/sci/ - Science & Math

because then 0*j*A=A=0*j, and A/0=1/0=j implies A=1, else you have a contradiction. this makes more problems then it solves.

>>12360884
>negative square roots are still defined as i^2 = -1
no you shitstain, it's how i is defined

>>12360884
Is there a solution for the square root of -1 using cosines? Something about three phase power?

>>12360884
>You cannot take the square root of a negative number because there is no number squared that produces a negative result.
you're absolutely right. You can't take the square root of a negative number, because the domain of the square root is the non-negative reals. And there indeed is no real number that when squared gives you a negative number.
>Despite this, negative square roots are still defined as i^2 = -1.
That's why i is not a real number (real as in an element of the real numbers, not the schizo "hurr durr not real" way). If we use fancy language, complex numbers are an algebraic extension of the reals, that is the reals are a subset of the complex numbers that satisfy the same algebraic properties, but the converse is not true.
>Why can't I define division by zero as 0*j = 1?
You can do this only for the trivial ring {1}, i.e. if you define 0=1, and that is the only element of the ring (a set with addition and multiplication). For any other ring, for example the integers, this cannot hold, because that would contradict the axioms of addition and multiplication. Here is a proof:
There's an axiom that tells us that 0 is an identity element of addition. All that means is
0 + a = a + 0 = a
for any number a. Similarly 1 is an identity element of multiplication:
1*a = a*1 = a
We also have the distributive axiom
a*(b + c) = a*b + a*c
and the inverse axiom of addition that says that for any number a there exists a number (-a) such that
a + (-a) = 0
Therefore we can derive the following chain of equalities
a*1 = a
a*(0+1) = a
a*0 + a*1 = a
a*0 + a = a
a*0 + a + (-a) = a + (-a)
a*0 = 0
QED
of course then you can say "but why can't we define something with different axioms?". You sure can. Now whether or not what you define is first of all consistent with itself and most importantly useful at all is actually what matters. Hope this helps.

>>12360901
>this makes more problems then it solves.
that applies to any surjective function, not sure why that matters

>>12360901
>>12360923
Thank you. So from what I understand 0*j = 1 cannot exist because it is inconsistent but can exist within its own useless ring. My next question is, what allows things like imaginary numbers to exist? What do they need to satisfy in order to exist? Consistency?

>>12360951
Do bear in mind, I am a retarded engicuck trying to make sense of mathematics since all my life I've been told what to believe in when it comes to mathematics.

>>12360951
>what allows things like imaginary numbers to exist
we construct them. Literally. Same with the reals, the rationals, even the natural numbers can be constructed from nothing but empty sets (look up Neumann naturals). Whether something "exists" really depends on what you define as "exists". It's an ontological question (philosophy) and mathematics does not attempt to answer it.

>>12360951
>what allows things like imaginary numbers to exist
we construct them. Literally. The complex numbers, the reals, the rationals. Even the natural numbers can be constructed from nothing but empty sets (look up von Neumann ordinals).
The question of existence or what it even means to exist is that of ontology. That is a branch of philosophy. Math does not deal with these questions.

Is [math]-23+\frac{2}{3}\sqrt{i}[/math] a complex number?

>>12360978
[math] -23 + \frac{2}{3}\sqrt{i} = -23 + \frac{2}{3}(e^{i*pi/2})^{1/2} = -23 + \frac{2}{3*\sqrt{2}} (1 + i) [/math]. The imaginary part doesn't vanish, therefore it is a complex number. I expect some schizo shit in response, Tooker, give me your best.

>>12360951
it's only inconistent with the assumptions that:
>zero is the additive identity
>inf+1=inf.
These are both truisms that have been parroted for ceturies, without any foundation in logic. Spooks, if you will.

How would you go about drawing a angle with the same horizontal side, but twice as big?

>>12360884
>Why can't I define division by zero as 0*j = 1?
Of course you can do that. It's math, you can make up anything you want as long as you fix shit to make it self-consistent.
https://en.wikipedia.org/wiki/Wheel_theory

But then
2(0*j) = 2*1
And by commutativity of multiplication
=(2*0)*j = 2
=0*j=2

Then 2=1

>>12361161
Correction, link doesn't work as an example of this. Wheels do allow division by zero, but 0 * (1/0) is not 1. My mistake.

