BR Maths Corner-1

Vayutuvan · Post by **Vayutuvan** » 18 Sep 2013 01:11

SaiK wrote:Thanks for the inputs.. mat guru, could you pl give some 2 min brush on derivatives.. eons have past.

say pi = yq^2 + (x - c) q, and i want to know pi changes by varying q.

first derivative -> dpi/dq = ?
second derivative?

some steps. tia

You have already found the answer, I suppose. But it is simply

d pi/dq = 2qy + x - c
(pi)'' = 2y

Vayutuvan · Post by **Vayutuvan** » 18 Sep 2013 01:15

AmberG ji, that is an incredible visualization.

SaiK, as for Akamai, there is an article by Tom Leighton in CACM sometime in 2009. Also there was a survey article in ACM Computing Surveys around 2007 (IIRC).

Both would be worth looking at.

SaiK · Post by **SaiK** » 18 Sep 2013 03:04

yup thanks math guru.

amber g, nice!

btw, notice on the akamai visualization for chippanda nation. it is very late in the evenings they are heavy internet users.. any idea?

usa and india has similar day time usage pattern. some eu and africa nations is similar to chippanda as well.. but less intense.

may be it could be additional data points in the evening perhaps... but should be something these guys must be transacting heavy.

Amber G. · Post by **Amber G.** » 20 Sep 2013 08:21

For Serious Mathematicians, Physicists, Musicians ...
this is really very good.

Enjoy:
[youtube]2rjbtsX7twc#[/youtube]

Amber G. · Post by **Amber G.** » 21 Sep 2013 01:02

This is an amazing collection of maps showing racial diversity, or lack thereof, across US. Check out the maps of various cities and be sure to see the US map at the end. You can go anywhere in the country and zoom in almost to the street level. Each dot on the map is one person.

My sons had fun looking near neighborhood of MIT campus .. I can also zoom to our house.. a pale single red dot in the sea of blue...

Article

Interactive Map

SaiK · Post by **SaiK** » 21 Sep 2013 22:55

I think Indians would fill in those red Asian areas as well.

I was also looking Akamai documents.. trying to research more on what the company does.. got a report on them on chacha. wonder what platform other caching strategies and synch-up strategies to enable content distribution. They have to at least have a b-tree of nodes.

btw, do we know there was a math guru by the name kasturi lived here? where did he go?

brihaspati · Post by **brihaspati** » 26 Sep 2013 08:10

matrimc ji,
didnt visit this thread for a very long time.

Re: singularity of the Laplacian, for the graph - can you be so sure that just dropping one row and column will make it non-singular? I think it will simply mean you have eliminated one node and its edges. If this reduction makes the graph disjoint/disconnected completely [no connected components] then you can have full rank and therefore non-singularity. However, I think the number of zero eigenvalues corresponds to number of connected components - so just dropping one node need not give you full rank in the reduced dimensionality.

Vayutuvan · Post by **Vayutuvan** » 26 Sep 2013 10:18

Brihaspti ji, of course you are absolutely right. I had this note in my original post

But be careful - assume that the graph in question is connected. You can assume this without loss of generality.

Taking a recourse to physical reality, the scalar field is wrt to a ground node at infinity (or for heat equation, absolute zero). So in that sense all problems are within a single physical system - the universe we live in.

Added later:

In fact, let me expand on that a little bit. If the original graph (i.e. before dropping a row and a column) is connected, then the resulting graph could either be connected or disconnected which does does not matter. It will have a full rank. Let us consider a general electrical circuit. One can choose an arbitrary node as the ground node. The row and column corresponding to the ground node - WLOG say we choose node 1 as the ground node - dropping the row and column corresponding to the ground node will leave diagonal excesses at nodes which have conductances to the ground node. The resulting matrix is a symmetric positive definite diagonally dominant (SPDDD)* and hence has full rank. If one wants to solve a Laplace equation say on a cube made up of some conductor with uniform coefficient of conductance, i.e. isotropic material. Then an equally spaced finite difference grid will result in a special case of an electrical circuit where all conductances are of equal value and thus can be normalized to have unit conductances through out. What one would get is a Laplacian matrix and choosing an arbitrary ground node one can solve for the voltages at all nodes WRT to the ground node given injected currents at one or more nodes.

