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Prove that: Re z= (z+bar(z))/2, Im z= (...

Prove that:
`R_e z= (z+bar(z))/2, I_m z= (z-bar(z))/(2i)`.

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MODERN PUBLICATION-COMPLEX NUMBERS-EXERCISE
  1. Prove that : z= bar(z) iff z is real.

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  2. Prove that : z= -bar(z) iff z is either zero or purely imaginary.

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  3. Prove that: Re z= (z+bar(z))/2, Im z= (z-bar(z))/(2i).

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  4. Prove that : the sum of a complex number and its conjugate is real.

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  5. Prove that : the product of a complex number and its conjugate is real...

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  6. Give an example to show that subtraction of complex numbers is not com...

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  7. Find te sum and product of the complex numbers z1=-sqrt3+sqrt-2, z2=2s...

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  8. If z(1), z(2) are 1-i, -2 + 4i respectively, find I(m)((z(1)z(2))/(bar...

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  9. Find the values of x and y if : (x+iy)(1+i)=1-i where i=sqrt(-1)

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  10. Find the values of x and y if : ((1+i)x-2i)/(3+i)+((2-3i)y +i)/(3-i)...

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  11. If 1 +4 sqrt3i = (a+ ib)^2, prove that : a^2-b^2=1 and ab=2 sqrt3.

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  12. Show that if a, b,c, d in R, bar((a + ib) (c+ id)) = (a-ib)(c- id).

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  13. If x+iy=(a+ib)/(a-ib), then prove that x^2+y^2=1

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  14. If (a+i)^2/((2a-i))= p + iq, prove that : p^2+q^2 =(a^2+1)^2 /(4a^2+1)...

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  15. If x-iy= sqrt((a-ib)/(c-id)) prove that (x^2+y^2)^2= (a^2+b^2)/(c^2+d^...

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  16. If x-iy= sqrt((a-ib)/(c-id)), prove that (x^2+y^2)^2= (a^2+b^2)/(c^2+d...

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  17. If (x + iy)^3 = u + iv, then show that : u/x+v/y=4 (x^2-y^2).

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  18. Define addition and multiplication of two complex numbers z1 and z2. H...

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  19. Define addition and multiplication of two complex numbers z1 and z2. H...

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  20. Define addition and multiplication of two complex numbers z1 and z2. H...

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