If z is a complex number, show that the sum of z and its conjugate is a real number.

Q: Find all complex numbers z satisfying 4 = -81

A: Find all the complex numbers z satisfying z4= -81

Q: Find the value of real number if the complex number is 8 + 9j = 0

A: Given complex number is 8+9j=0 Real value of the given complex number is 8. Real number can be…

Q: Represent the complex number geometrically.(1 + 2i)2

A: Represent the complex number geometrically.(1 + 2i)2

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

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Q: Determine the absolute value of the complex number z = -1 - 2i.

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Q: What happens when a complex number is multiplied by its conjugate? It turns into a real number…

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

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Q: 2. If a and b are complex numbers, and z is a complex number that obeys |z| = 1, show that az + b =…

A: Given that a,b and z all are complex numbers.   let, a=c+idb=r+is  and z=x+iy, where c,d,r,s,x,y all…

Q: In the complex number system, find the complex zeros off(x) = x2 + 2x + 2

A: Given : fx = x2+2x+2

Q: 2+7i 3-3i

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Q: Determine whether the given statement "In the complex number system, x2 + y2 (the sum of two…

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

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Q: Prove:The addition of complex number is associative.

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Q: Consider the complex number z = 10250°. What is the smallest positive integer n such that z" is a…

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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: Find the conjugate, argument and modulus of the complex number Z=1+i3

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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: Pick two complex numbers (like a + bi and c + di) and consider if the following statement is True or…

A: Set of pure imaginary numbers

Q: The conjugate of complex number z satisfy: Oz+z=1 Oz-z=0 Oz-z=2ilmz | Oz+z=2Rez

A: let's consider a complex number , z=Re(z)+i Img(z)  i.e z=x+iy

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: 3. If z, and zz are two complex numbers, prove that

A: Let's find.

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

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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: 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: Which complex numbers z satisfy z = z? Explain.

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Q: bers are conjugate. If z, and z2 are two non-zero complex numbers st jal arg (2) ) - arg (z2 ).

A: Since we know that, | Z1 + Z2 | = |Z1| + |Z2| only when the the point Z1 , Z2 and origin are…

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

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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: 2. Identify the conjugate of each complex number, then plot both complex numbers and their…

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Q: A complex number can be a real number, an imaginary number, or the _____ of a realnumber and an…

A: Consider a complex number: z=x+iy……1   where: x=Real numbery=Real numberi=-1iy=Imaginary number

Q: Describe all complex numbers z such that V3

A: Given that, zz+zz=3 Take z=x+iy So, z2+z2zz=3x2-y2+2ixy+x2-y2-2ixyx2+y2=3

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: Given the complex number z such that z-4i = a+3zi , find the value of a if z is purely imaginary.

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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: 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: 2. It is easier to find the power or the root of a complex number if the numnber is written in form.

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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: 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: Show that the solution of the equation  4x2 + 24x + 37 = 0 are complex conjugates of each other.

A: Given quadratic equation: 4x2+24x+37=0

Q: Find all complex numbers z satisfying the equation 2|=|* + =' -5+iV3 =0.

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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: Find all complex numbers z satisfying = -i

A: i = -1(i)2 = -1

Q: Show that if z1 and z2 are complex numbers then (z1/22) = Zī/22.

A: Given, z1 and z2 are two complex numbers and we have to show,

Q: a. -16 + V-7 b. 47 ,601

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