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.
6 Key excerpts on "Imaginary Unit and Polar Bijection"
- Lawrence S. Leff, Christina Pawlowski-Polanish(Authors)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
...To give meaning to solutions of such equations, the imaginary unit i is defined as with i 2 = −1. If x 2 + 1 = 0, then. A number that has the form a + bi, where a and b stand for real numbers, is called a complex number. Because any real number a can be written in the form a + 0 · i, the set of real numbers is a subset of the set of complex numbers. Thus, 5 is a complex number because 5 = 5 + 0 · i. LESSONS IN CHAPTER 5 • Lesson 5-1: Complex Numbers • Lesson 5-2: Multiplying and Dividing Complex Numbers • Lesson 5-3: Completing the Square • Lesson 5-4: The Quadratic Formula Lesson 5-1: Complex Numbers KEY IDEAS If a and b are real numbers, then a + bi, where, is called a complex number. When a complex number is written in the standard form a + bi, a is called the real part of the complex number and b is the imaginary part. Thus, the real part of the complex number 3 + 2 i is 3, and the imaginary part is 2. Arithmetic operations can be performed with complex numbers in much the same way that these operations are performed with binomials. THE IMAGINARY UNIT The imaginary unit i is the number whose square is −1. In other words, i 2 = −1, so. Square roots of negative numbers other than −1 can be expressed in terms of i by factoring out and replacing it with i. For example,. The number 2 i is called a pure imaginary number. A pure imaginary number is the product of any nonzero real number and i. SIMPLIFYING POWERS OF i The expression i n, where n is a positive whole number, can be reduced to either ±1 or ± i, as the following list suggests: Consecutive positive-integer powers of i follow a cyclic pattern that repeats every four integers. Using the list of powers of i, you should be able to predict that i 8 = 1, i 9 = i, i 10 = –1, i 11 = – i, …. A large power of i can be simplified by dividing the exponent by 4 and using the remainder as the new power of i...
- Afshin Samani(Author)
- 2019(Publication Date)
- CRC Press(Publisher)
...The i satisfies i 2 = −1 or sometimes written as i = − 1. This is why the i is called the “imaginary” unit. No real number satisfies this relationship, as the square of any real number is always non-negative. Very often, j can also be used to denote the imaginary unit. MATLAB recognizes both i and j as the imaginary unit, provided you have not already defined them as variables named i and j. To avoid confusion, newer versions of MATLAB introduced 1 i and 1 j to denote the imaginary unit. In the notation of a complex number c = a + i.b, a is the real part of c, and it is written as Re (c) = a, and i. b is the imaginary part of c : Im (c) = i. b. One can perform arithmetic operations with complex numbers, such as addition, subtraction and multiplication. If c 1 = a 1 + i.b 1 and c 2 = a 2 + i.b 2, then, c 1 + c 2 = (a 1 + a 2) + i. (b 1 + b 2) Subtraction is defined similarly. For multiplication, one can write: c 1. c 2 = (a 1 a 2 − b 1 b 2) + i. (a 1 b 2 + a 2 b 1) Note that in deriving the multiplication relationship, i 2 = −1 has been used. Figure A.2.1 The geometrical representation of a complex number with a real part (a) and imaginary part (b). The phase θ and the magnitude R of the complex number is used for polar representation of the complex number The complex numbers can also be represented in what is known as “polar” representation. Instead of expressing a complex number c in terms of its real and imaginary parts, one can represent it in terms of its magnitude (R) and phase (θ). c = R e i θ Where e iθ = cos (θ) + i.sin (θ), which is known as the Euler formula. Based on trigonometric equations, it is very straightforward to show that for a complex number c = a + i.b R = a 2 + b 2 and θ = atan (b a) ; This is how the real and imaginary parts of a complex number are related to its magnitude and phase. For a complex number like c = a + i.b, a complex-conjugate number is defined as c * = a − i.b. Here, c * is the complex-conjugate (or just conjugate) of c...
eBook - ePub- Cary Wolfe, Adam Nocek, Cary Wolfe, Adam Nocek(Authors)
- 2021(Publication Date)
- Routledge(Publisher)
...One solution is the real value 2. The other two solutions can only be constructed as expressions involving the square root of a negative number. This forces the consideration of numbers that, when multiplied by themselves, yield a negative number, but there is no real number that multiplied by itself yields a negative number! Imagine a garden A meters by A meters square, with a total area of A × A = A 2. If A were such a quantity involving the square root of a negative quantity, lengthening side of length A would yield less area, not more. The mechanical application of a standardized algebraic procedure meant to simply find a solution to an abstract equation contradicts our physical spatial experience and classical geometric interpretation. Small wonder that Descartes called these entities imaginary. Yet, a geometric understanding of imaginary numbers is possible in an enlarged mathematical ontology. Adjoining the square root of negative one, √−1, written as i, to the real numbers, lifts the algebraic operations from the one dimension of the real numbers to the two-dimensional field of what mathematicians call complex numbers. This is a structural operation that extends the universe of entities and makes the Fundamental Theorem of Algebra: every polynomial of degree n has exactly n roots – not true in the domain of real numbers – true in the extended domain of complex numbers. Intricately extending the mathematical ontology in order to render true a theorem is one of the creative motors of mathematical ontogenesis. Moreover, whole universes of new properties, techniques, and theorems in what later became the mathematical field of complex analysis open up that have no analogs in the prior domain of the functions on the “reals” ℝ...
