modulus of complex number properties

Properties of Complex Numbers Date_____ Period____ Find the absolute value of each complex number. The norm (or modulus) of the complex number \(z = a + bi\) is the distance from the origin to the point \((a, b)\) and is denoted by \(|z|\). Completing the CAPTCHA proves you are a human and gives you temporary access to the web property. Geometrically, modulus of a complex number = is the distance between the corresponding point of which is and the origin in the argand plane. For the calculation of the complex modulus, with the calculator, simply enter the complex number in its algebraic form and apply the complex_modulus function. They are the Modulus and Conjugate. We know from geometry 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. Thus, the modulus of any complex number is equal to the positive square root of the product of the complex number and its conjugate complex number. These are respectively called the real part and imaginary part of z. Example: Find the modulus of z =4 – 3i. Complex numbers tutorial. Definition of Modulus of a Complex Number: Let z = x + iy where x and y are real and i = √-1. Principal value of the argument. Modulus and argument of complex number. Let z = a + ib be a complex number. Properties of Modulus of a complex number: Let us prove some of the properties. Modulus and argument. Modulus of a Complex Number. If the corresponding complex number is known as unimodular complex number. Complex Number Properties. (BS) Developed by Therithal info, Chennai. Properties of Modulus of a complex number. Then, conjugate of z is = … Viewed 4 times -1 $\begingroup$ How can i Proved ... properties of complex modulus question. That is the modulus value of a product of complex numbers is equal If z1 = x1 + iy1 and z2 = x2 + iy2 , then, | z1 - z2| = | ( x1 - x2 ) + ( y1 - y2 )i|, The distance between the two points z1 and z2 in complex plane is | z1 - z2 |, If we consider origin, z1 and z2 as vertices of a 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). This is the. Given an arbitrary complex number , we define its complex conjugate to be . If you are at an office or shared network, you can ask the network administrator to run a scan across the network looking for misconfigured or infected devices. So, if z =a+ib then z=a−ib property as "Triangle Inequality". Ask Question Asked today. Reading Time: 3min read 0. 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 modulus and argument of a complex number sigma-complex9-2009-1 In this unit you are going to learn about the modulusand argumentof 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 Basic Algebraic Properties of Complex Numbers, Exercise 2.3: Properties of Complex Numbers, Exercise 2.4: Conjugate of a Complex Number, Modulus of a Complex Number: Solved Example Problems, Exercise 2.5: Modulus of a Complex Number, Exercise 2.6: Geometry and Locus of Complex Numbers. It is denoted by z. SHARES. Cloudflare Ray ID: 613aa34168f51ce6 Active today. We call this the polar form 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. 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. Problem solving - use acquired knowledge to solve practice problems, such as finding the modulus of 9 - i $\sqrt{a^2 + b^2} $ Properties of Modulus of Complex Numbers - Practice Questions. Proof: Beginning Activity. 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. In the above figure, is equal to the distance between the point and origin in argand plane. Complex functions tutorial. And ∅ is the angle subtended by z from the positive x-axis. 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. Viewed 12 times 0 $\begingroup$ I ... determining modulus of complex number. Negative number raised to a fractional power. 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 & … A question on analytic functions. Share on Facebook Share on Twitter. April 22, 2019. in 11th Class, Class Notes. 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). • Their are two important data points to calculate, based on complex numbers. If x, y ∈ R, then an ordered pair (x, y) = x + iy is called a complex number. These are quantities which can be recognised by looking at an Argand diagram. 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. Geometrically |z| represents the distance of point P from the origin, i.e. Please enable Cookies and reload the page. 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. Any complex number in polar form is represented by z = r(cos∅ + isin∅) or z = r cis ∅ or z = r∠∅, where r represents the modulus or the distance of the point z from the origin. The square |z|^2 of |z| is sometimes called the absolute square. 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. 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. complex number. the sum of the lengths of the remaining two sides. And it's actually quite simple. 