TP1 Números Complejos [1-10]

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TP1 Números Complejos [1-10] 
 
1) Realizar los cálculos indicados 
 
a) (3 − 4𝑖)(6 + 2𝑖) = 3 ∙ 6 − (−4) ∙ 2 + (3 ∙ 2 + (−4) ∙ 6)𝑖 = 26 − 18𝑖 
 𝑖(6 − 2𝑖) + (1 + 𝑖) = 6𝑖 − 2𝑖2 + 1 + 𝑖 = 3 + 7𝑖 
b) 
2+𝑖
4−7𝑖
=
(2+𝑖)(4−7𝑖)̅̅ ̅̅ ̅̅ ̅̅ ̅
|4−7𝑖|2
=
(2+𝑖)(4+7𝑖)
42+(−7)2
=
8−7+(14+4)𝑖
65
 
 =
1+18𝑖
65
=
1
65
+
18
65
𝑖 
 
(2+𝑖)−(3−4𝑖)
(5−𝑖)(3+𝑖)
=
2−3+(1+4)𝑖
15+1+(5−3)𝑖
=
−1+5𝑖
16+2𝑖
=
(−1+5𝑖)(16+2𝑖)̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅
|16+2𝑖|2
 
 =
(−1+5𝑖)(16−2𝑖)
162+22
=
−16+10+(2+80)𝑖
256+4
 
 =
−6+82𝑖
260
= −
3
130
+
41
130
𝑖 
c) 𝑖3 − 4𝑖2 + 2 = −𝑖 + 4 + 2 = 6 − 𝑖 
 (1 + 𝑖)4 = [(1 + 𝑖)2]2 = (1 + 2𝑖 + 𝑖2)2 = (1 + 2𝑖 − 1)2 = (2𝑖)2 = 4𝑖2 = −4 
 (
−6+2𝑖
1−8𝑖
)
2
= (
(−6+2𝑖)(1+8𝑖)
|1−8𝑖|2
)
2
= (
−6−16+(−48+2)𝑖
12+(−8)2
)
2
 
 = (
−22−46𝑖
65
)
2
=
484+2024𝑖−2116
4225
=
−1632
4225
+ 
2024
4225
𝑖 
2) Probar que para todo 𝑛 ∈ ℕ : 
 
• 𝑖4𝑛 = 1; 𝑖4𝑛 = (𝑖4)𝑛 = [(𝑖2)2]𝑛 = [(−1)2]𝑛 = 1𝑛 = 1 
 
• 𝑖4𝑛+1 = 𝑖; 𝑖4𝑛+1 = 𝑖4𝑛 𝑖 = 1 𝑖 = 𝑖 
 
• 𝑖4𝑛+2 = −1; 𝑖4𝑛+2 = 𝑖4𝑛 𝑖2 = 1 𝑖2 = −1 
 
• 𝑖4𝑛+3 = −𝑖; 𝑖4𝑛+3 = 𝑖4𝑛 𝑖3 = 1𝑖3 = −𝑖 
 
 
3) Usando la fórmula 𝑖𝑛 = 𝑖𝑟(r es el resto de dividir n por 4), calcular 
 
• 𝑖57; 𝑖57 = 𝑖57𝑚𝑜𝑑4 = 𝑖1 = 𝑖 
• 𝑖2020; 𝑖2020 = 𝑖2020𝑚𝑜𝑑4 = 𝑖0 = 1 
• (1 + 𝑖)8; (1 + 𝑖)8 = [(1 + 𝑖)2]4 = (2𝑖)4 = 24𝑖4 = 16𝑖0 = 16 
• (1 + 𝑖)14; (1 + 𝑖)14 = [(1 + 𝑖)2]7 = (2𝑖)7 = 27𝑖7 = 128 𝑖3 
 = 128(−𝑖) = −128𝑖 
 
4) Calcular Re (
1+𝑖
3−𝑖
), 
 
1+𝑖
3−𝑖
=
(1+𝑖)(3+𝑖)
(3−𝑖)(3+𝑖)
=
3−1+(1+3)𝑖
9+1+(3−3)𝑖
=
2+4𝑖
10
=
1
5
+
2
5
𝑖 
Luego 
 Re(
1+𝑖
3−𝑖
) =
1
5
 
