1. Complex numbers:
Let's probe associative, commutative and distributive properties for any three complex numbers:
Let Z1=a+ix, Z2=b+iy and Z3=c+iz
a. Associative property: (z1+z2)+z3=z1+(z2+z3)
Then:
(z1+z2)+z3=((a+ix)+(b+iy))+c+iz
=(a+b+ix+iy)+c+iz
=(a+b+i(x+y))+c+iz
=a+b+c+i(x+y+z))
=a+b+c+ix+iy+iz
=a+ix+(b+c+iy+iz)
=a+ix+((b+iy)+(c+iz))
=z1+(z2+z3)
Hence, complex numbers follow the associative property.
b. Commutative property: z1+z2=z2+z1
Then:
z1+z2=(a+ix)+(b+iy)
=a+ix+b+iy
=b+iy+a+ix
=(b+iy)+(a+ix)
=z2+z1
Hence, complex numbers follow the commutative property.
c. Distributive property: z1(z2+z3)=z1*z2+z1*z3
z1*(z2+z3)=(a+ix)[(b+iy)+(c+iz)]
=(a+ix)*(b+iy)+(a+ix)(c+iz)
=z1*z2+z1*z3
Hence, complex numbers follow the distributive property.
2. Polynomials:
Let P=ax+b, Q=ax^2+bx+c, R=bx+c
a. Associative property: (P+Q)+R=P+(Q+R)
(P+Q)+R=[(ax+b)+(ax^2+bx+c)]+bx+c
=[ax^2+ax+bx+b+c]+bx+c
=(ax+b)+[(ax^2+bx+c)+(bx+c)]
=P+(Q+R)
Polynomials follow the associative property
b. Commutative property: P+R=R+P
P+R=(ax+b)+(bx+c)
=ax+b+bx+c
=bx+c+ax+b
=(bx+c)+ax+b)
=R+P
Polynomials follow the commutative property.
d. Distributive property: P(Q+R)=PQ+QR
P(Q+R)=(ax+b)[(ax^2+bx+c)+(bx+c)]
=ax^3+abx^2+acx+abx^2+acx+abx^2+b^2x+bc+b^2x+bc
=ax^3+3abx^2+2acx+2b^2x+2bc
PQ+QR=(ax+b)(ax^2+bx+c)+(ax+b)(bx+c)
=ax^3+abx^2+acx+abx^2+b^2x+bc+abx^2+acx+b^2x+bc
=ax^3+3abx^2+2acx+2b^2x+2bc
Then:
P(Q+R)=PQ+PR
Polynomials follow the distributive property.