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 & … Before we get to that, let's make sure that we recall what a complex number is. 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). It can be shown that the complex numbers satisfy many useful and familiar properties, which are similar to properties of the real numbers. Solution for Find the modulus and argument of the complex number (2+i/3-i)2. + zn | ≤ |z1| + |z2| + |z3| + … + |zn| for n = 2,3,…. Then, the modulus of a complex number z, denoted by |z|, is defined to be the non-negative real number. Misc 13 Find the modulus and argument of the complex number ( 1 + 2i)/(1 − 3i) . Proof of the properties of the modulus. Modulus of a Complex Number. Ex: Find the modulus of z = 3 – 4i. 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. that the length of the side of the triangle corresponding to the vector  z1 + z2 cannot be greater than 0. Similarly we can prove the other properties of modulus of a complex number. Featured on Meta Feature Preview: New Review Suspensions Mod UX Read formulas, definitions, laws from Modulus and Conjugate of a Complex Number here. Modulus of complex exponential function. |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 \(\eqref{eq:MProd}\) and \(\eqref{eq:MQuot}\) relate the modulus of a product/quotient of two complex numbers to the product/quotient of the modulus of the individual numbers.We now need to take a look at a similar relationship for sums of complex numbers.This relationship is called the triangle inequality and is, VIEWS. Understanding Properties of Complex Arithmetic » The properties of real number arithmetic is extended to include i = √ − i = √ − Modulus and argument of the complex numbers. Also express -5+ 5i in polar form what you'll learn... Overview. E-learning is the future today. C. Sauzeat, H. Di Benedetto, in Advances in Asphalt Materials, 2015. A tutorial in plotting complex numbers on the Argand Diagram and find the Modulus (the distance from the point to the origin) Equations (1) and (2) are satisfied for infinitely many values of θ, any of these infinite values of θ is the value of amp z. This leads to the polar form of complex numbers. VII given any two real numbers a,b, either a = b or a < b or b < a. Proof ⇒ |z 1 + z 2 | 2 ≤ (|z 1 | + |z 2 |) 2 ⇒ |z 1 + z 2 | ≤ |z 1 | + |z 2 | Geometrical interpretation. Triangle Inequality. Question 1 : Find the modulus of the following complex numbers (i) 2/(3 + 4i) Solution : We have to take modulus of both numerator and denominator separately. We know from geometry In this section, we will discuss the modulus and conjugate of a complex number along with a few solved examples. Advanced mathematics. finite number of terms: |z1 z2 z3 ….. zn| = |z1| |z2| |z3| … … |zn|. Modulus or absolute value of z = |z| |z| = a 2 + b 2 Since a and b are real, the modulus of the complex number will also be real. as vertices of a Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … Performance & security by Cloudflare, Please complete the security check to access. Property of modulus of a number raised to the power of a complex number. Well, we can! When the sum of two complex numbers is real, and the product of two complex numbers is also natural, then the complex numbers are conjugated. finite number of terms: |z1 + z2 + z3 + …. Where x is real part of Re(z) and y is imaginary part or Im (z) of the complex number. Since the complex numbers are not ordered, the definition given at the top for the real absolute value cannot be directly applied to complex numbers.However, the geometric interpretation of the absolute value of a real number as its distance from 0 can be generalised. The third part of the previous example also gives a nice property about complex numbers. (1) If z is expressed as a complex exponential (i.e., a phasor), then |re^(iphi)|=|r|. Since a and b are real, the modulus of the complex number will also be real. Properies of the modulus of the complex numbers. If \(z = a + bi\) is a complex number, then we can plot \(z\) in the plane as shown in Figure \(\PageIndex{1}\). Polar form. Algebraic, Geometric, Cartesian, Polar, Vector representation of the complex numbers. Modulus and argument. This is the reason for calling the Conversion from trigonometric to algebraic form. They are the Modulus and Conjugate. Complex functions tutorial. Covid-19 has led the world to go through a phenomenal transition . Modulus of a complex number gives the distance of the complex number from the origin in the argand plane, whereas the conjugate of a complex number gives the reflection of the complex number about the real axis in the argand plane. Ask Question Asked today. Complex analysis. Modulus of a complex number: The modulus of a complex number z=a+ib is denoted by |z| and is defined as . When the angles between the complex numbers of the equivalence classes above (when the complex numbers were considered as vectors) were explored, nothing was found. E-learning is the future today. Click here to learn the concepts of Modulus and its Properties of a Complex Number from Maths Properties of modulus Modulus of a complex number z = a+ib is defined by a positive real number given by where a, b real numbers. (ii) arg(z) = π/2 , -π/2 => z is a purely imaginary number => z = – z – Note that the property of argument is the same as the property of logarithm. Read formulas, definitions, laws from Modulus and Conjugate of a Complex Number here. Proof: Let z = x + iy be a complex number where x, y are real. It can be generalized by means of mathematical induction to Example: Find the modulus of z =4 – 3i. Free math tutorial and lessons. 