead of √ _ a i to avoid any confusion when defining a radical. EXAMPLE 1 Using the Imaginary Unit i to Simplify Radicals Simplify using imaginary numbers. a. √ _ −9 b. √ _ −8 Solution a. √ _ −9 = i √ _ 9 = 3i b. √ _ −8 = i √ _ 8 = i · 2 √ _ 2 = 2i √ _ 2 Your Turn Simplify √ _ −144 . Answer 12i a new set of numbers based on a number, called the imaginary unit, which when squared would give the negative quantity −1. This new set of numbers is called imaginary numbers. 408 CHAPTER 8 Complex Numbers, Polar Coordinates, and Parametric Equations A complex number written as a + bi is said to be in standard form. If a = 0 and b ≠ 0, then the resulting complex number bi is called a pure imaginary number. If b = 0, then a + bi = a is a real number. The set of all real numbers and the set of all pure imaginary numbers are both subsets of the set of complex numbers.
eBook - PDFMathematics NQF3 SB
TVET FIRST
- M Van Rensburg, I Mapaling, M Trollope A Thorne(Authors)
- 2017(Publication Date)
- Macmillan(Publisher)
22 Module 1 Summary of Module 1 1. Imaginary numbers are square roots of negative numbers represented as i or j e.g. √ ___ − 3 = √ __ 3 i . 2. A complex number is a combination of a real and imaginary number e.g. 2 + 3 i . 3. i is the imaginary unit in i -notation: i 2 = − 1 and i = ± √ ___ − 1. 4. Powers of i (for example i 5 ) can be simplified by writing it in terms of i or i 2 (for example i 5 = i 2 × i 2 × i ). 5. Imaginary numbers in fractions can be simplified by multiplying the numerator and denominator by i in the case of a single-term denominator. 6. The Argand diagram of z = a + bi in a complex plane is represented as: O real axis imaginary axis bi a z Figure 1.17: Argand diagram of z = a + bi 7. Any point in a complex plane is a complex number that can be described by the rectangular coordinates or ( a ; b ), or polar coordinates ( r ; i ). 8. The standard or rectangular form of a complex number is a + bi. 9. Any point in a complex plane can also be described by the polar coordinates ( r ; θ ). 10. The polar form of a complex number is z = r (cos θ + i sin θ ), which can be written as r cis θ or r θ where: l r is the modulus (the distance from the origin to the point). l θ is the argument (the angle between the modulus and the positive real axis). 11. The Argand diagram of the polar form of a complex number can be represented as: O R-axis i -axis r θ Figure 1.18: Argand diagram of z = r(cos θ + i sin θ ) 12. To convert from polar to rectangular form: a = r cos θ and b = r sin θ 13. To convert from rectangular to polar form: θ = tan − 1 b __ a (depending on the quadrant) and r = √ ______ a 2 + b 2 23 Module 1 Summative assessment of Module 1 1. Simplify the following and write in the form a + bi . 1.1 i 17 1.4 i 16 1.2 (3 i ) 3 × i 6 1.5 i 2 + 3 i 3 − 2 i 4 1.3 3 i ___ 9 i 3 1.6 i 4 ÷ 2 i 6 (2 marks × 6)[12] 2. Simplify without using a calculator and write in standard form.
eBook - PDF- Cynthia Y. Young(Author)
- 2017(Publication Date)
- Wiley(Publisher)
We will discuss the polar (trigonometric) form of complex numbers and operations on complex numbers. We will then introduce the polar coordinate system, which is often a preferred coordinate system over the rectangular system. We will graph polar equations in the polar coordinate system and finally discuss parametric equations and their graphs. COMPLEX NUMBERS, POLAR COORDINATES, AND PARAMETRIC EQUATIONS 8.1 COMPLEX NUMBERS 8.2 POLAR (TRIGONOMETRIC) FORM OF COMPLEX NUMBERS 8.3 PRODUCTS, QUOTIENTS, POWERS, AND ROOTS OF COMPLEX NUMBERS; DE MOIVRE’S THEOREM 8.4 POLAR EQUATIONS AND GRAPHS 8.5 PARAMETRIC EQUATIONS AND GRAPHS • The Imaginary Unit i • Adding and Subtracting Complex Numbers • Multiplying Complex Numbers • Dividing Complex Numbers • Raising Complex Numbers to Integer Powers • Complex Numbers in Rectangular Form • Complex Numbers in Polar Form • Products of Complex Numbers • Quotients of Complex Numbers • Powers of Complex Numbers • Roots of Complex Numbers • Polar Coordinates • Converting Between Polar and Rectangular Coordinates • Graphs of Polar Equations • Parametric Equations of a Curve 406 CHAPTER 8 Complex Numbers, Polar Coordinates, and Parametric Equations SKILLS OBJECTIVES ■ ■ Write radicals with negative radicands as imaginary numbers. ■ ■ Add and subtract complex numbers. ■ ■ Multiply complex numbers. ■ ■ Divide complex numbers. ■ ■ Raise complex numbers to powers. CONCEPTUAL OBJECTIVES ■ ■ Understand that real numbers and imaginary numbers are subsets of complex numbers. ■ ■ Recognize the real and imaginary parts of a complex number. ■ ■ Recognize that the square of i is 21. ■ ■ Understand how to eliminate imaginary numbers in denominators. ■ ■ Understand why i raised to a positive integer power can be reduced to 1, 21, i, or 2i. 8.1 COMPLEX NUMBERS 8.1.1 The Imaginary Unit i For some equations like x 2 5 1, the solutions are always real numbers, x 5 61.
