Complex number proof

[keen lance]

what happened in the marked place...please explain this and the remaining steps

[solemn cave]

Wait

How do you define the absolute value of complex numbers?

So that I can help

@keen lance

[keen lance]

z bar

[keen lance]

.

[solemn cave]

Wdym z bar?

Like

$\abs{a + bi} = a - bi$ or smth?

[iron swan]

iaminfinityiq

[keen lance]

like this

[solemn cave]

Okay... i see

I think

I don't really know much

[plain relic]

$|a + b \cdot \iota| = \sqrt{a^2 + b^2}$

[iron swan]

HitenTandon

[solemn cave]

But i will do the algaebraic way

Let $z_1 := a_1 + b_1i$

[iron swan]

iaminfinityiq

[keen lance]

no..i wanna understand this @solemn cave

[solemn cave]

Oh...

[versed wyvern]

conjugate of the conjugate of a number is just that number

conjugate of a+bi is a-bi so if you take the conjugate again it will be back to a+bi

[keen lance]

ok then next step?

[versed wyvern]

if you don’t see it, there is two bars on top

[keen lance]

and why we did conjugate here

is this valid to do like this

[solemn cave]

$z_2 := a_2 + b_2i$

[iron swan]

iaminfinityiq

[versed wyvern]

I don’t understand what you are confused with

[solemn cave]

=> $z_1 + z_2 = (a_1 + a_2) + (b_1 + b_2)i$

[iron swan]

iaminfinityiq

[solemn cave]

So if im correct

$\abs{z_1 + z_2} = (a_1 + a_2) - (b_1 + b_2)i$?

[iron swan]

iaminfinityiq

[plain relic]

No

[plain relic]

.

[plain relic]

$|(a_1 + a_2) + (b_1 + b_2)\cdot\iota| = \sqrt{(a_1 + a_2)^2 + (b_1 + b_2)^2}$

[iron swan]

HitenTandon

[keen lance]

basically i only understood the first line only...
idk why they conjugate(the highlighted part) the conjugate...
idk how (2Re(z1.z2) came

[solemn cave]

Why is this iota 😭😭😭

Okay okay

$\abs{z_1 + z_2} = \sqrt{a_1^2 + a_2^2 + b_1^2 + b_2^2 + 2(a_1a_2 + b_1b_2)}$

[iron swan]

iaminfinityiq

[plain relic]

exactly

[solemn cave]

Okay.. this ain't going anywhere ngl

[keen lance]

.how this

i got the second line too

how the third line

[solemn cave]

Im self-learning 😭😭😭

[plain relic]

Square this expression

You'll find

[solemn cave]

Yeah

[plain relic]

$\abs{z_1 + z_2}^2 = a_1^2 + a_2^2 + b_1^2 + b_2^2 + 2(a_1a_2 + b_1b_2)$

[iron swan]

HitenTandon

[solemn cave]

That's obvious

[plain relic]

Yes

If you re-write this

[keen lance]

guys you are supposed to solve my query. please help me

[plain relic]

Sorry I mixed up who was the OP lol

[solemn cave]

LOL

Srru

Well

If i prove some problems about complex numbers or smtj

[plain relic]

That said this is also the solution to your query

[solemn cave]

Ill just convert to $a + bi$ and do some algebra

[iron swan]

iaminfinityiq

[keen lance]

how this came?

[solemn cave]

This one... 😭😭😭😭😭😭

[keen lance]

is this not doable to you?

[solemn cave]

What's Re?

[keen lance]

real

[solemn cave]

Im self-learning complex number 😭😭😭

[keen lance]

no issue..its fine..but also appreciate other people time

[solemn cave]

Anyways, where does this come from?
And also, what's Re?

In the proof you sent me

[keen lance]

if z = a + ib
then Re(z) = a
(i.e. real number)
and Im(z) = b
(i.e. imaginary number)

[solemn cave]

Made sense

[plain relic]

$$\abs{z_1}^2 = a_1^2 + b_1^2$$
similarly,
$$\abs{z_2}^2 = a_2^2 + b_2^2$$
$$z_1 \cdot z_2 = (a_1 \cdot a_2 - b_1 \cdot b_2) + (a_1 \cdot b_2 + b_1 \cdot a_2)\cdot\iota$$

[iron swan]

HitenTandon

[solemn cave]

Yeah

And $2Re(z_1 \cdot \con{z_2})$

[iron swan]

iaminfinityiq
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[solemn cave]

How do you write conjugate function in Latex 😭😭😭

Anyways

[plain relic]

$$\Re(z_1 \cdot z_2) = a_1 \cdot a_2 - b_1 \cdot b_2$$

[iron swan]

HitenTandon

[solemn cave]

So according to this

[plain relic]

We're missing a negative sign somewhere lol

Classic

[solemn cave]

Lol

Wait

[solemn cave]

$z_2$ is conjugated

[iron swan]

iaminfinityiq

[solemn cave]

Maybe it's that classic @plain relic

[plain relic]

lol

It is

$z_1 \cdot \overline{z_2} = (a_1 \cdot a_2 + b_1 \cdot b_2) + (a_2 \cdot b_1 - a_1 \cdot b_2)\cdot \iota$

[solemn cave]

I don't know how to write the conjigate function in LaTeX 😭😭😭

[iron swan]

HitenTandon

[solemn cave]

So uh

Yeah

$Re(z_1 \cdot \bar{z_2}) = a_1a_2 + b_1b_2$

[iron swan]

iaminfinityiq

[plain relic]

Using this, if we evaluate the expression
$$\abs{z_1}^2 + \abs{z_2}^2 + 2\cdot\Re(z_1\cdot\overline{z_2})$$

[iron swan]

HitenTandon

[plain relic]

We get,
$$a_1^2 + b_1^2 + a_2^2 + b_2^2 + 2\cdot(a_1\cdot a_2 + b_1 \cdot b_2)$$

[iron swan]

HitenTandon

[plain relic]

Which is exactly what the original statement is

@keen lance

does this help?

[solemn cave]

You just need to convert from complex numbers

To the form of $a + bi$

[iron swan]

iaminfinityiq

[solemn cave]

Then do some algebra and apply special properties