One can obviously setup an experiment, say in India, with a conducting cube made up of say ASI steel, ground a corner and connect a battery across some surface and measure voltages at various points on and in the cube. Another can conduct the same experiment say in "puRANA vilAyati" (old Blighty). If the measurements match - my educated guess is that they would but surely one has to conduct that and several other experiments - then we have to conclude that there is a unique solution implying that the matrix is non-singular.

* diagonals are all positive and off-diagonals are negative with diagonal dominance guaranteed due to dropping of the row and column corresponding to the ground node.

Vayutuvan · Post by **Vayutuvan** » 02 Oct 2013 09:04

Bankrupting Physics: How Today's Top Scientists Are Gambling Away Their Credibility
By Unzicker, Alexander
Author Jones, Sheilla

With assistance from science writer Jones, theoretical physicist and neuroscientist Unzicker compares the current state of theoretical physics to a bubble economy.

Publisher Comments

The recently celebrated discovery of the Higgs boson has captivated the public's imagination with the promise that it can explain the origins of everything in the universe. It's no wonder that the media refers to it grandly as the "God particle." Yet behind closed doors, physicists are admitting that there is much more to this story, and even years of gunning the Large Hadron Collider and herculean number crunching may still not lead to a deep understanding of the laws of nature. In this fascinating and eye-opening account, theoretical physicist Alexander Unzicker and science writer Sheilla Jones offer a polemic . They question whether the large-scale, multinational enterprises actually lead us to the promised land of understanding the universe. The two scientists take us on a tour of contemporary physics and show how a series of highly publicized theories met a dead end. Unzicker and Jones systematically unpack the recent hot theories such as "parallel universes," "string theory," and "inflationary cosmology," and provide an accessible explanation of each. They argue that physics has abandoned its evidence-based roots and shifted to untestable mathematical theories, and they issue a clarion call for the science to return to its experimental foundation.

Amber G. · Post by **Amber G.** » 02 Oct 2013 21:04

xpost - Of interest here...
Perhaps the most popular and important learning material of my generation ...
Feynman lectures now online

http://www.feynmanlectures.caltech.edu/

Caltech and The Feynman Lectures Website are pleased to present this online edition of The Feynman Lectures on Physics. Now, anyone with internet access and a web browser can enjoy reading a high-quality up-to-date copy of Feynman's legendary lectures. This edition has been designed for ease of reading on devices of any size or shape; text, figures and equations can all be zoomed without degradation.

I have attended many of his lectures and I think he was one of the greatest teacher. He knew his audience well and whether it was an undergraduate class or a hall full of professional physicists (not to mention an audience listening to his drums, or mardi gra crowd watching his band), he made everything look so easy.

I still remember when I first time visited Caltech (late sixties) and saw the parking space reserved for "Prof. Feynman" (Only one or two spaces available right near the docking lot of Physics building), it hit me that Feynman was actually a real human being and not just some super hero who produced those lectures.

Yayavar · Post by **Yayavar** » 02 Oct 2013 23:03

wow! Amber -- that must have been a great experience. I got those lectures when they were made available in Indian publication (much much cheaper) almost 20 years back. With me still. Thanks for the link.

Amber G. · Post by **Amber G.** » 04 Oct 2013 06:40

Thanks Viv,
****
Looks like what I posted here is gone viral ...

Canadian Physics Student’s Rendition of Bohemian Rhapsody Goes Viral

There is a similar one for Higgs boson too..

Amber G. wrote: For Serious Mathematicians, Physicists, Musicians ...
this is really very good.

Enjoy:
http://www.youtube.com/watch?feature=pl ... rjbtsX7twc

For the God's particle Higgs Boson ...

Amber G. · Post by **Amber G.** » 06 Oct 2013 20:57

Abraham Nemeth, 94, developer of Braille math code, dies

Abraham Nemeth, a blind mathematician and college professor who developed a widely used Braille system that made it easier for other blind people to become proficient in mathematics and science, died Oct. 2 at his home in Southfield, Mich. He was 94.