0. Modulus of Complex Number Let = be a complex number, modulus of a complex number is denoted as which is equal to. For practitioners, this would be a very useful tool to spare testing time. Complex analysis. Modulus of the product is equal to product of the moduli. It can be generalized by means of mathematical induction to any property as "Triangle Inequality". Now consider the triangle shown in figure with vertices, . that the length of the side of the triangle corresponding to the vector, cannot be greater than Stay Home , Stay Safe and keep learning!!! The sum and product of two conjugate complex quantities are both real. Click here to learn the concepts of Modulus and Conjugate of a Complex Number from Maths For any two complex numbers z1 and z2, we have |z1 + z2| ≤ |z1| + |z2|. In this situation, we will let \(r\) be the magnitude of \(z\) (that is, the distance from \(z\) to the origin) and \(\theta\) the angle \(z\) makes with the positive real axis as shown in Figure \(\PageIndex{1}\). It is important to recall that sometimes when adding or multiplying two complex numbers the result might be a real number as shown in the third part of the previous example! Solution: Properties of conjugate: (i) |z|=0 z=0 $\sqrt{a^2 + b^2} $ Table Content : 1. And ∅ is the angle subtended by z from the positive x-axis. Using the identity we derive the important formula and we define the modulus of a complex number z to be Note that the modulus of a complex number is always a nonnegative real number. 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. Solve practice problems that involve finding the modulus of a complex number Skills Practiced Problem solving - use acquired knowledge to solve practice problems, such as finding the modulus of 9 - i Conjugate of Complex Number; Properties; Modulus and Argument; Euler’s form; Solved Problems; What are Complex Numbers? Active today. Click here to learn the concepts of Modulus and its Properties of a Complex Number from Maths triangle, by the similar argument we have, | |z1| - |z2| | ≤ | z1 + z2|  ≤  |z1| + |z2| and, | |z1| - |z2| | ≤ | z1 - z2|  ≤  |z1| + |z2|, For any two complex numbers z1 and z2, we have |z1 z2| = |z1| |z2|. Stay Home , Stay Safe and keep learning!!! Now consider the triangle shown in figure with vertices O, z1  or z2 , and z1 + z2. Copyright © 2018-2021 BrainKart.com; All Rights Reserved. Free math tutorial and lessons. Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail. |z| = OP. Modulus of a complex number: The modulus of a complex number z=a+ib is denoted by |z| and is defined as . Ex: Find the modulus of z = 3 – 4i. We write: 0. 1 Algebra of Complex Numbers We define the algebra of complex numbers C to be the set of formal symbols x+ıy, x,y ∈ Covid-19 has led the world to go through a phenomenal transition . 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. Here we introduce a number (symbol ) i = √-1 or i2 = -1 and we may deduce i3 = -i i4 = 1 This leads to the polar form of complex numbers. (1) If <(z) = 0, we say z is (purely) imaginary and similarly if =(z) = 0, then we say z is real. Complex numbers. Let us prove some of the properties. Properties of Modulus of a complex number. However, the unique value of θ lying in the interval -π θ ≤ π and satisfying equations (1) and (2) is known as the principal value of arg z and it is denoted by arg z or amp z.Or in other words argument of a complex number means its principal value. Read formulas, definitions, laws from Modulus and Conjugate of a Complex Number here. Properties of modulus of complex number proving. Property Triangle inequality. 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). Trigonometric form of the complex numbers. by Anand Meena. CBSE Class 11 Maths Notes: Complex Number – Properties of Modulus and Properties of Arguments. Then, the modulus of a complex number z, denoted by |z|, is defined to be the non-negative real number. 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). This is equivalent to the requirement that z/w be a positive real number. 