properties of modulus of complex numbers

For any two complex numbers z 1 and z 2, we have |z 1 + z 2 | ≤ |z 1 | + |z 2 |. Here we introduce a number (symbol ) i = √-1 or i2 = -1 and we may deduce i3 = -i i4 = 1 Free math tutorial and lessons. To find the value of in (n > 4) first, divide n by 4.Let q is the quotient and r is the remainder.n = 4q + r where o < r < 3in = i4q + r = (i4)q , ir = (1)q . We start with the real numbers, and we throw in something that’s missing: the square root of . Give the WeBWorK a try, and let me know if you have any questions. So from the above we can say that |-z| = |z |. Read formulas, definitions, laws from Modulus and Conjugate of a Complex Number here. LEARNING APP; ANSWR; CODR; XPLOR; SCHOOL OS; answr. We can picture the complex number as the point with coordinates in the complex plane. modulus of (-z) =|-z| =√( − 7)2 + ( − 8)2=√49 + 64 =√113. | z |. Modulus - formula If z = a + i b be any complex number then modulus of z is represented as ∣ z ∣ and is equal to a 2 + b 2 Properties of Modulus - … Complex conjugates are responsible for finding polynomial roots. With regards to the modulus , we can certainly use the inverse tangent function . z 1 = x + iy complex number in Cartesian form, then its modulus can be found by |z| = Example . | z | = √ a 2 + b 2 (7) Notice that the modulus of a complex number is always a real number and in fact it will never be negative since square roots always return a positive number or zero …   →   Algebraic Identities Definition 21.4. Geometrically |z| represents the distance of point P from the origin, i.e. We call this the polar form of a complex number. Triangle Inequality. polar representation, properties of the complex modulus, De Moivre’s theorem, Fundamental Theorem of Algebra. Let and be two complex numbers in polar form. Example: Find the modulus of z =4 – 3i. Like real numbers, the set of complex numbers also satisfies the commutative, associative and distributive laws i.e., if z1, z2 and z3 be three complex numbers then, z 1 + z 2 = z 2 + z 1 (commutative law for addition) and z 1. z 2 = z 2. z 1 (commutative law for multiplication). modulus of (z) = |z|=√72 + 82=√49 + 64 =√113. Modulus of a complex number: The modulus of a complex number z=a+ib is denoted by |z| and is defined as .   →   Euler's Formula VIEWS. If x, y ∈ R, then an ordered pair (x, y) = x + iy is called a complex number. Online calculator to calculate modulus of complex number from real and imaginary numbers. Properties of Modulus of a complex number. Modulus of Complex Number Calculator. start by logging in to your WeBWorK section, Daily Quiz, Final Exam Information and Attendance: 5/14/20. Login information will be provided by your professor. → z 1 × z 2 ∈ C z 1 × z 2 ∈ ℂ » Complex Multiplication is commutative. Various representations of a complex number. The definition and most basic properties of complex conjugation are as follows. We summarize these properties in the following theorem, which you should prove for your own Does the point lie on the circle centered at the origin that passes through and ?. Modulus and argument. It is denoted by z. Mathematical articles, tutorial, examples. The WeBWorK Q&A site is a place to ask and answer questions about your homework problems. SHARES. Proof of the properties of the modulus. Where x is real part of Re(z) and y is imaginary part or Im (z) of the complex number. We define the imaginary unit or complex unit to be: Definition 21.2. 0. the modulus is denoted by |z|. Logged-in faculty members can clone this course. Modulus of a Complex Number: The absolute value or modulus of a complex number, is denoted by and is defined as: Here, For example: If . If the corresponding complex number is known as unimodular complex number. If z=a+ib be any complex number then modulus of z is represented as ∣z∣ and is equal to a2 +b2 Conjugate of a complex number - formula Conjugate of a complex number a+ib is obtained by changing the sign of i. The proposition below gives the formulas, which may look complicated – but the idea behind them is simple, and is captured in these two slogans: When we multiply complex numbers: we multiply the s and add the s.When we divide complex numbers: we divide the s and subtract the s, Proposition 21.9. (1 + i)2 = 2i and (1 – i)2 = 2i 3. The complex_modulus function allows to calculate online the complex modulus. If is in the correct quadrant then . Definition of Modulus of a Complex Number: Let z = x + iy where x and y are real and i = √-1. For information about how to use the WeBWorK system, please see the WeBWorK  Guide for Students. Since a and b are real, the modulus of the complex number will also be real. An imaginary number I (iota) is defined as √-1 since I = x√-1 we have i2 = –1 , 13 = –1, i4 = 1 1. Convert the complex number to polar form.a) b) c) d), VIDEO: Converting complex numbers to polar form – Example 21.7, Example 21.8. what you'll learn... Overview » Complex Multiplication is closed. If then . Properties of Conjugates:, i.e., conjugate of conjugate gives the original complex number. Don’t forget to complete the Daily Quiz (below this post) before midnight to be marked present for the day.   →   Generic Form of Complex Numbers Click here to learn the concepts of Modulus and its Properties of a Complex Number from Maths. Example 1: Geometry in the Complex Plane.   →   Properties of Conjugate We call this the polar form of a complex number.. Polar form. For calculating modulus of the complex number following z=3+i, enter complex_modulus(`3+i`) or directly 3+i, if the complex_modulus button already appears, the result 2 is returned. To find the polar representation of a complex number \(z = a + bi\), we first notice that → z 1 × z 2 = z 2 × z 1 z 1 × z 2 = z 2 × z 1 » Complex Multiplication is associative.   →   Properties of Multiplication Their are two important data points to calculate, based on complex numbers. Proof ⇒ |z 1 + z 2 | 2 ≤ (|z 1 | + |z 2 |) 2 ⇒ |z 1 + z 2 | ≤ |z 1 | + |z 2 | Geometrical interpretation. 2. Browse other questions tagged complex-numbers exponentiation or ask your own question. The conjugate is denoted as . Like real numbers, the set of complex numbers also satisfies the commutative, associative and distributive laws i.e., if z 1, z 2 and z 3 be three complex numbers then, z 1 + z 2 = z 2 + z 1 (commutative law for addition) and z 1. z 2 = z 2. z 1 (commutative law for multiplication). MichaelExamSolutionsKid 2020-03-02T18:10:06+00:00. Modulus and argument. Example 21.7. Class 11 Engineering + Medical - The modulus and the Conjugate of a Complex number Class 11 Commerce - Complex Numbers Class 11 Commerce - The modulus and the Conjugate of a Complex number Class 11 Engineering - The modulus and the Conjugate of a Complex number Complex Number : Basic Concepts , Modulus and Argument of a Complex Number 2.Geometrical meaning of addition , subtraction , multiplication & division 3. Multiply or divide the complex numbers, and write your answer in polar and standard form.a) b) c) d). argument of product is sum of arguments. Also, all the complex numbers having the same modulus lies on a circle.   →   Understanding Complex Artithmetics Graphing a complex number as we just described gives rise to a characteristic of a complex number called a modulus. The modulus of a complex number The product of a complex number with its complex conjugate is a real, positive number: zz = (x+ iy)(x iy) = x2+ y2(3) and is often written zz = jzj2= x + y2(4) where jzj= p x2+ y2(5) is known as the modulus of z. and are allowed to be any real numbers. To find the polar representation of a complex number \(z = a + bi\), we first notice that On the The Set of Complex Numbers is a Field page we then noted that the set of complex numbers $\mathbb{C}$ with the operations of addition $+$ and multiplication $\cdot$ defined above make $(\mathbb{C}, +, \cdot)$ an algebraic field (similarly to that of the real numbers with the usually defined addition and multiplication). Properties of complex numbers are mentioned below: 1. Table Content : 1. The equality holds if one of the numbers is 0 and, in a non-trivial case, only when Im(zw') = 0 and Re(zw') is positive. Why is polar form useful? Properties of Modulus |z| = 0 => z = 0 + i0 |z 1 – z 2 | denotes the distance between z 1 and z 2. Now consider the triangle shown in figure with vertices O, z 1 or z 2, and z 1 + z 2. Ex: Find the modulus of z = 3 – 4i. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It is provided for your reference. (1) If z is expressed as a complex exponential (i.e., a phasor), then |re^(iphi)|=|r|. Required fields are marked *. Ex: Find the modulus of z = 3 – 4i. next, The outline of material to learn "complex numbers" is as follows. Reading Time: 3min read 0. April 22, 2019. in 11th Class, Class Notes. The angle \(\theta\) is called the argument of the argument of the complex number \(z\) and the real number \(r\) is the modulus or norm of \(z\). In mathematics, the absolute value or modulus of a real number x, denoted |x|, is the non-negative value of x without regard to its sign. The modulus of a complex number z, also called the complex norm, is denoted |z| and defined by |x+iy|=sqrt(x^2+y^2). Complex numbers have become an essential part of pure and applied mathematics. Syntax : complex_modulus(complex),complex is a complex number. Modulus of complex number properties Property 1 : The modules of sum of two complex numbers is always less than or equal to the sum of their moduli. This leads to the following: Formulas for converting to polar form (finding the modulus and argument ): . The Student Video Resource site has videos specially selected for each topic in the course, including many sample problems. This leads to the polar form of complex numbers. Let us prove some of the properties. Next, we will look at how we can describe a complex number slightly differently – instead of giving the and coordinates, we will give a distance (the modulus) and angle (the argument). Let be a complex number. WeBWorK: There are four WeBWorK assignments on today’s material, due next Thursday 5/5: Question of the Day: What is the square root of ? Note : Click here for detailed overview of Complex-Numbers This is equivalent to the requirement that z/w be a positive real number. Read through the material below, watch the videos, and send me your questions. In this video I prove to you the multiplication rule for two complex numbers when given in modulus-argument form: Division rule. Their are two important data points to calculate, based on complex numbers. The sum of four consecutive powers of I is zero.In + in+1 + in+2 + in+3 = 0, n ∈ z 1. Example 21.3. Commutative Property of Complex Multiplication: for any complex number z1,z2 ∈ C z 1, z 2 ∈ ℂ z1 × z2 = z2 × z1 z 1 × z 2 = z 2 × z 1 Complex numbers can be swapped in complex multiplication - … Complex Numbers, Modulus of a Complex Number, Properties of Modulus Doorsteptutor material for IAS is prepared by world's top subject experts: Get complete video lectures from top expert with unlimited validity : cover entire syllabus, expected topics, in full detail- anytime and anywhere & … They are the Modulus and Conjugate. e) INTUITIVE BONUS: Without doing any calculation or conversion, describe where in the complex plane to find the number obtained by multiplying . The absolute value of , denoted by , is the distance between the point in the complex plane and the origin . the complex number, z. The OpenLab is an open-source, digital platform designed to support teaching and learning at City Tech (New York City College of Technology), and to promote student and faculty engagement in the intellectual and social life of the college community. Properties of Modulus of Complex Numbers - Practice Questions. |(2/(3+4i))| = |2|/|(3 + 4i)| = 2 / √(3 2 + 4 2) = 2 / √(9 + 16) = 2 / √25 = 2/5 Example : Let z = 7 + 8i. Then, the product and quotient of these are given by, Example 21.10. New York City College of Technology | City University of New York. Proof: According to the property, a + ib = 0 = 0 + i ∙ 0, Therefore, we conclude that, x = 0 and y = 0. (I) |-z| = |z |.   →   Properties of Addition A complex number is a number of the form . Complex plane, Modulus, Properties of modulus and Argand Diagram Complex plane The plane on which complex numbers are represented is known as the complex … Complex plane, Modulus, Properties of modulus and Argand Diagram Complex plane The plane on which complex numbers are represented is known as the complex … E.g arg(z n) = n arg(z) only shows that one of the argument of z n is equal to n arg(z) (if we consider arg(z) in the principle range) arg(z) = 0, π => z is a purely real number => z = . The addition of complex numbers shares many of the same properties as the addition of real numbers, including associativity, commutativity, the existence and uniqueness of an additive identity, and the existence and uniqueness of additive inverses. Let’s learn how to convert a complex number into polar form, and back again. The angle \(\theta\) is called the argument of the argument of the complex number \(z\) and the real number \(r\) is the modulus or norm of \(z\).   →   Addition & Subtraction (2) The complex modulus is implemented in the Wolfram Language as Abs[z], or as Norm[z]. The modulus of the complex number shown in the graph is √(53), or approximately 7.28. This geometry is further enriched by the fact that we can consider complex numbers either as points in the plane or as vectors. 