(Prof. Nemeth is a giant among teachers and advocates who made math easier to learn for Blind people..Great loss )

Amber G. · Post by **Amber G.** » 12 Oct 2013 23:20

Okay, just for fun, here is a recent problem from one of India's national math contest. (For high-school students).

Given that a,b,c are all positive (real) numbers
and
abc(a+b+c) = 3
Prove that (a+b)(b+c)(c+a) >= 8

(equality happens when a=b=c=1)

(Hint: Using algebra here is not too hard but a very beautiful solution exist if one is familiar with Brahmagupta's formula)

ArmenT · Post by **ArmenT** » 13 Oct 2013 12:35

Amber G. wrote: (Hint: Using algebra here is not too hard but a very beautiful solution exist if one is familiar with Brahmagupta's formula)

Which one? His area of a cyclic quadrilateral formula?

Amber G. · Post by **Amber G.** » 13 Oct 2013 12:59

^^^ Yes, actually a special case when one side is zero..(that is for a triangle).

SaiK · Post by **SaiK** » 13 Oct 2013 21:34

first i didn't understand ( my math is below high school now).. now you say the d is zero, me thinking onlee on the multiplier how to prove.

any value greater than 1 should definitely make it > 8?

Amber G. · Post by **Amber G.** » 13 Oct 2013 23:03

Saikji - All I was doing was to confirm ArmenT's post. I was talking about, what is also known as Heron's formula (see wiki - if unfamiliar) which is a special case of Brahmgupta's formula. (Again see wiki or any other good math source, for details.). How this fits in the problem is not at all obvious - which makes the problem beautiful.

What I was hinting at was that one can prove what I asked by geometric means in the sense that the result, if looked at the right way, becomes quite obvious if one thinks in geometric terms.

This may look confusing, but see if you can be prove the result by algebra (one does not even need calculus).

***
(BTW to satisfy abc(a+b+c)=3, and if a>1 you would prob. need one of (b or c or both) to be less than 1)
so I don't know exactly what you mean by

any value greater than 1 should definitely make it > 8?

SaiK · Post by **SaiK** » 14 Oct 2013 02:51

sorry I was being stupid, any one of the variable value being > 1, while keeping the other two variables at 1. and more stupidity onlee. pliss to ignore.. am going back to learning mode.

SaiK · Post by **SaiK** » 12 Nov 2013 09:09

unrelated, but indirect - besides losing the court case against limelight, akamai appears to be suing for DNS query.

whatever, it is a great read. never read technology court case so far
http://www.cafc.uscourts.gov/images/sto ... 9-1372.pdf

Amber G. · Post by **Amber G.** » 21 Nov 2013 22:58

Have not visited BRF for some time ...

Amber G. wrote:Okay, just for fun, here is a recent problem from one of India's national math contest. (For high-school students).

Given that a,b,c are all positive (real) numbers
and
abc(a+b+c) = 3
Prove that (a+b)(b+c)(c+a) >= 8

(equality happens when a=b=c=1)

(Hint: Using algebra here is not too hard but a very beautiful solution exist if one is familiar with Brahmagupta's formula)

Here is a simple proof using algebra ---
(We know that arithmetic mean is greater than or equal to geometric mean, IOW (a+b)/2 >= sqrt(ab))
anyway
LHS = (a+b)(b+c)(c+a)=(a+b)(bc+ab+c^2+ac)=(a+b)(ab+(a+b+c)c)
=(a+b)(ab+3/ab) (since (a+b+c) = 3/abc)
=(a+b)(ab+1/ab+1/ab+1/ab)
Now AM-GM inequality gives
(a+b)/2>=sqrt(ab)
(ab+1/ab+1/ab+_1/ab)/4 >= fourth_root_of(ab/((ab)(ab)(ab))) = sqrt(1/ab)
so LHS >= 2 sqrt(ab)*4*sqrt(1/1b) = 8
QED.
***
But geometric solution is quite neat!
(Equilateral triangle has the largest are inscribed in a given circle)

If one is familiar with Brhamgupta (or Heron's) formula for area of triangle...
whose sides are (x,y,z) where x=a+b,y=b+c,z+c+a)
(so that a+b+c = (1/2)(x+y+z) = s (say))

Then abc(a+b+c) is nothing but ((s(s-x)(s-y)(s-z)) or (square of) area of triangle of the sides (x,y,z), and
we have to prove the product of sides is greater than on equal to 8.. (Obvious result if one remembers the radius of the circumcircle of the triangle is nothing but (xyz/4A))

ramana · Post by **ramana** » 22 Nov 2013 00:05

AmberG, What are fractional derivatives and what iare the analog in physical world?

