MTH632 Solved Exercises

Q5. Prove that multiplication of complex numbers is commutative.

Proof:

Let and z z be two complex numbers such that z x iy and z x iy where x x y y 

z z x iy x iy x x iy iy x iy x x x iy iy x iy iy x x i x y i y x i i y y

x x i y x i x y i i y y

As real numbers are commutative

x x i x y i y x i i y y x x iy iy x iy x iy x iy z z

Hence complex numbers are commutative.

Q6. Verify the associative law for addition of complex numbers

Proof:

Let , z and z z be three complex numbers such that z x iy z x iy and z x iy where x x x y y y 

z z z x iy x iy x iy x x i y y x iy x x i y y x iy

As real numbers are commutative x x x i y y iy x x x i y y y x x x i y y y x x x iy i y y x iy x x i y y z z z

Q7. Ans:

Q8. a):

Write ( , ) x y u v x y and point out how it follows that the complex number 0=(0,0) is unique as an additive identity.

Answer: x y u v x y x iy u iv x iy x u i y v x iy

Comparing the real and imaginary numbers on both sides, x u x and y v y

x u x and y v y u x x and v y y u and v

Hence (0,0) u v =

Suppose there is another additive ( c id identity then

x y c d x y x iy c i x iy x c i y d x iy

x c x and y d y c x x and d y y c and d Hence ( , ) (0,0) c d u v proved that additive identity is a unique number.

Q9. use -1=(-1,0) and z=(x,y) to show that (-1)z=-z

Answer: Given that 1 ( 1,0) i and z x y x iy

( 1) (-1,0)( , ) (-1 0)( (-1)( 0( L H S z x y i x iy x iy i x iy x iy i x iy z R H S

Q10. (0,1)

( ,0) i and y y to verify that –(iy)=(-i)y. Thus show that additive inverse of a complex number z=x+iy can be written –z=-x-iy without ambiguity.

Proof:

(0,1) ( ,0) i and y y

- (0,1)( ,0) - (0 0) - (0)( 0) 0) - 0 -0 - (0-0) 0) L H S iy y i y i y i i y i i iy i y i y R H S

Additive inverse gives the zero number in complex numbers If z x iy then let w u iv be the additive inverse of the z , so

z w i x iy u iv i x u i y v i