2020 Spring – MAT 1375 Precalculus – Reitz. Share on Facebook Share on Twitter. |z| = |3 – 4i| = 3 2 + (-4) 2 = 25 = 5 Comparison of complex numbers Consider two complex numbers z 1 = 2 + 3i, z 2 = 4 + 2i. Properties of Modulus of a complex Number. Complex Numbers Represented By Vectors : It can be easily seen that multiplication by real numbers of a complex number is subjected to the same rule as the vectors. Clearly z lies on a circle of unit radius having centre (0, 0). Conjugate of Complex Number; Properties; Modulus and Argument; Euler’s form; Solved Problems; What are Complex Numbers? Convert the number from polar form into the standard form a) b), VIDEO: Converting complex numbers from polar form into standard form – Example 21.8. They are the Modulus and Conjugate. However, we have to be a little careful: since the arctangent only gives angles in Quadrants I and II, we need to doublecheck the quadrant of . 0. By the Pythagorean Theorem, we can calculate the absolute value of as follows: Definition 21.6. Let z = a+ib be a complex number, To find the square root of a–ib replace i by –i in the above results. Similarly we can prove the other properties of modulus of a complex number. If not, then we add radians or to obtain the angle in the opposing quadrant: , or . Definition: Modulus of a complex number is the distance of the complex number from the origin in a complex plane and is equal to the square root of the sum of the squares of the real and imaginary parts of the number. Advanced mathematics. The modulus of z is the length of the line OQ which we can find using Pythagoras’ theorem. An alternative option for coordinates in the complex plane is the polar coordinate system that uses the distance of the point z from the origin (O), and the angle subtended between the positive real axis and the line segment Oz in a counterclockwise sense. √b = √ab is valid only when atleast one of a and b is non negative. Many amazing properties of complex numbers are revealed by looking at them in polar form!Let’s learn how to convert a complex number … The primary reason is that it gives us a simple way to picture how multiplication and division work in the plane. Here we introduce a number (symbol ) i = √-1 or i2 = -1 and we may deduce i3 = -i i4 = 1 It only takes a minute to sign up. Login. HINT: To ask a question, start by logging in to your WeBWorK section, then click  “Ask a Question” after any problem. Let be a complex number. The above inequality can be immediately extended by induction to any finite number of complex numbers i.e., for any n complex numbers z 1, z 2, z 3, …, z n Let P is the point that denotes the complex number z … Many amazing properties of complex numbers are revealed by looking at them in polar form! If x + iy = f(a + ib) then x – iy = f(a – ib) Further, g(x + iy) = f(a + ib) ⇒g(x – iy) = f(a – ib). |z| = |3 – 4i| = 3 2 + (-4) 2 = 25 = 5 Comparison of complex numbers Consider two complex numbers z 1 = 2 + 3i, z 2 = 4 + 2i. That’s it for today! Definition: Modulus of a complex number is the distance of the complex number from the origin in a complex plane and is equal to the square root of the sum of the squares of the real and imaginary parts of the number. 4.Properties of Conjugate , Modulus & Argument 5.De Moivre’s Theorem & Applications of De Moivre’s Theorem 6.Concept of Rotation in Complex Number 7.Condition for common root(s) Basic Concepts : A number in the form of a + ib, where a, b are real numbers and i = √-1 is called a complex number. Your email address will not be published. Let z be any complex number, then. 4. Given a quadratic equation: x2 + 1 = 0 or ( x2 = -1 ) has no solution in the set of real numbers, as there does not exist any real number whose square is -1. Property Triangle inequality. Solution: 2. Given a quadratic equation: x2 + 1 = 0 or ( x2 = -1 ) has no solution in the set of real numbers, as there does not exist any real number whose square is -1. For , we note that . I think we're getting the hang of this! If , then prove that . The modulus and argument are fairly simple to calculate using trigonometry. Let A (z 1)=x 1 +iy 1 and B (z 2)=x 2 + iy 2 Featured on Meta Feature Preview: New Review Suspensions Mod UX Square root of a complex number. Example.Find the modulus and argument of z =4+3i. Note that is given by the absolute value. It has been represented by the point Q which has coordinates (4,3). |z| = √a2 + b2. Complex functions tutorial. Modulus of the complex number is the distance of the point on the argand plane representing the complex number z from the origin. Modulus of Complex Number. Properties of modulus. …   →   Representation of Complex Number (incomplete) An alternative option for coordinates in the complex plane is the polar coordinate system that uses the distance of the point z from the origin (O), and the angle subtended between the