5) Sea 𝑧 = 𝑥 + 𝑖𝑦. Expresar: (a) Re(𝑧2), Im(𝑧2) en términos de 𝑥 y 𝑦. 
 𝑧2 = (𝑥 + 𝑖𝑦)2 = 𝑥2 + 2𝑥𝑖𝑦 + (𝑖𝑦)2 = 𝑥2 − 𝑦2 + 𝑖2𝑥𝑦 
Luego 
 𝑅𝑒(𝑧2) = 𝑥2 − 𝑦2, 𝐼𝑚(𝑧2) = 2𝑥𝑦 
 
6) Sea 𝑧 = 𝑥 + 𝑖𝑦. Expresar: (a) |𝑧 − 1 − 3𝑖|2 en términos de 𝑥 y 𝑦. 
 |𝑧 − 1 − 3𝑖|2 = |𝑥 + 𝑖𝑦 − 1 − 3𝑖|2 = |(𝑥 − 1) + 𝑖(𝑦 − 3)|2 
 = [√(𝑥 − 1)2 + (𝑦 − 3)2]
2
= (𝑥 − 1)2 + (𝑦 − 3)2 
 
7) Hallar 𝛾 ∈ ℝ tal que |
𝛾−3𝑖
𝛾+4𝑖
| = √
13
20
 
(Usar la propiedad |
𝑧
𝑤
| =
|𝑧|
|𝑤|
 ) 
Como |
𝛾−3𝑖
𝛾+4𝑖
| =
|𝛾−3𝑖|
|𝛾+4𝑖|
=
√𝛾2+(−3)2
√𝛾2+42
=
√𝛾2+9
√𝛾2+16
= √
𝛾2+9
𝛾2+16
 
Tenemos 
 √
𝛾2+9
𝛾2+16
= √
13
20
 ⇒
𝛾2+9
𝛾2+16
=
13
20
 
Luego 
 (𝛾2 + 9) 20 = (𝛾2 + 16) 13 ⇒ 20 𝛾2 + 180 = 13 𝛾2 + 208 
 ⇒ 7 𝛾2 = 28 ⇒ 𝛾2 = 4 
 ⇒ 𝛾 = 2 o 𝛾 = −2 
 
8) Probar que 𝑅𝑒(𝑧 + 𝑤) = 𝑅𝑒(𝑧) + 𝑅𝑒(𝑤) 
 
Sean 𝑧 = 𝑥 + 𝑖𝑦, 𝑤 = 𝑢 + 𝑖𝑣 
 
 𝑅𝑒(𝑧 + 𝑤) = 𝑅𝑒(𝑥 + 𝑖𝑦 + 𝑢 + 𝑖𝑣) = 𝑅𝑒(𝑥 + 𝑢 + 𝑖(𝑦 + 𝑣)) 
 = 𝑥 + 𝑢 = 𝑅𝑒(𝑧) + 𝑅𝑒(𝑤) 
 
9) Probar que 𝑧 + 𝑤̅̅ ̅̅ ̅̅ ̅̅ = 𝑧̅ + �̅� 
 
Sean 𝑧 = 𝑥 + 𝑖𝑦, 𝑤 = 𝑢 + 𝑖𝑣 
 
 𝑧 + 𝑤̅̅ ̅̅ ̅̅ ̅̅ = 𝑥 + 𝑖𝑦 + 𝑢 + 𝑖𝑣̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ = 𝑥 + 𝑢 + 𝑖(𝑦 + 𝑣)̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅ 
 = 𝑥 + 𝑢 − 𝑖(𝑦 + 𝑣) = 𝑥 + 𝑢 − 𝑖𝑦 − 𝑖𝑣 
 = 𝑥 − 𝑖𝑦 + 𝑢 − 𝑖𝑣 = 𝑧̅ + �̅� 
 
10) Probar que 𝑅𝑒(𝑧)= 
𝑧+�̅�
2
 , 
 
Sea 𝑧 = 𝑥 + 𝑖𝑦 
 
 𝑧 + 𝑧̅ = 𝑥 + 𝑖𝑦 + 𝑥 − 𝑖𝑦 = 2𝑥 = 2𝑅𝑒(𝑧) ⇒ 𝑅𝑒(𝑧)=
𝑧+�̅�
2