Modulus and Phase

8 Key excerpts on "Modulus and Phase"

• 

  eBook - ePub

  Trigonometric Functions and Complex Numbers

  • Desheng Yang, Chunhui Shen;;;(Authors)
  • 2016(Publication Date)
  • WCPC

    (Publisher)

  PART II

  Complex Number

  Passage contains an image

  Chapter 7

  Concept of Complex Number

  Imaginary Unit, Complex Number, Modulus of Complex Number

    1.

  i2 = −1, the number i is called imaginary unit; suppose a, b ∈ R, any number of the form z = a + bi is called a complex number, a is the real part of z, and b is the imaginary part of z. We write a = Re z, b = Im z. To complex number z = a + bi, when b = 0, z is the real number; when b = 0, z is called an imaginary number; when a = 0 and b = 0, z is called a purely imaginary number. A collection of all complex numbers is called complex set, and we write it C.

    2.

  In the plane rectangular coordinate system, Z(a, b) expresses complex number a + bi. The distance from Z to origin is called modulus of complex number. The modulus of z = a + bi (a, b ∈ R) is written |z| or |a + bi|, and

    3.

  To learn the concept of complex number, pay attention to the following:

  (1) When we write the algebraic form of complex number a + bi, do not leave out a, b ∈ R. If not, a, b do not necessarily express the real and imaginary parts of the complex number, and in this case, the definition of equality of complex numbers and formula for modulus of complex numbers cannot be used. Do not think that a complex number must be an imaginary number, and do not even think a complex number is certainly not a real number. Should master the following relations: complex number a + bi (a, b ∈ R),

  (2) If two complex numbers are not real numbers, the size of them cannot be compared. That is two imaginary numbers cannot be compared the size and a real number and an imaginary number cannot be compare the size. Only two real numbers can be compared the size.

  (3)

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Quantum Computing for Computer Scientists

  • Noson S. Yanofsky, Mirco A. Mannucci(Authors)
  • 2008(Publication Date)
  • Cambridge University Press

    (Publisher)

   We cannot continue without mentioning another property of conjugation: c × c = |c| 2 . (1.49) In words, the modulus squared of a complex number is obtained by multiplying the number with its conjugate. For example, (3 + 2i ) × (3 − 2i ) = 3 2 + 2 2 = 13 = |3 + 2i | 2 . (1.50) We have covered what we need from the algebraic perspective. We see in the next section that the geometric approach sheds some light on virtually all topics touched on here. Programming Drill 1.2.1 Take the program that you wrote in the last programming drill and make it also perform subtraction and division of complex numbers. In ad- dition, let the user enter a complex number and have the computer return its modulus and conjugate. 1.3 THE GEOMETRY OF COMPLEX NUMBERS As far as algebra is concerned, complex numbers are an algebraically complete field, as we have described them in Section 1.2. That alone would render them invaluable as a mathematical tool. It turns out that their significance extends far beyond the algebraic domain and makes them equally useful in geometry and hence in physics. To see why this is so, we need to look at a complex number in yet another way. At the beginning of Section 1.2, we learned that a complex number is a pair of real 16 Complex Numbers Figure 1.1. Complex plane. numbers. This suggests a natural means of representation: real numbers are placed on the line, so pairs of reals correspond to points on the plane, or, equivalently, correspond to vectors starting from the origin and pointing to that point (as shown in Figure 1.1). In this representation, real numbers (i.e., complex numbers with no imaginary part) sit on the horizontal axis and imaginary numbers sit on the vertical axis. This plane is known as the complex plane or the Argand plane. Through this representation, the algebraic properties of the complex numbers can be seen in a new light. Let us start with the modulus: it is nothing more than the length of the vector.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Understand Electrical and Electronics Maths

  • Owen Bishop(Author)
  • 2013(Publication Date)
  • Newnes

    (Publisher)

