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Let's complex bash Part 2:P

 Okie so continuing from the previous post ( sorry for huge time gap, got stuck in Allen stuff)

A warning though if anyone from the "anti-bash" community is reading, sorry in advance and R.I.P. 


Notes:-

1. We have $z_1=r_1e^{i\theta_1}$ and $z_2=r_2e^{i\theta_2}$ then we have $z_1z_2=r_1r_2e^{i(\theta_1+\theta_2)}$ and we get $|z_1z_2|=|z_1||z_2|$ and $\arg z_1z_2=\arg z_1+\arg z_2 .$

SPIRAL SIMILARITIES and TRANSFORMATIONS:-

2. How to rotate a point about origin by 90.

  • If $90^{\circ}$ anti-clockwise then multiply the number by $i$ 
  • If $90^{\circ}$ clockwise then multiply by $-i.$
Proof:- Note that $i= 0+1\cdot i.$ So $|i|= \sqrt{0^2+1^2}=1.$ And when we locate $i$ in complex plane, clearly $\arg i=\pi/2.$

Similarly for the second, but note that $\arg -i=-\pi /2.$ (Since angles are measured anti-clockwise)

Example:- Here we have $z=1+2i$ then when we multiply $i$ we get $zi=i-2.$

Now, what about we want to dialate a point from one to another and scale it? (Note that that is spiral similarity only :P)



Like say we want to rotate point $z=4+5i$ about point $x=2+2i$ with dilation of $\pi/2.$

In this case what we do is we shift the whole diagram such that $x$ to $(0,0).$ Here we have $(2,2)\rightarrow (0,0),$ so $z$ goes to $z=(4,5)\rightarrow z'=(4-2,5-2)=(2,3)= 2+3i.$

Now we perform the same thing, so we will multiply $i$ and we get $z''=i-3=(-3,2).$
Then we transform it back, so we get $z''=(-3,2)\rightarrow z'''=(-3+2,2+2)=(-1,4).$



So in summary, what we are doing is shifting the $x$ to $O$ i.e in all the coordinates we subtracting $x$'s coordinates.. and then we get the new image of $z,$ we multiply $i,$ get new image, of $(z-x)$ then add the coordinates back .

So we get $\boxed{z\rightarrow i(z-x)+x.}$

Here $i$'s modulus is one, so there hasn't been a scaling thingy but we can do that too. 

$$z\rightarrow \alpha(z-w)+w$$ 

This is a spiral similarity centrerd at $w$ dilating $z$ which rotates by $\alpha $ and dilates by $|\alpha |.$

COMPLEX REFLECTION:- 

Let $W$ be the reflection of $Z$ over a given $AB.$ Then

$$w=\frac{  (a-b)\overline{z} + \overline{a}b - a\overline{b}}{(\overline{a} - \overline{b})}$$

Proof:- It's in egmo:P.

We first transform $z$ to $z-a$ ( We have the motive to turn $\overline{AB}$ in to the segment between $0$ and $1$ on the real axis, so that we can then apply complex conjugate thing that we have on reflection about real axis)

Then we down everything by $b-a.$

So we get $$\left(\frac{w-a}{b-a}\right) = \overline{\left(\frac{z-a}{b-a}\right)}$$

Using the basic conjugate properties, we get,

$$ \overline{\left(\frac{z-a}{b-a}\right)}=\left(\frac{\bar{z}-\bar {a}}{\bar{b}-\bar{a}}\right)$$

Then calcs done, we get $$w=\frac{  (a-b)\overline{z} + \overline{a}b - a\overline{b}}{(\overline{a} - \overline{b})}$$

Then we get this problem from egmo :P

Problem:- Show that the foot of the altitude from $Z$ to $\overline{AB}$ is given by

$$ \frac{(\overline{a} - \overline{b})z + (a-b)\overline{z} + \overline{a}b - a\overline{b}}{2(\overline{a} - \overline{b})} $$

