Pick two complex numbers (like a + bi and c + di) and consider if the following statement is True or False. "The set of pure imaginary numbers (like bi) are closed under subtraction."

Q: Suppose Zi and Z, are two given complex numbers. Construct graphically a. 3Z1 – 2Z2 b. (1/2)Z, +…

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Q: Define the Complex Number by using example inside this arc: (7- 3i )

A: Consider the given example of the complex number. C=7-3i The general form of the complex number is…

Q: Express the complex number in trigonometric form with 0 ≤ θ < 2π.-4 + 2i

A: z=-4+2i Now, r=-42+22=20=25 & θ=tan-1yx=tan-12-4=π-tan-112     ∵Given complex number lies in…

Q: Suppose z is a complex number that possesses a fourth root w that is neither real nor pure…

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Q: Write the complex number in polar form with argument 0 between 0 and 2n. 7 - 7i

A: The expression of the polar form is given by z=rcosθ+isinθ Given complex number is (x+iy)=7-7i. The…

Q: Express the complex number in exact simplified rectangular form.

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Q: Express the complex number in trigonometric form with 0 ≤ θ < 2π.-10 + 10i

A: We have given  z = -10 + 10i    Now, z can written as reiθ    r =  (-10)2+ 102              [ As if…

Q: Find the midpoint of the line segment joining the points corresponding to the complex numbers in the…

A: The point corresponding to -1 - (1/4)i is= (-1, -1/4)  The point corresponding to 3/2 + (3/4)i is=…

Q: How may the following complex numbers be obtained fromz = x + iy geometrically? Sketch. 1 / z

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Q: When finding the n-th root of a complex number in polar form, why are there n-1 roots (solutions)?

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Q: The distance between two points in the complex plane is the ________ of the difference of the…

A: Let (a,b) and (m,n) are two points in the complex plane. Then the difference of the complex numbers…

Q: 2) Suppose 1 and w are complex numbers, show that Aw =

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Q: This complex number is pure real? 1-2i/i and why?

A: For a complex number, z=x+iy, where x is the real part, y is the imaginary part. The complex number:…

Q: Find the distance between the complex numbers in the complex plane. 1 + 4i, -1 + 7i

A: The distance between the two complex numbers, z1=a+bi and z2=c+di in the complex plane is  In the…

Q: Show that if z1 and z2 are complex numbers, then zi + z2 = Zī + Z2.

A: Suppose the two complex numbers are z1 and z2 such that where a,c are real parts and b,d are…

Q: :The sum and product of two complex numbers are real if and only if they are conjugate of each…

A: Solve by complex no.

Q: True or False The polar form of a nonzero complex number is unique.

A: The polar form of a nonzero complex number is unique.

Q: Use De Moivre’s theorem to change the given complex number to the form a + bi, where a and b are…

A: 1-i10 Consider, z=1-i Now, z=12+-12=2 & θ=tan-1yx=tan-1-11=2π-tan-11     ∵The complex number 1-i…

Q: The 5th roots of a complex number would have arguments that differ from each otherby _______…

A: Given :5th roots of a complex numberzn=rncosα+isinαα=θ+2πkn

Q: If z is a complex number then the sum of z and its conjugate is an imaginary number. O true false

A: Let z be a complex number given by, z=a+ib where a is the real part and b is the complex part of z.

Q: If z is a complex number, show that the product of z and its conjugate is a realnumber.

A: Let z=a+ib be any complex number where a and b are real number. Then conjugate of z is z¯=a-ib

Q: Find the distance between the complex numbers in the complex plane. -з - 6і, 6 + 2i

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Q: Is the sum of a complex number and its complex conjugate a real number?

A: Consider the given statement as "Is the sum of a complex number and its complex conjugate a real…

Q: Evaluate b. Topic under Cartesian Complex Numbers

A: Given, -1j7

Q: Completely factor over the complex numbers (2x-5) (x+1) = 2

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Q: Present complex number z in algebraic form =* + 23D 35 1-3i

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Q: What is the complex conjugate of a complex number a  bi?

A: The given complex number is a+bi. Here a and  b  are real numbers. The real part is a and the…

Q: Graph the complex number on the complex plane. -2 + 3i 4- 3- 2- -3 -2 -1 -2- -3- 4. 3, 1. 4.

A: Topic = Complex number

Q: Write the complex number –2 in polar form with argument 0 between 0 and 27.

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Q: If a and b are complex numbers, prove that |a+b|<\a|+\b| .

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Q: Show that the conjugate of the sum (product, or quotient) of two complex numbers, z1 and z2 , is the…

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Q: 1. Write the complex number 2+ iv3 in the form of reio . (2 Marks)

A: Here we first find modulus and arg then convert this into polar form

Q: Express the complex number in trigonometric form with 0 ≤ θ &lt; 2π.2√3 + 2i

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Q: . Ifa complex number z has modulus 2 and argument , then

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Q: Express the complex number in trigonometric form with 0 ≤ θ &lt; 2π.2 - 2√3 i

A: Given complex number is: z=2-23i Let us consider, r cos θ=2r sin θ=-23 Then we get,…

Q: Find the difference of the complex numbers in the complex plane. (4 + 2i) - (5 + 4i)

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Q: Show that the conjugate of the sum (product, or quotient) of two complex numbers, z1 and z2 , is the…

A: Let z1 = a+ib and z2 = c+id be two complex numbers. Then we get, Conjugate of z1 is = a-ib Conjugate…

Q: Use de Moivre’s formula to find the Cartesian form of the complex number  (-2 + 2√3i)7

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Q: Every complex number is either a real number or an imaginary number.   True or False

A: Given statement is every complex number is either a real number or an imaginary number. To determine…

Q: The variable z is often used to denote a complex number and 7 is used to denote its conjugate. If z…

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Q: To divide two complex numbers in trigonometric form, ________ their moduli and_____ their arguments.

A: If z1=r1cosθ1+isinθ1=r1cisθ1 andz2=r2cosθ2+isinθ2=r2cisθ2 are two complex numbers intrigonometric…

Q: Suppose Z, and Z, are two given complex numbers. Construct graphically a. 3Z1- 2Z2 b. (1/2)Z,…

A: Suppose complex number Z1=a1+ib1=2+3i and Z2=a2+ib2=-1-2i (a) given complex number 3Z1-2Z2…

Q: Express the complex number in trigonometric form with 0 ≤ θ &lt; 2π.-6i

A: Express the complex number in trigonometric form with 0 ≤ θ &lt; 2π.z=-6i=0-6i

Q: The sum and product of two complex numbers are real if and only if they are conjugate of each other

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Q: Express the complex number in trigonometric form with 0 ≤ θ &lt; 2π.-5 - 5√3i

A: Express the complex number in trigonometric form with 0 ≤ θ &lt; 2π.-5 - 5√3i

Q: Write the complex number in polar form with the argument detha in domain [0. 360 degrees) z=9

A: For a Complex number, Z=a + bi the polar form is given by:

Q: Find all complex numbers z satisfying 23 = 27 i

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Q: Express the complex number in trigonometric form with 0 ≤ θ &lt; 2π.-2 - 2i

A: Solution: Consider the given complex number is: z = -2 - 2i The trigonometric form of a complex…