eBook - PDFThe Shape of Algebra in the Mirrors of Mathematics
A Visual, Computer-Aided Exploration of Elementary Algebra and Beyond (With CD-ROM)
- Gabriel Katz, Vladimir Nodelman;;;(Authors)
- 2011(Publication Date)
- WSPC(Publisher)
The Complex Numbers 105 Fig. 3.10 Your next task is to give the curve C a new analytical description: find formulas (in the polar coordinates { r , θ }) for a complex number w ( t ) so that, as t varies, the point u * w ( t ) will trace C . 3.6.3 A Short Historical Note The concept of a number that squares to -1 took hold slowly. The set of square roots of negative numbers were at first referred to as imaginary numbers—imaginary but useful. Geronimo Cardano, an Italian mathematician, used the concept in a published method of solving the cubic polynomial equation (1545). But this method was viewed with some suspicion because it depended on “imaginary” numbers for the calculations. Chapter 8 is devoted to the Cardano method and its ramifications. Early in the history of mathematics negative numbers were also termed “imaginary”, but this label eventually faded. Now the term “imaginary” is used to describe the set of vectors on the y -axis. They are of the form bi where b can be any real number. The Shape of Algebra 106 Recall that any number z of the form a + bi has its real part a = Re( z ) and its imaginary part b = Im( z ). The interpretation of complex numbers as vectors (points) in the Cartesian plane appears in the paper of Caspar Wessel published in 1799 in the Proceedings of Copenhagen Academy. In 1806, Jean-Robert Argand published an essay on the same subject. His paper is now viewed as the standard reference for the theoretic foundations of the vectorial representation of complex numbers. In 1831, this geometric representation was rediscovered and popularized by Carl Friedrich Gauss. With this ingenious interpretation, the mystery of the imaginary was gone. 3.6.4 Division, Conjugation and Absolute Value Each complex number has a conjugate complex number associated with it . The conjugate of a + bi is defined as a – bi . The conjugate of a complex number z is commonly denoted z In VisuMatica, the conjugate of z is denoted by “conj(z)”.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Orange Apple(Publisher)
A more abstract formalism for the complex numbers was further developed by the Irish mathematician William Rowan Hamilton, who extended this abstraction to the theory of quaternions. ________________________ WORLD TECHNOLOGIES ________________________ Complex numbers are used in a number of fields, including: engineering, electromagnetism, quantum physics, applied mathematics, and chaos theory. When the underlying field of numbers for a mathematical construct is the field of complex numbers, the name usually reflects that fact. Examples are complex analysis, complex matrix, complex polynomial, and complex Lie algebra. Complex numbers are plotted on the complex plane, on which the real part is on the horizontal axis, and the imaginary part on the vertical axis. Introduction and definition Complex numbers have been introduced to allow for solutions of certain equations that have no real solution: the equation x 2 + 1 = 0 has no real solution x , since the square of x is 0 or positive, so x 2 + 1 can not be zero. Complex numbers are a solution to this dilemma. The idea is to enhance the real numbers by adding a number i whose square is −1, so that x = i is a solution to the preceding equation. Definition A complex number is an expression of the form Here a and b are real numbers, and i is a mathematical symbol which is called imaginary unit . For example, -3.5 + 2 i is a complex number. The real number a of the complex number z = a + bi is called the real part of z and the real number b is the imaginary part .. They are denoted Re( z ) or ℜ ( z ) and Im( z ) or ℑ ( z ), respectively. For example, Some authors also write a + ib instead of a + bi . In some disciplines (in particular, electrical engineering, where i is a symbol for current), the imaginary unit i is instead written as j , so complex numbers are sometimes written as a + bj or a + jb .
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Library Press(Publisher)
A more abstract formalism for the complex numbers was further developed by the Irish mathematician William Rowan Hamilton, who extended this abstraction to the theory of quaternions. ________________________ WORLD TECHNOLOGIES ________________________ Complex numbers are used in a number of fields, including: engineering, electro-magnetism, quantum physics, applied mathematics, and chaos theory. When the underlying field of numbers for a mathematical construct is the field of complex numbers, the name usually reflects that fact. Examples are complex analysis, complex matrix, complex polynomial, and complex Lie algebra. Complex numbers are plotted on the complex plane, on which the real part is on the horizontal axis, and the imaginary part on the vertical axis. Introduction and definition Complex numbers have been introduced to allow for solutions of certain equations that have no real solution: the equation x 2 + 1 = 0 has no real solution x , since the square of x is 0 or positive, so x 2 + 1 can not be zero. Complex numbers are a solution to this dilemma. The idea is to enhance the real numbers by adding a number i whose square is −1, so that x = i is a solution to the preceding equation. Definition A complex number is an expression of the form Here a and b are real numbers, and i is a mathematical symbol which is called imaginary unit . For example, -3.5 + 2 i is a complex number. The real number a of the complex number z = a + bi is called the real part of z and the real number b is the imaginary part .. They are denoted Re( z ) or ℜ ( z ) and Im( z ) or ℑ ( z ), respectively. For example, Some authors also write a + ib instead of a + bi . In some disciplines (in particular, electrical engineering, where i is a symbol for current), the imaginary unit i is instead written as j , so complex numbers are sometimes written as a + bj or a + jb .