  The mathematics of imaginary and complex numbers is a useful tool for dealing with these phasors. Phasors are usually specified by length and by phase angle, for we most often know the amplitude and the phase angle of the sine wave they represent. It is therefore more convenient if the complex numbers that are to represent the phasors are expressed in polar form, as a magnitude and an angle. Figure 14.9 shows a complex number at point ζ in the complex number plane. The real part of ζ is a, and the imaginary part of ζ is b = a + }b (1) This is the way of expressing a complex number in rectangular form. The figure also shows a vector (or phasor) drawn from the origin to point z. The length of the vector is r and its phase angle is Θ. From the definirion of the trig rarios (page x): = r cos θ (2) and b = r sin θ (3) Subsrituring these values for a and b in equation (1): ζ = r (cos θ + j sin Θ) This is the polar form of the complex number. Converting from rectangular to polar form Given a and Z?, the real and imaginary parts of z, the conversion is exactly the same as on page 100. The fact that b is the imaginary part of ζ (though not an imaginary number itself) makes no difference to the geometry. 250 UNDERSTAND ELECTRICAL AND ELECTRONICS MATHS z = 3+j4 θ = 108.43° θ = -71.57° θ = 300.%° 3 -5 - V -z = 3-j5 Figure 14.10 Examples Convert the complex number ^ = 3 + j4 into polar form (Figure 14.10a). r = y]3^ + 42 = ^ 9 + 16 = ^25 = 5 θ = tan^ 4/3 = 53.13° (2 dp, by pocket calculator) In polar form, ζ = 5 (cos 53.13° + j sin 53.13°) Convert ζ — 2 + j6 into polar form (Figure 14.10b). r = V2^ + 6^ = V4 + 36 = V40 = 6.32 (2 dp) θ = tan^ 6/-2 = -71.57° (2 dp, by pocket calculator) A calculator does not necessarily show the correct result. We might take -71.57° to mean that the point is in die fourth quadrant. But Figure 14.10b shows that the number is in the second quadrant. Both the angles shown have a tangent of 3. The calculator does not tell us which angle we require.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  A Concise Handbook of Mathematics, Physics, and Engineering Sciences

  • Andrei D. Polyanin, Alexei Chernoutsan(Authors)
  • 2010(Publication Date)
  • CRC Press

    (Publisher)

  The geometric meaning of the operations of addition and subtraction of complex num-bers is as follows: the sum and the difference of complex numbers z 1 and z 2 are the vectors equal to the directed diagonals of the parallelogram spanned by the vectors z 1 and z 2 (Fig. M9.2). The following inequalities hold (Fig. M9.2): | z 1 + z 2 | ≤ | z 1 | + | z 2 | , | z 1 – z 2 | ≥ vextendsingle vextendsingle | z 1 | – | z 2 | vextendsingle vextendsingle . (M9. 1 . 2 . 1 ) 229 230 F UNCTIONS OF C OMPLEX V ARIABLE Inequalities (M9.1.2.1) become equalities if and only if the arguments of the complex numbers z 1 and z 2 coincide (i.e., arg z 1 = arg z 2 ; see Subsection M9.1.2) or one of the numbers is zero. The product z 1 z 2 of complex numbers z 1 = x 1 + iy 1 and z 2 = x 2 + iy 2 is defined to be the number z 1 z 2 = ( x 1 x 2 – y 1 y 2 ) + i ( x 1 y 2 + x 2 y 1 ). The product of a complex number z = x + iy by its conjugate is always nonnegative: z ¯ z = x 2 + y 2 . If z 2 ≠ 0 , then the quotient of z 1 and z 2 is defined as z 1 z 2 = x 1 x 2 + y 1 y 2 x 2 2 + y 2 2 + i x 2 y 1 – x 1 y 2 x 2 2 + y 2 2 . (M9. 1 . 1 . 1 ) Relation (M9.1.1.1) can be obtained by multiplying the numerator and the denominator of the fraction z 1 /z 2 by ¯ z 2 . M9.1.2. Trigonometric Form of Complex Numbers. Powers and Radicals ◮ Modulus and argument of a complex number. There is a one-to-one correspondence between complex numbers z = x + iy and points M with coordinates ( x , y ) on the plane with a Cartesian rectangular coordinate system OXY or with vectors −−→ OM connecting the origin O with M (Fig. M9.1). The length r of the vector −−→ OM is called the modulus (also magnitude and absolute value ) of the number z and is denoted by r = | z | , and the angle ϕ formed by the vector −−→ OM and the positive direction of the OX -axis is called the argument (also phase ) of the number z and is denoted by ϕ = Arg z .

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Numbers and Proofs

  • Reg Allenby(Author)
  • 1997(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  The important concepts of modulus and argument were introduced via polar representation of complex numbers, and their more prominent properties - in particular the triangle inequality - were established. The neat polar form of complex multiplication leads, naturally, to De Moivre's theorem with its applications to finding roots and obtaining trigonometrical identities. Complex numbers, in particular the theory of functions of a complex variable, are important in many branches of mathematics, physics and engineering where complex functions are able to express, mathematically, the idea of flows and potentials. 176 and

