With addition as given of (complex numbers) prove the following statement:-   The additive inverse of (a, b) ϵ C is (-a,-b) ϵ C.

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

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

Q: Determine the argument and the modulus of the given complex (-2-4i)(5i) number: (3+8i)

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Q: The division of a complex number z, by a non - zero complex number z, is defined as the…

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Q: If z1,72,., are complex numbers such that = =.==,| =1,Prove that 1. 1 1

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Q: N=-7+1li %3D

A: Given complex number Z=-7+11i

Q: 38. Let a be a complex number that is algebraic over Q and let r be a rational number. Show that a'…

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Q: express  in the form z = x +yi the complex number z whose modulus and argument are given.  |z| =…

A: z= x+iyz=12,    arg(2) =-π3modulus of z =z = x2+y2 =12⇒x2+y2 =12     -(i)

Q: Express each of the folowing complex numbers in the form x + yi: (3 - 4)5 (4 + 37)

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Q: Let a be a complex number that is algebraic over Q. Show that√a is algebraic over Q.

A: Consider the polynomial: Clearly,  Hence, we have following: Let f(x) be the minimal polynomial…

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

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Q: Suppose the product z1z2 of two complex numbers is a nonzero real constant. Sho 2 kz1, where k is a…

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Q: Recall that the symbol -z represnets the complex conjugate of z. If z = a + bi and w = c - di, show…

A: Recall that the symbol z¯ represents the complex conjugate of z. If z = a + bi and w = c - di, show…

Q: The number a - bi is called the complex __________ of a + bi. The product of a + biand a - bi will…

A: Complex number  z=a+bi Conjugate of complex number z=a-bi And product of (a+bi)(a-bi) is…

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: If z=3-4i then express these following complex numbers in the form of a+ib: (a) (=+1)(=+i) 2z (b)

A: If z = a + ib and w = c + id then z+w = (a+ib) + (c+id)       = (a+c) + i(b+d)

Q: Let a be a complex number that is algebraic over Q and let r be arational number. Show that ar is…

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Q: p and q are complex numbers such that pl = 7√3 and lp + ql = 12√2. - On what interval must q fall…

A: Use  Cauchy inequalities and solve the question.

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 sum of z and its conjugate is a real number.

A: Given z is a complex number. Let z=x+iy We know that conjugate of a complex number is given by…

Q: If z1:2., are complex numbers such that =|= =. = |=,| =1, Prove that

A: To prove: z1+z2+…+zn=1z1+1z2+…+1zn Given: z1=z2=…=zn=1 Since, (1) zz¯=|z|2 (2) z=z¯

Q: Let C be a fixed nonzero complex number and note that i is a multiple valued, what restriction must…

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Q: If 2 complex numbers z, and z, have the same modulus then they must be equal. Select one: O True O…

A: Let two complex numbers 2+i and  z1=2+i=22+12=5z2=2-i=22-12=5

Q: Is it true that the quotient of two complex number is obtained by dividing their moduli and…

A: Given question says "Is it true that the quotient of two complex number is obtained by dividing…

Q: If 4,,2 are two complex numbers then : ollz|-|z||>|z1-z |z,+z2ls\z]\+|z3l |

A: If z1,z2 are two complex numbers, then (a) z1+z2=z1+z2 (b) z1-z2>z1-z2 (c) z1+z2≤z1+z2 (d)…

Q: Given that Z1 and Z2 are complex numbers, show geometrically or otherwise that |Z1| - |Z2| is less…

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Q: . For a sequence of complex numbers (zn), show that lim,-0o Zn = 0 if and only if lim,-o IZnl = 0.

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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: Find non-zero real number values of a, b, c, and d such that the sum (a + bi) + (c + di) is... -a…

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

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Q: Recall that the complex conjugate of a = a + bi E C is a = a - bi. Show that aßaß for every a, B e…

A: Given;  Alpha = a +b i.

Q: If |x = 3i| <5, then prove that the complex number z also satisfies the inequality i(z+1)+1| <25.

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Q: Express in exponential form: –8 + 8i/3. 31 Represent graphically the set of values of z for which <…

A: -8+8i3

Q: For every real number a > 0, Find all complex numbers z, sati sfying the equati on z|-|+ a +i= 0.

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Q: B- The Modulus and Argument of a Complex Number Z is (8, 90°) respectively find Vz4

A: We will find out the required value.

Q: he division of a complex number z, by a non - zero complex number z, is defined s the multiplicati…

A: The given statement is

Q: Given that A is an element of complex numbers, there exists a B that is equal or greater than 0, and…

A: As there is no A in the expressed I assume that is D.

Q: Find the modulus, Izl, and argument, 8, of the complex number. -1+i One charaes 4.6

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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: 10. Use i = (0, 1) and y = (y, 0) to verify that -(iy) = (-i)y. Thus show that the addi- tive…

A: i=0 ,1y=y ,0 verify :   -iy = -iy  To find additive inverse of z=x+iy

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: 17. The argument of the complex number z. where com CO is 1- i 13

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Q: 8. The number of roots of a complex is equal to the index of the radical sign True O False

A: The radical operation can also be performed for a given complex number. The nth radical of a complex…

Q: What is the argument of the complex number z= 1?

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Q: p and q are complex numbers such that |a| = 4√5 and lp + ql = 9√/2. - On what interval must lp fall…

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Q: (a) Given complex numbers, z, =-V3+2i and z, =-/ /2i. (i) Express z, and z, in exponential form.…

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Q: 3.5. If |2| = 1, then show that az + b 1 bz +a for all complex numbers a and b. ||

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Q: Enter in form a + bi the complex number z which has an absolute value 2 and argument 2/3 π.

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Q: The division of a complex number z, by anon- zero complex number z, is defined as the multiplication…

A: Let, Z1= a1+ib1 and, Z2=a2+ib2