Prove, using complex numbers, that you cannot draw an equilateral triangle in the plane with all three vertices with rational coordinates

Q: when dividing complex numbers you must find the complex conjugate of the ________

A: When dividing the complex numbers, we must find the complex conjugate.

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

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Q: My work with complex numbers verified that the only possible cube root of 8 is 2.Determine whether…

A: Concept: The calculus helps in understanding the changes between values that are related by a…

Q: Does the Theorem (Roots) have to be used to find the nth roots of a complex number, or can other…

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Q: Find all complex solutions of 3x+x+4%3D0. (If there is more than one solution, separate them with…

A: We have to find complex solutions of given equation.

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

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Q: solve the complex number using De Moivre's Theorem 483 ( - V3 2

A: The general form of a complex number is given by z=a+ib, where a,b are two real numbers and i=-1.…

Q: Find the cube root of unity using De-Moivre's theorem and apply the same theorem to find the square…

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Q: Find all complex solutions of 2x +3x+6= 0. (If there is more than one solution, separate them with…

A: use quadratic formula for finding complex solution.

Q: A point in the complex plane with both coordinates rational is called a rational point. The set of…

A: The set of rational points on the unit circle will be denoted by Cℚ=a+bi ∈ℂ |a, b∈ℝ and a2+b2=1. The…

Q: A point in the complex plane with both coordinates rational is called a rational point. The set of…

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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: Prove:The addition of complex number is associative.

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Q: Consider a complex number z such that Arg (z) = - 120º. Then, the image that represents all the…

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Q: Use DeMoivre’s theorem to find (–8)1/3. Express your answer in complex form.

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Q: What is the complex conjugate of -6 + 4i?

A: Complex number is -6+4ithen complex conjugate is -6-4i

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: 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: Write down the definition of the complex conjugate, ž, of z = a + ib where a, b are real number…

A: Complex conjugate of a complex number z is defined as a complex number which has same real part as z…

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: A point in the complex plane with both coordinates rational is called a rational point. The set of…

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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: 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: Write down the definition of the complex conjugate, z, of z = a + ib where a, b are real numbers.…

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

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Q: To convert a complex number z = x + yi from standard form into trigono metricform, use the…

A: The modulus of a complex number z=x+yi is r=x2+y2 & the amplitude or argument is,…

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

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

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Q: Explain geometrically or algebraically how magnitude of a complex number is equivalent to…

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Q: Show that multiplying a complex number by i rotates it through 90° without changing its length. Find…

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Q: Let z be a complex number such that z, z3 and z5 form an equilateral triangle on the complex plane.…

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Q: Use DeMoivre’s theorem to find z4, if z is 1 + 1i. Express your answer in complex form

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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: Find all complex solutions of 5x+5x+3=0. (If there is more than one solution, separate them with…

A: Given that: 5x2+5x+3=0

Q: What are all the cube roots of -1 in the complex plane?

A: answer is in next step

Q: Find and graph all the roots in the complex plane. (3 + i4)5

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Q: With addition and multiplication as in definition of complex numbers, prove that the right…

A: According to the question, Let C represents the set of all ordered pairs a,b of real numbers a and…

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: 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 the cube roots of the complex number. Express in standard form

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Q: Find all complex solutions of 2x + 3x+ 6 = 0. (If there is more than one solution, separate them…

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Q: Determine the complex number shown in the complex plane.

A: Complex plane: - In a complex projective plane, the complex plane (also known as the Gauss plane or…

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: Find and graph all the roots in the complex plane. (3 + i4)

A: this is a problem of complex analysis.