eBook - ePubMechanical Vibration
Analysis, Uncertainties, and Control, Fourth Edition
- Haym Benaroya, Mark Nagurka, Seon Han(Authors)
- 2017(Publication Date)
- CRC Press(Publisher)
...Appendix A Mathematical Concepts for Vibration “We are reminded of the essential rules.” This appendix presents mathematical concepts that may be helpful to those studying vibration. It is not meant to be a substitute for courses covering linear algebra, calculus, and differential equations. We assume that the reader has had some experience with the topics. A.1 Complex Numbers A complex number z = x + iy is the sum of a real part x and an imaginary part iy, with x and y both real and i = - 1. The complex conjugate of z is x - iy. Complex numbers, which arise in many scientific disciplines, are needed in the general solution polynomial equations, including of quadratic equations. Before the 18th century, complex numbers and the square root of minus one were in regular use. 1 The equation 2 e i θ = cos θ + i sin θ is credited to Euler and is known as Euler ’ s formula. A.1.1 Complex Number Operations Two complex numbers are added or subtracted by simply adding or subtracting their corresponding real and imaginary parts. For example, the sum of 8 + 3 i and 6 + 3 i is 14 + 6 i. Complex number multiplication is similar to real number multiplication. For example, (x + i y) (u + i v) = (x u - y v) + i (x v + y u). In division of complex numbers, the convention is to express the fraction such that there is no imaginary part in the denominator. This can be accomplished by multiplying the numerator and denominator of the complex fraction by the complex conjugate of the denominator u - iv to find (A.1) x + i y u + i v = (x u + y v) + i (- x v + y u) u 2 + v 2 Example A.l Rewrite the complex fraction 1 - 2 i 7 + 3 i into its real and imaginary parts. Solution: To express the fraction in terms of its real and imaginary parts, multiply its numerator and denominator by the complex conjugate of the denominator and expand as...
eBook - ePub- Arsen Melkumian(Author)
- 2012(Publication Date)
- Routledge(Publisher)
...If a point A represents z = a + bi on the complex plane, then the distance from the origin O to the point A is the modulus of z Note that compex conjugates have the same modulus: E XAMPLE 11.4 (a) |3 + 4 i | = 5 (b) |3 − 4 i | = 5 (c) (d) Using complex numbers we can solve any quadratic equation. Recall that the solution for ax 2 + bx + c = 0 is given by where We can use the formula over the complex numbers even when the discriminant D is negative. E XAMPLE 11.5 Solve Solution : E XAMPLE 11.6 Prove that. Solution : Let z 1 = a + bi and z 2 = c + di. Then E XAMPLE 11.7 Show that. Solution : Let z 1 = a + bi and z 2 = c + di. Then 11.2 Polar and trigonometric form of complex numbers So far we have discussed the so-called Cartesian representation of the complex numbers. In essence, we have assigned to each complex number a point on the Cartesian (or complex) plane. However, complex numbers have also a representation in polar coordinates. A complex number z = a + bi is assigned coordinates (r, θ) in the polar system, where and θ is such that as shown in Figure 11.3. The pair (r, θ) specifies a unique point on the complex plane. However, a given point on the complex plane does not have a unique polar representation. In fact, a point z = (r, θ) has infinitely many polar representations of the form z = (r, θ + 2 πn), with n ∈ ℤ. A number in the form (θ + 2 πn) is called an argument of z. The argument of z lying in the range (−π, π] is referred to as the principal argument of z or Arg(z). For instance, the principal argument of 2 i is and the principal argument of (2 i + 2) is Figure 11.3 A complex number z can also be written in a trigonometric form: The trigonometric form of complex numbers is used extensively in time-series analysis. E XAMPLE 11.8 Find the trigonometric form of z = 2 + 2 i. Solution : It follows that Hence Complex numbers can also be written in an exponential form...
eBook - ePubDigital Signal Processing 101
Everything You Need to Know to Get Started
- Michael Parker(Author)
- 2017(Publication Date)
- Newnes(Publisher)
...This will become important in digital communication, because it can be used to compute the power of a complex signal. To summarize, we have tried to show that the imaginary numbers which are used to form things called complex numbers are really not so complex, and imaginary is really a very misleading description. What we have really been after is to create a two dimensional number plane, and define a set of expanded arithmetic rules to manipulate the numbers in it. Now we are ready to move onto the next topic, the complex exponential. Figure 2.4 Complex conjugate diagram. 2.6. The Complex Exponential The complex exponential has an intimidating sound to it, but in reality, it is very simple to visualize. It is simply the unit circle (radius = 1) on the complex number plane (Fig. 2.5). Any point on the unit circle can be represented by “e” or raised to the power (j·angle) or more also expressed e jΩ, which is called a complex exponential function. A few examples should help. Let the angle Ω = 0°degree. Anything raised to the power 0 is equal to 1. This checks out, since this is the Point 1 on the positive real axis. Let angle Ω = 90°degrees. The complex exponential is e j90. This is the point j on the positive imaginary axis. We need a way to evaluate the complex exponential to show this. This leads to the Euler equation. This equation can easily be derived using series Taylor expansion for exponential, but we have promised to minimize the math. But the result is: e j Ω = cos (Ω) + j sin (Ω) Let us try exp(j90) again. Using Euler equation e j 90 = cos (90) + j sin (90) = 0 + j · 1 = j Imagine the point Z = e jΩ with the angle Ω starting at 0°degree and gradually increasing to 360°degrees. This will start at the point +1 on real axis, and move counter-clockwise around the circle until it ends up where it started, at 1 again...
