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Further Complex Numbers (DP IB Applications & Interpretation (AI))

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FrontGeometry of Complex Numbers
What happens geometrically to a complex number, $z$ , when the complex number $a + b i$ is added to it?

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What happens geometrically to a complex number, $z$ , when the complex number $a + b i$ is added to it?
When the complex number $a + b i$ is added to $z$ , $z$ is translated on the Argand diagram by the vector $(\begin{matrix} a \\ b \end{matrix})$ .
What happens geometrically to a complex number, $z$ , when the complex number $a + b i$ is subtracted from it?
When the complex number $a + b i$ is subtracted from $z$ , $z$ is translated on the Argand diagram by the vector $(\begin{matrix} - a \\ - b \end{matrix})$ .
Describe the geometric relationship between the origin of an Argand diagram and the complex numbers $z$ , $w$ and $z + w$ .
The origin of an Argand diagram and the complex numbers $z$ , $w$ and $z + w$ form a parallelogram. $z + w$ is the vertex opposite the origin.
Describe the geometric relationship between the origin of an Argand diagram and the complex numbers $z$ , $w$ and $z - w$ .
The origin of an Argand diagram and the complex numbers $z$ , $w$ and $z - w$ form a parallelogram. $z - w$ is the vertex opposite the origin.
What two geometrical transformations happen to a complex number, $z$ , when it is multiplied by the complex number $w$ ?
When $z$ is multiplied by the complex number $w$ :
- it is rotated counter-clockwise by $\arg (w)$ ,
- it is stretched from the origin by scale factor $|w|$ .
What two geometrical transformations happen to a complex number, $z$ , when it is divided by the complex number $w$ ?
When $z$ is divided by the complex number $w$ :
- it is rotated clockwise by $\arg (w)$ ,
- it is stretched from the origin by scale factor $\frac{1}{|w|}$ .
Describe the geometrical relationship between $z$ and $z *$ .
On an Argand diagram, $z$ and $z *$ are reflections in the real axis.
What does a complex number look like when written in modulus-argument (polar) form?
A complex number that is written in modulus-argument (polar) form looks like $z = r (\cos θ + isin θ)$ where $r$ is the modulus and $θ$ is the argument.
What is denoted by $r cis (θ)$ ?
$r cis θ = r (\cos θ + isin θ)$ .
What is the complex conjugate of $r cis (θ)$ ?
The complex conjugate of $r cis (θ)$ is $r cis (- θ)$ .
True or False?
$2 (\cos (\frac{π}{3}) - isin (\frac{π}{3}))$ is written in modulus-argument (polar) form.
False.
$2 (\cos (\frac{π}{3}) - isin (\frac{π}{3}))$ is not written in modulus-argument (polar) form. There should be a "+" in front of $i$ .
Write the complex number $r (\cos θ - isin θ)$ in modulus-argument (polar) form.
The complex number $r (\cos θ - isin θ)$ in modulus-argument (polar) form is $r (\cos (- θ) + isin (- θ))$ .
If you know the modulus ( $r$ ) and argument ( $θ$ ) of a complex number, how can you find the real part?
If you know the modulus ( $r$ ) and argument ( $θ$ ) of a complex number, the real part is equal to $r \cos θ$ .
If you know the modulus ( $r$ ) and argument ( $θ$ ) of a complex number, how can you find the imaginary part?
If you know the modulus ( $r$ ) and argument ( $θ$ ) of a complex number, the imaginary part is equal to $r \sin θ$ .
What does a complex number look like when written in exponential (Euler) form?
A complex number that is written in exponential (Euler) form looks like $z = r e^{i θ}$ where $r$ is the modulus and $θ$ is the argument.
What is the complex conjugate of $r e^{i θ}$ ?
The complex conjugate of $r e^{i θ}$ is $r e^{- i θ}$ .
How can you write $z = r e^{i θ}$ in Cartesian form?
You can write $z = r e^{i θ}$ in Cartesian form as $z = r \cos θ + i r \sin θ$ .
True or False?
$e^{iπ} = - 1$ .
True.
$e^{iπ} = - 1$ .
True or False?
You can multiply and divide complex numbers written in exponential (Euler) form by using the laws of indices.
For example, $2 e^{\frac{π}{4} i} \times 3 e^{\frac{π}{2} i} = 6 e^{(\frac{π}{4} + \frac{π}{2}) i}$ .
True.
You can multiply and divide complex numbers written in exponential (Euler) form by using the laws of indices.
For example, $2 e^{\frac{π}{4} i} \times 3 e^{\frac{π}{2} i} = 6 e^{(\frac{π}{4} + \frac{π}{2}) i}$ .
How do you divide complex numbers written in modulus-argument (polar) form?
To divide complex numbers written in modulus-argument (polar) form, you divide the moduli and subtract the arguments.
$\frac{r_{1} (\cos (θ_{1}) + isin (θ_{1}))}{r_{2} (\cos (θ_{2}) + isin (θ_{2}))} = \frac{r_{1}}{r_{2}} (\cos (θ_{1} - θ_{2}) + isin (θ_{1} - θ_{2}))$ .
True or False?
When written in modulus-argument form, $z = r e^{i θ}$ , the real part relates to the cos part.
True.
When written in modulus-argument form, $z = r e^{i θ}$ , the real part relates to the cos part.
What is Euler's form for $a \sin (b x + c)$ ?
Euler's form for $a \sin (b x + c)$ is $Im (a e^{i (b x + c)})$ .
True or False?
$a \cos (b x + c) = Im (a e^{i (b x + c)})$
False.
$a \cos (b x + c) \neq Im (a e^{i (b x + c)})$
$a \cos (b x + c) = Re (a e^{i (b x + c)})$
When using complex numbers to add together two sinusoidal functions, what must be the same in both functions?
When using complex numbers to add together two sinusoidal functions, they must have the same frequency.
However, it does not matter if the amplitudes or phase shifts are different.
E.g. You can add together $4 \sin (2 x + 1)$ and $5 \sin (2 x - 7)$ as both functions have a frequency of $2$ .

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