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Calculus, Volume 1

  • Tom M. Apostol(Author)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  . . , a n are arbitrary real numbers, with a n ≠ 0, has a solution among the complex numbers if n ≥ 1. Moreover, even if the coefficients a 0 , a 1 , . . . , a n are complex, a solution exists in the complex-number system. This fact is known as the fundamental theorem of algebra. † It shows that there is no need to construct numbers more general than complex numbers to solve polynomial equations with complex coefficients. 9.5 Geometric interpretation. Modulus and argument Since a complex number (x, y) is an ordered pair of real numbers, it may be represented geo- metrically by a point in the plane, or by an arrow or geometric vector from the origin to the point (x, y), as shown in Figure 9.1. In this context, the xy-plane is often referred to as the complex plane. The x-axis is called the real axis; the y-axis is the imaginary axis. It is customary to use the words complex number and point interchangeably. Thus, we refer to the point z rather than the point corresponding to the complex number z. The operations of addition and subtraction of complex numbers have a simple geometric inter- pretation. If two complex numbers z 1 and z 2 are represented by arrows from the origin to z 1 and z 2 , respectively, then the sum z 1 + z 2 is determined by the parallelogram law. The arrow from the origin to z 1 + z 2 is a diagonal of the parallelogram determined by 0, z 1 , and z 2 , as illustrated by the example in Figure 9.2. The other diagonal is related to the difference of z 1 and z 2 . The arrow from z 1 to z 2 is parallel to and equal in length to the arrow from 0 to z 2 − z 1 ; the arrow in the opposite direction, from z 2 to z 1 , is related in the same way to z 1 − z 2 . † A proof of the fundamental theorem of algebra can be found in almost any book on the theory of functions of a complex variable. For example, see K. Knopp, Theory of Functions, Dover Publications, New York, 1945, or E. Hille, Analytic Function Theory, Vol. I, Blaisdell Publishing Co., 1959.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Precalculus

  A Prelude to Calculus

  • Sheldon Axler(Author)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  Section 7.4 The Complex Plane 519 When thinking of complex numbers as vectors in the complex plane, the absolute value of a complex number is simply the magnitude of the corresponding vector. Example 1 Evaluate |2 + 3i |. solution |2 + 3i | = √ 2 2 + 3 2 = √ 13 ≈ 3.60555 Example 2 Show that | cos θ + i sin θ | = 1 for every real number θ. solution | cos θ + i sin θ | = p cos 2 θ + sin 2 θ = √ 1 = 1 Recall that the complex conjugate of a complex number a + bi, where a and b are real numbers, is denoted by a + bi and is defined by a + bi = a - bi. In terms of the complex plane, the operation of complex conjugation is the same as flipping across the real axis. The figure here shows a complex number and its complex conjugate. 2 + i and its complex conjugate 2 - i. A nice formula connects the complex conjugate and the absolute value of a complex number. To derive this formula, suppose z = a + bi, where a and b are real numbers. Then z z = (a + bi )(a - bi ) = a 2 - b 2 i 2 = a 2 + b 2 = |z| 2 . We record this result as follows. Complex conjugates and absolute values If z is a complex number, then z z = |z| 2 . Geometric Interpretation of Complex Multiplication and Division As we will soon see, using polar coordinates with complex numbers can bring extra insight into the operations of multiplication, division, and raising a complex number to a power. Suppose z = x + yi, where x and y are real numbers. We identify z with the point Here we think of a complex number as a point in the complex plane and then use the polar coordinates that were developed in Section 7.1. ( x, y) in the complex plane. If ( x, y) has polar coordinates (r, θ ), then x = r cos θ and y = r sin θ. Thus z = x + yi = r cos θ + ir sin θ = r(cos θ + i sin θ ). The equation above leads to the following definition.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Mathematics NQF3 SB

  TVET FIRST

  • M Van Rensburg, I Mapaling, M Trollope A Thorne(Authors)
  • 2017(Publication Date)
  • Macmillan

    (Publisher)

  14 Module 1 1.4.1 The polar form of a complex number A complex number can be represented in polar form from an Argand diagram. This is done by connecting the coordinate point ( z or P ) to the origin (O) to create a right-angled triangle as shown in Figure 1.11: O P a b θ r z = a + bi real axis imaginary axis Figure 1.11: The relationship between rectangular and polar form In the right-angled triangle in Figure 1.11: a = real part of the complex number b = imaginary part of the complex number r = modulus , which is the absolute value or the magnitude of the complex number θ = argument , which is the amplitude, ‘maximum extent’ or ‘phase angle’ of the complex number. To obtain P from O there are two possibilities: l Rectangular form: Move a units along the real axis (east) and b units directly north. This is the rectangular form a + bi. l Polar form: From line a (vector) on the real axis, rotate line r anti-clockwise (in a positive direction) from the real axis to create an angle q . This is the polar form r θ . Modulus The modulus ( r ) is the longest side (hypotenuse) of a right-angled triangle. It is the absolute or maximum value of a complex number and is sometimes written in the form | r |. The || lines around r mean that r is positive. From the Theorem of Pythagoras: r 2 = a 2 + b 2 r = ± √ ______ a 2 + b 2 r = ± √ _______________________ (real part) 2 + (imaginary part) 2 Note: The value of the modulus is always positive : r = √ ______ a 2 + b 2 . Argument The argument ( θ ) is the angle (or direction) of the line measured anti-clockwise from the positive real axis . Note: The fixed point of reference is the origin (O), which is called the pole . Note: The polar form is sometimes called the trigonometric or trig.form because it uses the Theorem of Pythagoras.

  Sign up to read

  Learn more about book