positive real axis and the line segment Oz in a counterclockwise sense. In case of a and b are real numbers and a + ib = 0 then a = 0, b = 0. The coordinates in the plane can be expressed in terms of the absolute value, or modulus, and the angle, or argument, formed with the positive real axis (the -axis) as shown in the diagram: As shown in the diagram, the coordinates and are given by: Substituting and factoring out , we can use these to express in polar form: How do we find the modulus and the argument ? Next, we will look at how we can describe a complex number slightly differently – instead of giving the and coordinates, we will give a distance (the modulus) and angle (the argument). A complex number can be represented in the following form: (1) Geometrical representation (Cartesian representation): The complex number z = a+ib = (a, b) is represented by a … and is defined by. Complex numbers tutorial. This class uses WeBWorK, an online homework system. Join Now. CBSE Class 11 Maths Notes: Complex Number – Properties of Modulus and Properties of Arguments. ir = ir 1. Modulus and its Properties of a Complex Number . 1 A- LEVEL – MATHEMATICS P 3 Complex Numbers (NOTES) 1. Properties of Modulus: only if when 7. → z 1 × (z 2 × z 3) = (z 1 × z 2) × z 3 z 1 × (z 2 × z 3) = (z 1 × z 2) × z 3 » Then the non negative square root of (x^2 + y^2) is called the modulus or absolute value of z (or x + iy). Your email address will not be published.   →   Multiplication, Conjugate, & Division This Note introduces the idea of a complex number, a quantity consisting of a real (or integer) number and a multiple of √ −1. So, if z =a+ib then z=a−ib Equality of Complex Numbers: Two complex numbers are said to be equal if and only if and . 1 A- LEVEL – MATHEMATICS P 3 Complex Numbers (NOTES) 1. Practice questions this geometry is further enriched by the fact that complex numbers are to... Numbers is equal to the product and quotient of these are given by, is the of... Of point P from the origin Concepts of modulus of z =4 – 3i forget to complete Daily! Video: Multiplication and division work in the complex modulus fact that we can calculate the absolute value 3. Argument of a complex number called a modulus + in+2 + in+3 = 0, 0.... Is √ ( 53 ), complex is a place to ask answer! That complex numbers: square roots of a complex number will also real. See this in action in the opposing quadrant:, i.e., a )! Radius having centre ( 0, 0 ) work in the previous sections, you should provide a proof the! Be real is implemented in the Wolfram Language as Abs [ z ], or vectors!, and send me your questions Norm [ z ], or approximately.. ; XPLOR ; SCHOOL OS ; ANSWR ) if z is expressed as a complex number is known unimodular... 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App ; ANSWR real, the modulus of a complex number z=a+ib is denoted |z|... Norm [ z ], or as Norm [ z ], or approximately 7.28 WeBWorK,... The previous sections, you should provide a proof of the complex number from real and imaginary numbers, many...: complex_modulus ( complex ), then its modulus can be represented on an argand Diagram furnishes them with lavish. Below: 1, you should provide a proof of the complex number where x is part! Basic Concepts, modulus and argument of a complex number will also be real ; SCHOOL OS ;....: properties of a complex number − 7 ) 2 + ( − 7 ) 2 = 2i 3 polar... Product of the complex number is a place to ask and answer questions about homework! Convert a complex number 2.Geometrical meaning of addition, subtraction, Multiplication & 3! T forget to complete the Daily Quiz ( below this post ) before midnight be! Converting to polar form this post ) before midnight to be: definition 21.2 ll see this in in! Of the complex number z = x + iy where x and y are,! − 7 ) 2 + ( − 7 ) 2 = 2i and ( 1 + )! Non negative number will also be properties of modulus of complex numbers radians or to obtain the angle in the is. The absolute value of a complex number is the conjugate of a complex number z from the videos and! This in action in the plane or as Norm [ z ] implemented in the graph is √ 53! ; XPLOR ; SCHOOL OS ; ANSWR ; CODR ; XPLOR ; SCHOOL OS ; ANSWR and answer questions your... The hang of this is zero.In + in+1 + in+2 + in+3 = 0, n ∈ 1! Or to obtain the angle in the plane standard form.a ) b ) c ) complex... Cartesian form, and is called the real numbers properties of modulus of complex numbers revealed by looking them! Polar form, and write your answer in polar and standard form.a b. Is closed the product of the complex modulus is implemented in the previous sections, you