Answer:
Let be a rational complex number of the form [tex]z = \frac{a + i\,b}{c + i\,d}[/tex], we proceed to show the procedure of resolution by algebraic means:
1) [tex]\frac{a + i\,b}{c + i\,d}[/tex] Given.
2) [tex]\frac{a + i\,b}{c + i\,d} \cdot 1[/tex] Modulative property.
3) [tex]\left(\frac{a+i\,b}{c + i\,d} \right)\cdot \left(\frac{c-i\,d}{c-i\,d} \right)[/tex] Existence of additive inverse/Definition of division.
4) [tex]\frac{(a+i\,b)\cdot (c - i\,d)}{(c+i\,d)\cdot (c - i\,d)}[/tex] [tex]\frac{x}{y}\cdot \frac{w}{z} = \frac{x\cdot w}{y\cdot z}[/tex]
5) [tex]\frac{a\cdot (c-i\,d) + (i\,b)\cdot (c-i\,d)}{c\cdot (c-i\,d)+(i\,d)\cdot (c-i\,d)}[/tex] Distributive and commutative properties.
6) [tex]\frac{a\cdot c + a\cdot (-i\,d) + (i\,b)\cdot c +(i\,b) \cdot (-i\,d)}{c^{2}-c\cdot (i\,d)+(i\,d)\cdot c+(i\,d)\cdot (-i\,d)}[/tex] Distributive property.
7) [tex]\frac{a\cdot c +i\,(-a\cdot d) + i\,(b\cdot c) +(-i^{2})\cdot (b\cdot d)}{c^{2}+i\,(c\cdot d)+[-i\,(c\cdot d)] +(-i^{2})\cdot d^{2}}[/tex] Definition of power/Associative and commutative properties/[tex]x\cdot (-y) = -x\cdot y[/tex]/Definition of subtraction.
8) [tex]\frac{(a\cdot c + b\cdot d) +i\cdot (b\cdot c -a\cdot d)}{c^{2}+d^{2}}[/tex] Definition of imaginary number/[tex]x\cdot (-y) = -x\cdot y[/tex]/Definition of subtraction/Distributive, commutative, modulative and associative properties/Existence of additive inverse/Result.
Step-by-step explanation:
Let be a rational complex number of the form [tex]z = \frac{a + i\,b}{c + i\,d}[/tex], we proceed to show the procedure of resolution by algebraic means:
1) [tex]\frac{a + i\,b}{c + i\,d}[/tex] Given.
2) [tex]\frac{a + i\,b}{c + i\,d} \cdot 1[/tex] Modulative property.
3) [tex]\left(\frac{a+i\,b}{c + i\,d} \right)\cdot \left(\frac{c-i\,d}{c-i\,d} \right)[/tex] Existence of additive inverse/Definition of division.
4) [tex]\frac{(a+i\,b)\cdot (c - i\,d)}{(c+i\,d)\cdot (c - i\,d)}[/tex] [tex]\frac{x}{y}\cdot \frac{w}{z} = \frac{x\cdot w}{y\cdot z}[/tex]
5) [tex]\frac{a\cdot (c-i\,d) + (i\,b)\cdot (c-i\,d)}{c\cdot (c-i\,d)+(i\,d)\cdot (c-i\,d)}[/tex] Distributive and commutative properties.
6) [tex]\frac{a\cdot c + a\cdot (-i\,d) + (i\,b)\cdot c +(i\,b) \cdot (-i\,d)}{c^{2}-c\cdot (i\,d)+(i\,d)\cdot c+(i\,d)\cdot (-i\,d)}[/tex] Distributive property.
7) [tex]\frac{a\cdot c +i\,(-a\cdot d) + i\,(b\cdot c) +(-i^{2})\cdot (b\cdot d)}{c^{2}+i\,(c\cdot d)+[-i\,(c\cdot d)] +(-i^{2})\cdot d^{2}}[/tex] Definition of power/Associative and commutative properties/[tex]x\cdot (-y) = -x\cdot y[/tex]/Definition of subtraction.
8) [tex]\frac{(a\cdot c + b\cdot d) +i\cdot (b\cdot c -a\cdot d)}{c^{2}+d^{2}}[/tex] Definition of imaginary number/[tex]x\cdot (-y) = -x\cdot y[/tex]/Definition of subtraction/Distributive, commutative, modulative and associative properties/Existence of additive inverse/Result.