>>12360951
The complex numbers historically exist so every polynomial of the nth degree has exactly n roots. In other words, as a ring, the reals aren't algebraically closed. The complex numbers are.

They historically exist because some dude was trying to solve cubic equations and couldn't get to the real-valued solution without invoking complex numbers in the process

>>12360989
seems ok to me. I never noticed that identity for sqrt{i}.

>>12360951
>what allows things like imaginary numbers to exist?
the fact that i*i = -1 is consistent with the field axioms. on the other hand, 0*j = 1 is not as shown here >>12361162

I am op with another stupid question, what's the point of the radical/root? As far as im aware it can be written as the inverse of an exponent, and the reason why it works is because of the laws of indices. So why does it need to exist?

>>12361322
is your question why do we write [math]\sqrt{x}[/math] when we can write [math]x^{\frac{1}{2}}[/math]?

>>12361371
yes, or more generally x^(1/n) for the n-th root.

Instead of trying to understand complex numbers in terms of abstract square roots you should learn what they mean geometrically. Complex numbers extend the number line in the up and down directions to make a plane of numbers. They're very useful for making trigonometric stuff simpler.

>>12361319
I wonder if OP knows what "field axioms" means.

For example, this satisfies the field axioms, OP. It is a perfectly valid field with perfectly valid addition, subtraction, multiplication, and division operations. Most of the rules for these operations that you are familiar with will still work.

>>12361375
Is there something wrong with having two ways of writing the same thing?

>>12361435
schizo shit

>>12361454
Cryptography depends on that sort of shit.
Much less schizo than the "real" numbers anyway or even the integers; you can write down the rules without having to use any abstruse second-order statements.

>>12361454
this schizo shit makes private online communication possible, anon

take the polynomial ring R[T] and the ideal I=(T^2+1), then C is R[T]/I and i is the class of T. hope this helps

>>12361553
this is how that probably reads to OP:
>take the polynomial durbub X^X#$ and the mongo M=(Z^2+1), then % is CF$# and i has the same teacher as @@. hope this helps
anyone who understands what you wrote would not be asking OP's question, nor would they be particularly impressed by you knowing it

>>12361435
you say "division," but the pic you posted has a suspiciously non-numerical marking under the mult. inverse of 0

>>12361576
0 doesn't have a multiplicative inverse, unless the field is trivial

>>12361584
>t.

>>12360914
i = arccos(2)/(log(2+√3)) I guess.

But no, there isn't really a definition the way you probably want it.

>>12361607
0*0=0 0/0=?

>>12361154
Call the angle ABC, where A is on the upper ray and C is on the lower ray as you drew it. Construct a line perpendicular to ray BA. (For example, you can use the classic construction for a perpendicular bisector of segment AB.) Label the point where this line intersects ray BA as O and the point where it intersects ray BC as P. Construct a circle centered at O through P and label the other point where it intersects line OP as Q. Construct ray BQ. Then angle CBQ is twice angle ABC.

To prove this, see that triangles PBO and QBO are congruent by SAS. They have segment BO in common, angles BOP and BOQ are both right because OP was constructed to be perpendicular to BO, and segments OP and OQ are congruent since they are both radii of the same circle.

>>12361676
That works. Can you do it without the compass?

>>12361375
I think it is confusing notation. We have three essentially different notations for closely related operators: ^, √, and log. The reasons are historical and don't make a ton of sense anymore. Superscripts were used for indices before exponents, and you can imagine the index as representing the degree of a formal variable for instance or the number of times to multiply the base. But that only works for natural number exponents. Still, we are stuck with it. I don't know where the radical sign comes from. It is visually appealing and clearly groups together values that will naturally fall within parentheses otherwise, but it is hard to type into most text boxes, so it's not my favorite. log is used mostly because mathematicians see log x as a function of just one variable, not a binary operator. Traditionally, you would look up logarithms in tables since they were impractical to calculate on the spot The addition of the subscript to indicate the base was more of an afterthought. In most fields, the subscript e will just be dropped and "log" always means the natural logarithm. In other fields, it is usually obvious from context or irrelevant. For science, they're still doing their own thing with base 10.

Basically, it's a mix of arbitrary historical reasons and notations that are useful to professionals but confusing to students, plus little regard for typesetters.

