Exprese lo siguiente en la forma $ a + bi \; $ donde $ a $ y $ b $ son reales.

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Luis Hernandez

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6 years, 2 months ago

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Luis Hernandez 6 years, 2 months ago

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Luis Hernandez 6 years, 2 months ago

Created this as a copy of Arithmetics of complex numbers I.

Name	Status	Author	Last Modified
Arithmetics of complex numbers I	draft	Newcastle University Mathematics and Statistics	20/11/2019 14:50
Complex Numbers Ex Sheet 1	draft	Violeta CIT	17/10/2017 19:49
Complex Numbers Ex Sheet 1	draft	Joshua Harnwell	02/11/2017 08:36
Exprese lo siguiente en la forma $a + bi \;$ donde $a$ y $b$ son reales.	Ready to use	Luis Hernandez	08/11/2020 01:10
Maria's copy of Arithmetics of complex numbers I	draft	Maria Aneiros	25/05/2019 08:20
Rechnen mit komplexen Zahlen I	Ready to use	Ulrich Görtz	08/02/2021 19:57
Multiplication of complex numbers (i)	Ready to use	Shaheen Charlwood	27/10/2022 18:53
Paul's copy of Arithmetics of complex numbers I	Ready to use	Paul McFadden	08/11/2024 14:56

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Question statement

Exprese lo siguiente en la forma $a + bi \;$ donde $a$ y $b$ son reales.

No incluya decimales en sus respuestas, solo fracciones o enteros. Tampoco incluya corchetes en sus respuestas. recuerde que $i^2=-1$

Give any introductory information the student needs.

			Name	Type	Generated Value

a	number	7 - 8i
b	number	-2 + 9i
f6	integer	3
s3	integer	-1
s2	integer	-1
s1	integer	1
d3	integer	4
s6	integer	-1
s4	integer	1
a1	number	1 + 2i
c3	integer	-7
a3	integer	7
b3	integer	-7
d6	integer	3
e6	integer	-1
z1	number	-5 + 3i
z2	number	-4 + 3i
z3	number	-5 - i

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Name

Data type

Value

xxxxxxxxxx
 
s1*random(1..9)+s2*random(1..9)*i

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Description

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Generated value: `number`

7 - 8i

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Parts

Part a) Gap-fill
1. Gap Mathematical expression
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Part b) Gap-fill
1. Gap Mathematical expression
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Part c) Gap-fill
1. Gap Mathematical expression
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Part d) Gap-fill
1. Gap Mathematical expression
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Gap-fill

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$(\simplify[std]{{a}})(\simplify[std]{{b}})\;=\;$ Unnamed gap.

Advice

a)
La fórmula para multiplicar números complejos es

$\begin{eqnarray*}\simplify[]{Re((a + ib)(c + id))} &=& ac -bd \\ \simplify[]{Im((a + ib)(c + id))} &=& ad +bc \end{eqnarray*}$

Entonces tenemos:

$\begin{eqnarray*}\simplify[]{Re({a}*{b})} &=& \simplify[]{{Re(a)}*{Re(b)} - {Im( a)}*{Im(b)} = {Re(a*b)}}\\ \simplify[]{Im({a}*{b})} &=& \simplify[]{{Re(a)}*{Im(b)} + {Im( a)}*{Re(b)} = {Im(a*b)}} \end{eqnarray*}$
Hence the solution is :

$(\simplify[std]{{a}})(\simplify[std]{{b}})=\var{a*b}$
b)

Esto se calcula de manera similar una vez que la expresión se escribe como:

$(\simplify[std]{{a1}})^2= (\simplify[std]{{a1}}) (\simplify[std]{{a1}})$ then we find:

$\begin{eqnarray*}(\simplify[std]{{a1}})^2&=& (\simplify[std]{{a1}}) (\simplify[std]{{a1}})\\ &=& \simplify[]{({Re(a1)}*{Re(a1)} - {Im(a1)}*{Im(a1)})+ ({Re(a1)}*{Im(a1)} + {Im(a1)}*{Re(a1)})i}\\ &=& \simplify[std]{{a1^2}} \end{eqnarray*}$
c)
Sabemos que

$i^2 = -1$ lo que da

$i^3 = i^2 i = -i$ .

Por lo tanto:

$\begin{eqnarray*} \simplify[std,!otherNumbers]{{a3} + {b3} * i + {c3} * i ^ 2 + {d3} * i ^ 3}&=&\simplify[std]{{a3} + {b3} * i -{c3} -({d3} * i)}\\ &=&\simplify[std]{ {a3} -{c3} + ({b3} -{d3}) * i}\\ &=&\simplify[std]{{a3 -c3} + {b3 -d3} * i} \end{eqnarray*}$
d)

Esto se puede calcular usando la fórmula dos veces, primero para multiplicar los dos primeros conjuntos de paréntesis,
y luego multiplicar el resultado de ese cálculo por el tercer conjunto de paréntesis.

Así obtenemos:

$\begin{eqnarray*} (\var{z1})(\var{z2})(\var{z3})&=&((\var{z1})(\var{z2}))(\var{z3})\\ &=&(\var{z1*z2})(\var{z3})\\ &=&\var{z1*z2*z3} \end{eqnarray*}$

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Exprese lo siguiente en la forma $a + bi \;$ donde $a$ y $b$ son reales.

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Luis Hernandez 6 years, 2 months ago

Luis Hernandez 6 years, 2 months ago

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Exprese lo siguiente en la forma a+bi a + bi \; donde a a y b b son reales.

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Luis Hernandez 6 years, 2 months ago

Luis Hernandez 6 years, 2 months ago

Error in variable testing condition

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Generated value: number

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Gap-fill

Exprese lo siguiente en la forma $a + bi \;$ donde $a$ y $b$ son reales.

Generated value: `number`