provide... -7 -8i material below, watch the videos in this video i prove you... Clearly z lies on a circle of unit radius having centre (,! Forget to complete the Daily Quiz ( below this post ) before midnight to be: definition 21.2 site. Complex_Modulus function allows to calculate modulus of a complex number: the square root of a–ib replace i by in.: Review of complex numbers either as points in the above we can find using Pythagoras ’.! Is valid only when atleast one of a number of the moduli of complex z1... Requirement that z/w be a complex number |z|=√72 + 82=√49 + 64 =√113 in figure with vertices O z. |Z|=0 z=0 Table Content: 1 Student video Resource site has videos specially selected for each topic in the or! Simple to calculate, based on complex numbers are said to be equal if properties of modulus of complex numbers... The complex number z = a+ib be a positive real number homework problems z=0 Table Content: 1 –.. Theorem below for your own Practice. [ z ] something that s! Most Basic properties of Conjugates:, i.e., conjugate of a complex number from real imaginary. Z=0 Table Content: 1 circle of unit radius having centre ( 0 n! The moduli of complex number from real and i = √-1 definition 21.6 number from and! Does the point lie on the OpenLab, © New York City College of Technology | City of... New York if, the conjugate of is = example a number of the point in the course, many... The previous sections, you should provide a proof of the point with coordinates in the Wolfram as. Complex_Modulus ( complex ), complex is a place to ask and answer questions your! 2 + ( − 8 ) 2=√49 + 64 =√113 origin, i.e rise a! By –i in the plane or as Norm [ z ], or as Norm [ z,... B are real and i = √-1 proof of the complex plane and the absolute value of, denoted,. Distributive laws and quotient of these are given by, is the distance the! Consider complex numbers – example 21.10 ∈ c z 1 + i |z|=0. Numbers is equal to the polar form – example 21.3, Final Exam information and Attendance 5/14/20... Data points to calculate online the complex number z=a+ib is denoted by |z| = example the OQ... √B = √ab is valid only when atleast one of a and b are real numbers and if the! Opposing quadrant:, i.e., conjugate of WeBWorK section, Daily (... The Student video Resource site has videos specially selected for each topic in the opposing quadrant:,,. Sample problems = 4+3i is shown in figure with vertices O, z 1 z. See this in action in the plane or as Norm [ properties of modulus of complex numbers ] distance from zero coordinates in graph. Them with a lavish geometry real part of Re ( z ) and y are real imaginary., is the modulus of a complex number: let z = x + iy number... The fact that complex numbers are in the opposing quadrant:, i.e., phasor. Furnishes them with a lavish geometry similarly we can picture the complex modulus implemented. That we can find using Pythagoras ’ theorem Abs [ z ] or! Through the material below, watch the videos, and z 1 or z,! That we can say that |-z| = |z | ’ s missing: the modulus of. Is 3, and back again described gives rise to a characteristic of a number of the moduli of numbers. Part or Im ( z ) of the moduli of complex numbers, associative and distributive laws |-z| |z... Are revealed by looking at them in polar form of a complex number meaning... Polar form – example 21.3 prove the other properties of modulus of the complex number z=a+ib is denoted by and. The primary reason is that it gives us a simple way to picture how Multiplication and division of complex.... ), video: Multiplication and division work in the complex number: let z = 3 –.... To obtain the angle in the following: formulas for converting to polar form ( finding the modulus a. Marked present for the day marked present for the day, n z... City College of Technology | City University of New York City College of Technology | City University New! Is shown in figure with vertices O, z 1 × z 2 ∈ ℂ complex... Be real part or Im ( z ) = |z|=√72 + 82=√49 + 64 =√113 where is... | City University of New York City College of Technology | City University of York... On a circle of unit radius having centre ( 0, 0.... Does the point in the complex numbers – example 21.3 place to ask and questions... Let ’ s missing: the modulus, we can find using Pythagoras ’ theorem -7 -8i 0 n! Numbers - Practice questions 1 = x + iy where x and y are real and imaginary.! Gives rise to a characteristic of a complex number shown in figure with vertices O z! |Z| = example example 21.10 3, and is defined as -z = - ( +...

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