>>12361454
It's just arithmetic modulo 5. That table tells you what you will get if you add or multiply 2 numbers and then divide by 5, taking the remainder. For instance, it says 4×4=1 because 4×4=16, and 16/5 = 3 remainder 1. This is an example of a field that is different from the field of real numbers, but it's still useful.

>>12361699
You definitely can't do it with only a straight edge, if that's what you mean.

>>12361715
But you can. Not for a general angle of course, but for an angle drawn using three points at the intersections of grid lines, you can.

>>12361720
Well, that particular angle has slope 1/2, so you can use the double angle identity for tangent to find that an angle with twice the measure would have slope 4/3 and use the grid to graph it. I suppose with an infinite grid, that technique will always work if the slope is rational.

>>12361747
Correct.
Now let's say we've forgotten all the double/half angle and sum/difference identities. How would you go about deriving them? Put another way, can we use simple geometry to find where to draw the line?

>>12361775
IDK, I don't see any method easier than the one I already described. You just don't need the compass anymore since you can use the grid to draw a perpendicular line and get segments of equal length.

>>12360978
is infinity hat a complex number

>>12361795
Yep, that works. This is a good solution.

>>12361834
Here's another way of seeing the picture. We can make the yellow triangle by scaling up the red triangle. This has the additional nice property of ensuring the point we draw the orange line through itself lies on a grid line intersection.

>>12361859
The scaling of the red triangle to make the yellow triangle corresponds with complex number multiplication.

We go 2 + i (two left, one up) twice, which we can write as (2 + i) * 2.
(2 + i) * 2 = 4 + 2i (four right, two up)

Then we do the 2 + i thing again but rotated by 90 degrees counterclockwise, which is what multiplication by i is.
(2 + i) * i = 2i - 1 (two up, one left)

Putting them together, what we have calculated is (2 + i) * (2 + i).
(2 + i) * (2 + i)
= (2 + i) * 2 + (2 + i) * i
= (4 + 2i) + (2i - 1)
= 3 + 4i (three right, four up)

>>12361882
forgot the pic

>>12361822
no it's 100% tin

>>12360951
They do not exist.

They exist in the way the greek gods exist. We can describe them and they are useful to use to explain or talk about certain things or inspire people.

but they do not actually exist

>>12361910
same as any other kind of number, it might be good to add

>>12361914
>inb4 2 apples show me i apples

>>12361921
>inb4 the only number is 1. 2 apples is just 1 apple and 1 apple.

>>12361925
If I have 0.9999.... apples, do I have 1 apple or 0.9999.... apples.

>>12361929
both

>>12361921
Simple, using a standard cat box, prepare a fruit bowl in the state
[eqn]| \psi_0 \rangle = \frac{1-i}{\sqrt 3} | {\rm 0~apples} \rangle + \frac{i}{\sqrt 3} | {\rm 1~apple} \rangle.[/eqn]
Then perform weak measurement on the number of apples.
Last, use strong measurement to determine whether the bowl is in the state
[eqn]| \psi_A \rangle = \frac{1}{\sqrt 2} | {\rm 0~apples} \rangle + \frac{1}{\sqrt 2} | {\rm 1~apple} \rangle[/eqn]
or
[eqn]| \psi_B \rangle = \frac{1}{\sqrt 2} | {\rm 0~apples} \rangle - \frac{1}{\sqrt 2} | {\rm 1~apple} \rangle[/eqn]
and postcondition the weak measurement on the former outcomes.
The expected value of the weak measurement is
[eqn]\frac{\langle \psi_A | \rm{apples} | \psi_0 \rangle}{\langle \psi_A | \psi_0 \rangle} = i[/eqn].

>>12361914
Sure a natural number can be viewed as an equivalence class of sets of objects.
Do equivalence classes actually exist?
Do sets actually exist?
Do objects actually exist?
At one level of physics, objects are sets, perhaps rather fuzzy sets, of particles.

>>12360884
whether you agree or not division by zero approaches infinity, root of a negative doesn't approach anything so it's its own dimension in a way.
i isn't the only imaginary number

>>12360955
Even from an engicuck point of view, you can easily understand that some mathematical constructions are begging to be made while some are useless at best. Assigning a value to sinc(0) makes more sense than assigning one to 1/0 because there is some continuity that allows it to somehow make sense.