Proof:-Consider the reflection of $z$ say $w.$ Then we know that the foot ( say $x$) will be the midpoint of $z$ and $w$ i.e $x=\frac{z+w}{2}.$

Now, using the reflection lemma, we get $$w+z=\frac{ (a-b)\overline{z} + \overline{a}b - a\overline{b}}{(\overline{a} - \overline{b})}+ z=\frac{(\overline{a} - \overline{b})z + (a-b)\overline{z} + \overline{a}b - a\overline{b}}{(\overline{a} - \overline{b})} $$

So $$\frac{w+z}{2}=  \frac{(\overline{a} - \overline{b})z + (a-b)\overline{z} + \overline{a}b - a\overline{b}}{2(\overline{a} - \overline{b})}$$

Properties:-
  • $z=\bar{z}$ iff $z$ is real
  • $z+\bar{z}=0 $ iff $z$ is pure imaginary  
perpendicular criterion:- The compleplex numbers $a,b,c,d$ have the property $AB\perp CD$ if and only if
$$\frac{d-c}{b-a} + \overline{\left(\frac{d-c}{b-a}\right)}=0$$

Proof:- We transform $d\rightarrow d-c, c \rightarrow 0.$ Now, this transformation takes line $CD$ to a line parallel to it, passing through origin.

Similarly, we transform $d\rightarrow d-c, c \rightarrow 0.$ Now, this transformation takes line $CD$ to a line parallel to it, passing through origin.

Now, remember in the previous spiral similarity transformation, we noticed that multiplying the complex number with $xi$ transforms line to a perpendicular line. So if $d-c$ and $b-a$ are perpendicular then  $\frac{d-c}{b-a}$ must be pure imaginary.

Using the properties, we get that  $$\frac{d-c}{b-a} + \overline{\left(\frac{d-c}{b-a}\right)}=0.$$

Collinearity Lemma:- Prove that complex numbers $z$, $a$ and $b$ are collinear iff

$$ \frac{z-a}{z-b} = \overline{\left(\frac{z-a}{z-b}\right)} $$

Proof:- We consider the transformation $z-a$ and $z-b.$ These vectors have tails in $O$ and heads as $z-a, z-b$ respectively. Now note that if $z,a,b$ are collinear then these two vectors ( because the transformation is mapping into parallel line through $O$) 
So we have the angles between these two vectors as $0, \pi.$

Hence the two vectors' quotient must be a real number. Using Proposition 6.4, we get that 

$$\frac{z-a}{z-b} + \overline{\left(\frac{z-a}{z-b}\right)}=0.$$

Complex shoelace Formula:- If $a,b,c$ are complex numbers, then the signed area of triangle $ABC$ is given by 





THE UNIT CIRCLE:-

The unit circle is the set of complex numbers $z$ with $|z|=1,$ which is centred at $0$ with radius $1.$

We have for any $z$ on the unit circle, $\boxed{\bar {z}=\frac{1}{z}}.$
This is because $z\cdot \bar{z}=|z|^2.$

----

Yeah now lemme do polynomials :P  

See ya next time! Also do check my other blog (if you get free time tmrw, new post coming there )

Sunaina 💜

Comments

  1. Where actually is the bash you just posted the thm's right?

    ReplyDelete
    Replies
    1. No bashing is the best,also even bashes are beautiful sometimes for eg check Mr Oreo Juice/nathantareep's (shameless advertising)/jj_ca888 solution to Romania TST 2007 Day 6 P2(by Cosmin Pohoata)

      Delete
    2. lol!! Now I know who you are :sunglasses:.. I tbh cant bash in an exam condition, so i felt it's not that useful to learn plus I dont even have specific interest! Also Oreo doing bash :O dang he must be high!

      Delete
    3. @bove lol tru :rotfl:!I ve never literally talked to him(maybe bcoz I don't use discord) but I have seen him write that in his posts :P.Yah I am not that good too in bashes.Just learning.

      Also complex bash is tough :(

      Delete

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