A complex number is a part of a number system that includes a real unit and an imaginary unit. All the statements are true except C and E.
<h3>What is a Complex Number?</h3>
A complex number is a part of a number system that includes the real numbers as well as a particular element labelled i, sometimes known as the imaginary unit, and which obeys the equation i² = 1.
As it is given that x=(a+bi), y=(c+di), and z=(f+gi), therefore, we need to plug in the values to check which all are true.
A.) x+y=y+x
![x+y=y+x\\a+bi+c+di=c+di+a+bi](https://tex.z-dn.net/?f=x%2By%3Dy%2Bx%5C%5Ca%2Bbi%2Bc%2Bdi%3Dc%2Bdi%2Ba%2Bbi)
As the two sides of the equation are equal, the equation is true.
B.) (x × y) × z = x × (y × z)
![(x \times y) \times z = x \times (y \times z)\\\\\ [ (a+bi) \times (c+di) ]\times (f+gi) = (a+bi) \times [(c+di) \times (f+gi)]\\\\(ac+adi+bci-bd) \times (f+gi)= (a+bi) \times (cf+cgi+fdi-dg)](https://tex.z-dn.net/?f=%28x%20%5Ctimes%20y%29%20%5Ctimes%20%20z%20%3D%20x%20%5Ctimes%20%20%28y%20%5Ctimes%20%20z%29%5C%5C%5C%5C%5C%20%5B%20%28a%2Bbi%29%20%20%5Ctimes%20%28c%2Bdi%29%20%5D%5Ctimes%20%28f%2Bgi%29%20%3D%20%28a%2Bbi%29%20%20%5Ctimes%20%5B%28c%2Bdi%29%20%5Ctimes%20%28f%2Bgi%29%5D%5C%5C%5C%5C%28ac%2Badi%2Bbci-bd%29%20%5Ctimes%20%28f%2Bgi%29%3D%20%20%28a%2Bbi%29%20%20%5Ctimes%20%28cf%2Bcgi%2Bfdi-dg%29)
![(ac+adi+bci-bd) \times (f+gi)= (a+bi) \times (cf+cgi+fdi-dg)\\\\\\acf+adfi+bcfi-bdf+acgi-adg-bcg-bdgi = acf+acgi+afdi-adg + bcfi-bcg-bfd-bdgi](https://tex.z-dn.net/?f=%28ac%2Badi%2Bbci-bd%29%20%5Ctimes%20%28f%2Bgi%29%3D%20%20%28a%2Bbi%29%20%20%5Ctimes%20%28cf%2Bcgi%2Bfdi-dg%29%5C%5C%5C%5C%5C%5Cacf%2Badfi%2Bbcfi-bdf%2Bacgi-adg-bcg-bdgi%20%3D%20acf%2Bacgi%2Bafdi-adg%20%2B%20bcfi-bcg-bfd-bdgi)
As the two sides of the equation are equal, the equation is true.
C.) x−y=y−x
![x-y=y-x\\\\(a+bi)-(c+di)=(c+di)-(a+bi)\\\\a+bi-c-di=c+di-a-bi](https://tex.z-dn.net/?f=x-y%3Dy-x%5C%5C%5C%5C%28a%2Bbi%29-%28c%2Bdi%29%3D%28c%2Bdi%29-%28a%2Bbi%29%5C%5C%5C%5Ca%2Bbi-c-di%3Dc%2Bdi-a-bi)
As the two sides of the equation are not equal, the equation is not true.
D.) (x+y)+z=x+(y+z)
![(x+y)+z=x+(y+z)\\\\(a+bi+c+di)+f+gi = a+bi+(c+di+f+gi)\\\\a+bi+c+di+f+gi = a+bi+c+di+f+gi](https://tex.z-dn.net/?f=%28x%2By%29%2Bz%3Dx%2B%28y%2Bz%29%5C%5C%5C%5C%28a%2Bbi%2Bc%2Bdi%29%2Bf%2Bgi%20%3D%20a%2Bbi%2B%28c%2Bdi%2Bf%2Bgi%29%5C%5C%5C%5Ca%2Bbi%2Bc%2Bdi%2Bf%2Bgi%20%3D%20a%2Bbi%2Bc%2Bdi%2Bf%2Bgi)
As the two sides of the equation are equal, the equation is true.
E.) (x−y)−z=x−(y−z)
![(x-y)-z=x-(y-z)\\\\(a+bi-c-di)-f-gi = a+bi-(c+di-f-gi)\\\\a+bi-c-di-f-gi =a+bi-c-di+f+gi)](https://tex.z-dn.net/?f=%28x-y%29-z%3Dx-%28y-z%29%5C%5C%5C%5C%28a%2Bbi-c-di%29-f-gi%20%3D%20a%2Bbi-%28c%2Bdi-f-gi%29%5C%5C%5C%5Ca%2Bbi-c-di-f-gi%20%3Da%2Bbi-c-di%2Bf%2Bgi%29)
As the two sides of the equation are not equal, the equation is not true.
Hence, all the statements are true except C and E.
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