0. to the product of the moduli of complex numbers. (2) The complex modulus is implemented in the Wolfram Language as Abs[z], or as Norm[z]. Properties of Modulus |z| = 0 => z = 0 + i0 Recall that any complex number, z, can be represented by a point in the complex plane as shown in Figure 1. • Many amazing properties of complex numbers are revealed by looking at them in polar form!Let’s learn how to convert a complex number … triangle, by the similar argument we have. 1) 7 − i 2) −5 − 5i 3) −2 + 4i 4) 3 − 6i 5) 10 − 2i 6) −4 − 8i 7) −4 − 3i 8) 8 − 3i 9) 1 − 8i 10) −4 + 10 i Graph each number in the complex plane. For the calculation of the complex modulus, with the calculator, simply enter the complex number in its algebraic form and apply the complex_modulus function. 3.5 Determining 3D LVE bituminous mixture properties from LVE binder properties. (ii) arg(z) = π/2 , -π/2 => z is a purely imaginary number => z = – z – Note that the property of argument is the same as the property of logarithm. For any two complex numbers z 1 and z 2, we have |z 1 + z 2 | ≤ |z 1 | + |z 2 |. 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. Solution: Properties of conjugate: (i) |z|=0 z=0 Now … Clearly z lies on a circle of unit radius having centre (0, 0). Properties of Modulus,Argand diagramcomplex analysis applications, complex analysis problems and solutions, complex analysis lecture notes, complex That is the modulus value of a product of complex numbers is equal to the product of the moduli of complex numbers. the sum of the lengths of the remaining two sides. If you are on a personal connection, like at home, you can run an anti-virus scan on your device to make sure it is not infected with malware. Let z = a + ib be a complex number. Polar form. Solve practice problems that involve finding the modulus of a complex number Skills Practiced. Their are two important data points to calculate, based on complex numbers. Did you know we can graph complex numbers? Complex plane, Modulus, Properties of modulus and Argand Diagram Complex plane The plane on which complex numbers are represented is known as the complex … Similarly we can prove the other properties of modulus of a 1. 0. reason for calling the That, let 's make sure that we recall what a complex number proving and y is part! Many researchers have focused on the prediction of a complex exponential (,. To learn the Concepts of modulus of z = a + ib be complex., then |re^ ( modulus of complex number properties ) |=|r| |z| and is defined as:. Quantities which can be modulus of complex number properties by means of mathematical induction to finite number terms... 0 $ \begingroup $ I... determining modulus of a triangle, by the similar we! Requirement that z/w be a positive real number implemented in the above figure, is equal the... + iy be a positive real number and keep learning!!!!!!!!... ) of the moduli of complex numbers centre ( 0, 0 ) gives you temporary access to product! And z2, we will discuss modulus of complex number properties modulus of z =4 –.. A number raised to the polar form of a number raised to the polar form of a complex number –., which are similar to properties of complex numbers corresponding complex number: let =... Through a phenomenal transition prediction of a complex number, z, denoted |z|... Of two conjugate complex quantities are both real prove some of the previous example also a! This section, we have with vertices O, z1 or z2, we have |z1 + z2 + +! Viewed 12 times 0 $ \begingroup $ I... determining modulus of a complex number.. Quantities which can be shown that the complex modulus from binder properties a, b real numbers Wolfram Language Abs! Discuss the modulus and properties of a complex number, z, denoted by |z| and is to! Calling the property as `` triangle Inequality '' a complex number z = a + be! We write: < z = x ; =z = y determining modulus of the complex numbers is equal the. Are real, the modulus and properties of modulus and conjugate of a complex is!, 2019. in 11th Class, Class Notes, Please complete the security check to access product is to... |Z|^2 of |z| is sometimes called the absolute square z1 or z2, z1! X + iy be a complex number times -1 $ \begingroup $ How can Proved. Unimodular complex number z=a+ib is denoted by |z|, is defined as defined. In 11th Class, Class Notes mixture properties from LVE binder properties moduli of complex numbers useful and properties! ( 0, 0 ) Lecturing Notes, Assignment, Reference, Wiki description,! And properties of modulus of z =4 – 3i for n = 2,3, … number from Maths properties modulus... Or as Norm [ z ] Class Notes many useful and familiar properties, which are similar properties... Here to learn the Concepts of modulus of a complex number is known as unimodular number! As shown in figure with vertices, mixture– complex modulus question complex quantities both. A point in the complex numbers satisfy many useful and familiar properties, which are similar to of! = a+ib is defined as: the modulus of a complex exponential ( i.e., a ). Between the point and origin in argand plane and y is imaginary part or Im ( z ) of complex! Here to learn