eBook - PDFComplex Variables
Principles and Problem Sessions
- A K Kapoor(Author)
- 2011(Publication Date)
- WSPC(Publisher)
Part I Principles This page intentionally left blank This page intentionally left blank Chapter 1 COMPLEX NUMBERS § 1.1 Introduction We begin with some elementary properties of complex numbers. After introducing geometric representation of complex numbers linear transformations of translation, scaling and rotation are described. These transformations together with inversion constitute building blocks of bilinear transformations which are important class of conformal mappings to be taken up in detail in the last two chapters. The point at infinity and related stereographic projection is outlined. All the important concepts, techniques, and results described here are illustrated by means of examples and solved problems. § 1.1.1 Complex conjugate The complex conjugate of the complex number z will be denoted as ¯ z . Thus, if z = x + iy , then ¯ z = x − iy , where x, y ∈ R . With | z | denoting the absolute value, x 2 + y 2 , some simple properties involving the complex conjugate are as follows: ( ∗ 1) (¯ z ) = z , ( ∗ 2) ( z 1 + z 2 ) = ¯ z 1 + ¯ z 2 , ( ∗ 3) ( z 1 z 2 ) = ¯ z 1 ¯ z 2 , ( ∗ 4) z 1 z 2 = ¯ z 1 ¯ z 2 , z 2 = 0, ( ∗ 5) (¯ z ) N = ( z N ), ( ∗ 6) Re ( z ) = 1 2 ( z + ¯ z ), ( ∗ 7) Im ( z ) = 1 2 i ( z − ¯ z ), ( ∗ 8) 1 z = ¯ z | z | 2 , z = 0, ( ∗ 9) | ¯ z | = | z | , ( ∗ 10) | z | 2 = z ¯ z . § 1.1.2 Polar form Geometrically the complex number z = x + iy is represented by a point, with ( x, y ) as the coordinates, in a two-dimensional plane called the complex plane or the Argand diagram . Let r and θ denote the polar coordinates of a point, z ; we have 3 4 Complex Variables: Principles and Problem Sessions Fig. 1.1 (Fig. 1.1) r = x 2 + y 2 , θ = tan − 1 y x , (1.1) and the complex number z itself can be written as z = r (cos θ + i sin θ ) . (1.2) The numbers r and θ are called the modulus and the argument of the complex number z and will be denoted as mod ( z ) and arg( z ), respectively. Note that mod( z ) is just | z | , introduced above.
eBook - PDF- Donald E. Marshall(Author)
- 2019(Publication Date)
- Cambridge University Press(Publisher)
Here is some additional notation: if z = x + iy is given in polar coordinates by the pair (r, θ ) then |z| = r = x 2 + y 2 is called the modulus or absolute value of z. Note that |z| is the distance from the complex number z to the origin 0. The angle θ is called the argument of z and is written θ = arg z. 1.1 Complex Numbers 5 The most common convention is that −π < arg z ≤ π , where positive angles are measured counter-clockwise and negative angles are measured clockwise. The complex conjugate of z is given by z = x − iy. The complex conjugate is the reflection of z about the real line R. It is an easy exercise to show the following: |zw| = |z||w|, |cz| = c|z| if c > 0, z/|z| has absolute value 1, z z = |z| 2 , Rez = (z + z)/2, Imz = (z − z)/(2i), z + w = z + w, zw = z · w, z = z, |z| = | z|, arg zw = arg z + arg w modulo 2π , arg z = − arg z = 2π − arg z modulo 2π . The statement modulo 2π means that the difference between the left- and right-hand sides of the equality is an integer multiple of 2π . The identity a + (z − a) = z expressed in vector form shows that z − a is (a translate of) the vector from a to z. Thus |z − a| is the length of the complex number z − a but it is also equal to the distance from a to z. The circle centered at a with radius r is given by {z : |z − a| = r} and the disk centered at a of radius r is given by {z : |z − a| < r}. The open disks are the basic open sets generating the standard topology on C. We will use D to denote the unit disk, D = {z : |z| < 1}, and use ∂ D to denote the unit circle, ∂ D = {z : |z| = 1}. Complex numbers were around for at least 250 years before good applications were found; Cardano discussed them in his book Ars Magna (1545). Beginning in the 1800s, and continu- ing today, there has been an explosive growth in their usage. Now complex numbers are very important in the application of mathematics to engineering and physics.
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