Imaginary numbers are not "imaginary", but rather think of them as numbers "orthogonal" to the real set. They are a natural continuation of real numbers and allow us to describe more complex systems as our grasp of reality evolves. We first had natural integers, then irrationals as we realized that we couldnt describe circular perimeters or diagonals without them, negative numbers to describe things such as debt, and as for complex numbers, phase, attenuation, angle transformations.

Some frameworks are also considering dealing with infinities in a way that would be continuous to our current mathematical framework : the notion of supersumming infinites for example comes to mind, because since the advent of transforms involving infinite sums, shit like this is starting to make more and more sense. Maybe, as we gain more insight on blackholes and the infinities involved in their properties, we might have to create frameworks about that too.

But the rule of thumb is, they have to be a smooth continuation of what we already have. The j you have in your OP fucks with everything we use in algebra so far and thus hardly makes sense.

>>12360884
Fellow engineer, it is time you stop looking at imaginary numbers as things that don't exist. Sure, they're not real, but they exist, you rely on them every day. I suggest you study signals, communications, and euler's formula (related: https://en.wikipedia.org/wiki/Amplitude_modulation#Simplified_analysis_of_standard_AM ) so that you can notice imaginary numbers are a great simplification we use in real world applications. You have two things and you want to combine them to get a third thing, call one real and the other imaginary, square each, add, take the square root.

>>12360884
>Why can't I define division by zero.

You absolutely can, and mathematicians do it in mathematically meaningful and interesting ways. The way you did it doesn't give rise to anything interesting so that's why they don't do it the way you did it but there is nothing wrong with creating definitions at all and seeing where they go, mathematicians do it all the time.

>>12363534
I wouldn't say mathematicians define division by zero "all the time." In order to even make a consistent definition, you need to radically change what "division" or "zero" means, or both. Someone earlier mentioned wheels, which have a unary operator / that in some ways resembles reciprocation, but it's not really division. There are rings with zero divisors, but they are not division rings, and zero divisors are not cancellable.

And there is a good reason. Because division by zero *should* be undefined.

Engineers divide by (arbitrarily close to) zero all the time. They call it infinity (maxint).

>>12361641
Good enough. It was a vague memory from an EE class thirty years ago.

>>12360914
it's not a "solution for the square root of -1" exactly, but you might be thinking of
[eqn]e^{i\theta} = \cos(\theta) + i \sin(\theta)[/eqn]

>>12363876
>>12360914
https://en.wikipedia.org/wiki/Euler%27s_formula

multiplication used in i*ci isn't the same multiplication used in the field of real numbers, it is defined as (a,b)*c(c,d)=(ab-cd,ad+bc), where a, b, c, d are real numbers.

>>12360884
It gives you paradoxes, like in pic related.

>>12363838
maxint and infinity are not the same. Maxint is the maximum expressible value in an integer data type. For bigints, there is no maxint. In either case, it is not infinity, which you only get for floats. And of course, you do need to consider the sign.

>>12360884
>You cannot take the square root of a negative number because there is no number squared that produces a negative result.
i^2 produces -1

>>12364088
Not a paradox
[math]1*\infty = 2*\infty = \infty [/math]

if it could be done, Euler would have figured it out. He did not, therefore it is impossible. QED.

>>12360884
Sure you can do that. No utility is created though. You basically say anything that touches j is also j. j + 1 is j, j/j is j, j*infinity is j. You don’t get any utility from this.

>>12364088
0*infinity is undefined. There is nothing wrong with defining 1/infinity = 0, afterall this is what the projective real numbers do, it just means you can't have 0*infinity.

>>12361963
what? measurements must give real results.

>>12364291
You also can't have ∞+∞, which feels weird.

>>12364346
(i*)(i) = 1.

>>12360884
0 * j = 1
multiply both sides by 2
2 * (0 * j) = 2
associative law
(2 * 0) * j = 2
0 * j = 2
but
0 * j = 1 by hypothesis
hence
1 = 2
If you end up proving 1 = 2, you've done something wrong, somewhere.

>>12364565
Technically, that just proves that multiplication by j isn't associative.

>>12364346
We're measuring the Aharonov–Albert–Vaidman weak value of the number of apples. This is a weak measurement and the rules of strong measurements do not apply here.

>>12364791
based