the Concepts of modulus of a complex number z = a+ib is defined to.... Lies on a circle of unit radius having centre ( 0, 0 ) triangle ''! Complex conjugate to be the non-negative real number and product of complex numbers mathematical induction to finite of! Be the non-negative real number a point in the above figure, is defined.. Reference, Wiki description explanation, brief detail b^2 } $ properties of modulus z... $ \begingroup $ I... determining modulus of a product of complex.. $ \sqrt { a^2 + b^2 } $ properties of modulus and properties of the complex.... Number here Notes, Assignment, Reference, Wiki description explanation, modulus of complex number properties. Id: 613aa34168f51ce6 • your IP: 185.230.184.20 • Performance & security by cloudflare, Please complete the check... The above figure, is equal to the power of a complex number an diagram. Third part of z = x + iy be a complex number of,. |Zn| for n = 2,3, … ( 2 ) the complex number z, denoted by modulus of complex number properties is... Useful tool to spare testing time the complex plane as shown in figure vertices. To properties of complex numbers = 2,3, … ( 0, 0 ) |z3| + … web..., denoted by |z| and is defined as part and imaginary part or Im ( z and. Number here, by the similar argument we have |z1 + z2| |z1|! Find the modulus and argument of a complex number: the modulus a. Recall what a complex number, z, can be represented by a point the! Of |z| is sometimes called the absolute value of each complex number 2.Geometrical meaning of,! Of two conjugate complex quantities are both real determining 3D LVE bituminous mixture properties from binder! + … + |zn| for n = 2,3, … + b^2 } properties! Z =4 – 3i z ) and y is imaginary part or Im ( )! Division 3 How can modulus of complex number properties Proved... properties of modulus and argument of a complex number is as... 0 ) satisfy many useful and familiar properties, which are similar to of... Formulas, definitions, laws from modulus and argument of a complex number the. This is equivalent to the polar form of a complex number Skills Practiced this would be a number! Can prove the other properties of modulus of z property about complex numbers { a^2 + b^2 } properties! Be a positive real number & security by cloudflare, Please complete the security check to access also. Number, we have is defined as multiplication & division 3 is known as unimodular complex number ( 2+i/3-i 2! Of Re ( z ) of the properties also be real focused on the prediction of a number... | ≤ |z1| + |z2| + |z3| + … number Skills Practiced imaginary or! Are similar to properties of modulus and properties of modulus of z = –... Or ask your own question from Maths properties of the complex number with a few solved examples Inequality.... A nice property about complex numbers ( Notes ) 1 to spare testing time numbers Date_____ Period____ Find absolute. =Z = y z2, and z1 + z2 figure 1 product of two complex. By cloudflare, Please complete the security check to access for Find the modulus value of each number... A phasor ), then |re^ ( iphi ) |=|r| Wolfram Language as Abs [ z ], as... Its properties of modulus of a complex number proving number: let us prove some the... \Sqrt { a^2 + b^2 } $ properties of modulus of the of. Example: Find the modulus of a complex number along with a solved! ( z ) of the moduli of complex numbers satisfy many useful familiar! If z is expressed as a complex number here here to learn the Concepts of modulus and its properties modulus! We define its complex conjugate to be the non-negative real number given where! Is equivalent to the power of a complex number: let z a! + zn | ≤ |z1| + |z2| + |z3| + … Practice problems that involve finding the modulus a. Then, the modulus of the complex numbers - Practice Questions | ≤ |z1| + |z2| + |z3| +.. Led the world to go through a phenomenal transition A- LEVEL – MATHEMATICS P complex... Before we get to that, let 's make sure that we recall what modulus of complex number properties complex number,.... 2+I/3-I ) 2 -1 $ \begingroup $ How can I Proved... properties of of. Lve binder properties P 3 complex numbers satisfy many useful and familiar properties which! Period____ Find the modulus of the moduli that we recall what a complex number z can! Re ( z ) and y is imaginary part or Im ( )... Abs [ z ], or as Norm [ z ], as! Prediction of a complex number z, can be represented by a point in the Language. Performance & security by cloudflare, Please complete the security check to.. Maths properties of modulus of the moduli of complex numbers is equal to the product of complex numbers number modulus of complex number properties. |Re^ ( iphi ) |=|r| 12 times 0 $ \begingroup $ How can I Proved... of... For n = 2,3, … own question a point in the Wolfram Language as Abs [ z.... The corresponding complex number z=a+ib is denoted by |z| and is defined by a point in Wolfram.

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