/sci/ - Science & Math

[Anonymous No.14485683]

But how do you make sense of multiplying a number iπ times?

[Anonymous No.14485710]

>>14485683
How can you make sense of doing anything an integer number of times?

[Anonymous No.14485726]

8008135 = Boobies

[Anonymous No.14485730]

>>14485683
Symmetry of multiplication collapses the form down to a negative domain.

[Anonymous No.14485738]

>>14485730
You don't have to go that far. What does it mean to multiply a number 2.6 times?

[Anonymous No.14485748]

>>14485683
>exponentiation = multiplying something n times
only for positive integers

[Anonymous No.14485755]

>>14485683
[math] \sqrt{2} = 2^{1/2} [/math]
>But how do you make sense of multiplying a number 1/2 times?

[Anonymous No.14485796]

>>14485683
Derivatives. It's all derivatives.
I was thinking about this all day. Then it eventually hit me.
e^x is its own derivative.
sin(x) is its 4th derivative

You WILL understand the beauty of Euler's identity.

[Anonymous No.14485798]

rotation by pi radians in the complex plane

[Anonymous No.14485814]

>>14485796 (cont)
another way to think of it is sin(x) being acceleration towards a center and exp(x) acceleration towards infinity.

Inwards and outwards acceleration. So incredibly beautiful. It's like we're at holomorphic dynamics already and we're only just defined the complex numbers.

[Anonymous No.14485872]

>>14485738
A multiplication table effectively implies multiplication is not repeated addition because it generates primes.

[Anonymous No.14485878]

>>14485872
>multiplication generates primes
wut?

[Anonymous No.14485892]

>>14485878
For all n,m >=2 a prime doesn't exist for 1 and only 1 interval for all n,m up to n=m . This would imply that primes are defined by a multiplication in their lack of presence in the set n*m.

[Anonymous No.14485998]

>>14485683
I can explain it to you rather simply.
I assume you are familar with the real number line.
The negative numbers are on the left.
The positive numbers are on the right.
Take any point on that line (this point represents a number on the line).
If you add a positive number , it will go to the right.
If you subtract a positive number, the point will go to the left.
If you multiply by a positive number, the point will go further away from the origin (the point corresponding to the zero value).
If you multiply by -1, you make the point turn by 180 degree to the other side of the real number line (the origin acts as the center of rotation when you multiply by -1).
So far so good, nothing new.
There was a time when mathematicians were convinced that all existing numbers could be found in this line.
Or so they thought.
Somone eventually asked if there is an answer to the square root -1.
Turns out there actually is.
This number has been given the name "i".
The particular thing with this number is that if you take again a point from the real number line and multiply it by i, that number will turn by 90 degrees (with the origin still acting as the center of rotation).
That point has left the real number line and has entered the complex plane.
The point has learned the existence of a new dimension.
Welcome to the complex plane, the realm of 2 dimensional numbers.
You can imagine the complex plane as a 2 dimensional XY plane, where i acts as the unit of the Y axis.
Much like 2D vectors, you can add a real number with an imaginary one (a bit like adding a vector on the x axis and a vector on the y axis).

[Anonymous No.14486002]

>>14485998
In this realm, functions we got accustomed to (linear, parabolic, inverse, etc) unveil new behaviors we have never observed when their usage was limited to the real numbers.
The exponential equation is a prime example of that.
Euler has discovered (and proved) a curious formula: e^(ix) = cos(x) + isin(x).
If you have ever studied the trigonometric cycle, you can easily see why this equation would plot a circle on the complex plane.
Very few people can intuitively understand this equation (me neither).
After all, it creates a bridge between exponents and trigonometric circles.
From this equation, you can see why e^(i*pi) = -1.

[Anonymous No.14486035]

>>14485683
>But how do you make sense of multiplying a number iπ times?

One positive number raised to another positive number on the imaginary number line.

How is this at all difficult???

Are you pretending to be stupid??

[Anonymous No.14486043]

>>14485683
multiplying a number by -1 rotates it by 180 degrees on the number line. multiplying it by -1 twice rotates it 360 degrees. By this line of reasoning, multiplying it by the square root of
-1, i should rotate it by 90 degrees.

We can extend this to any angle by the relation (-1)^(theta/pi)=cos(theta)+isin(theta). The taylor expansion of cos(theta) added to that of I*sin(theta) is equal to the taylor expansion of exp(i*theta), meaning cos(theta)+i*sin(theta)=exp(i*theta). For a rotation of pi or 180 degrees:
exp(i*pi)=cos(pi)+isin(pi)=-1
so exp(i*pi)+1=0.

[Anonymous No.14486068]

>>14486002
Good explanation layout. What I don't understand is how dimensional rotation over a number line is used as justification to use scalar factors to expand the domain of the even rooting function.

[Anonymous No.14486092]

>>14486068
The rotation is baked into negative numbers which are 180* flipped versions of their positive counterparts. If the angle of rotation is not confined to a discrete domain, its necessary that there must be a perpendicular axis to your number line, with which its spans a plane. Along this plane you can rotate by any angle and rotating by 90 degrees twice lands you at -1* whatever number you are rotating.

[Anonymous No.14486102]

It has the correct derivative.
[eqn]\frac{d}{dx}(e^{ix}) = i e^{ix}[/eqn]
[eqn]\frac{d}{dx}(\cos(x) + i \sin(x)) = i (\cos(x) + i \sin(x))[/eqn]

[Anonymous No.14486200]

>>14486035
i used to be like you

[Anonymous No.14486288]

>>14486002
>Very few people can intuitively understand this equation (me neither).
>After all, it creates a bridge between exponents and trigonometric circles.

Look at the Taylor series of exp(x) and sin(x)
e^x is its own derivative.
sin(x) is its 4th derivative

It is mindblowing when you truly comprehend and see this.

Now can we finally talk about the Euler-Mascheroni constant 0.5772156649... which is the first term in the Laurent series of the Riemann zeta function? Also it is the "renormalized" value if zeta(1) IF NOT even that value itself like in a way that 0^0=1.

Also it is the limit between log(x) and the harmonic series, but also the sum of the harmonic series itself??? (zeta(1))

What is all this magic?

[Anonymous No.14486297]

>>14486092
Is the 2nd dimensionality of the number line an arbitrary concept? Your points are very valid though.

[Anonymous No.14488107]

>>14485683
they're rotation

[Anonymous No.14488357]

>>14485878
This... I guess if you multiply by one...

[Anonymous No.14490148]

Complex exponentiation.

[Anonymous No.14491334]

>>14485726
science