Amber G. · Post by **Amber G.** » 23 Nov 2013 18:51

^^^ Checkout, for example: wolfram

vbedekar · Post by **vbedekar** » 24 Nov 2013 16:02

To honour Bhāskarācārya, India’s celebrated mathematician and astronomer, on the 900th anniversary of his birth, Vidya Prasarak Mandal, Thane, Maharashtra, is organizing an International Conference on Friday, 19th; Saturday, 20th and Sunday, 21st-September 2014 at Thane, Maharashtra, India - 400 601.

Bhāskarācārya was born in 1114 in a family of scholars who cultivated Jyotisa as a family tradition for several generations. He mastered all the traditional branches of learning and made valuable contributions to mathematics and astronomy through his writings. Comprehensive treatment of the subject, careful organization of the material, lucid exposition and high poetic quality of his works made them near-canonical in the subsequent centuries. Bhāskarācārya works were studied throughout the country and several commentaries were composed on them.
For more information visit website
http://www.vpmthane.org/bhaskara900/

ramana · Post by **ramana** » 03 Dec 2013 04:03

Anybody saw the pictures of the NYC train crash? it doesnt look like classic case of buckling.

Reason I post there is long ago in college there was book 'Applied Mathematics for Engineers and Scientists' by Louis Pipes where he studied the buckling equation for trains being impacted from one end and held at the other end. It ends up with a sine-wave deformation.

The NYC crash pictures look like they jumped the rails on a curve.

Most likely excess speed.

---
AmberG, Thanks. What physical phenomenon can we associate with fractional derivatives?

Amber G. · Post by **Amber G.** » 05 Dec 2013 09:38

^^^ NYC Train speed (82 mph) was almost three times the speed limit (30mph) at that sharp curve.

Amber G. · Post by **Amber G.** » 06 Dec 2013 04:43

FWIW: Some may find it interesting ...
From http://mathwithbaddrawings.com/

Math Experts Split the Check

Engineer: Remember to tip 18%, everybody.

Mathematician: Is that 18% of the pre-tax total, or of the total with tax?

Physicist: You know, it’s simpler if we assume the system doesn’t have tax.

Computer Scientist: But it does have tax.

Physicist: Sure, but the numbers work out more cleanly if we don’t pay tax and tip. It’s a pretty small error term. Let’s not complicate things unnecessarily.

Engineer: What you call a “small error,” I call a “collapsed bridge.”

Economist: Forget it. Taxes are inefficient, anyway. They create deadweight loss.

Mathematician: There you go again…

Economist: I mean it! If there were no taxes, I would have ordered a second soda. But instead, the government intervened, and by increasing transaction costs, prevented an exchange that would have benefited both me and the restaurant.

Engineer: You did order a second soda.

Economist: In practice, yes. But my argument still holds in theory.

The computer scientist lays a smart phone on the table.

Computer Scientist: Okay, I’ve coded a program to help us compute the check.

Mathematician: Hmmph. Any idiot could do that. It’s a trivial problem.

Computer Scientist: Do you even know how to code?

Mathematician: Why bother? Learning to code is also a trivial problem.

Engineer: Uh… your program says we each owe $8400.

Computer Scientist: Well, I haven’t de-bugged it yet, if that’s what you’re getting at.

Physicist: This is a waste of time. Let’s just split it evenly.

Economist: No! That’s so inefficient. Let’s each write down the amount we’re willing to put in, then auction off the remainder at some point on the contract curve.

Physicist: Huh?

Mathematician: Like most economics, that’s just gibberish with the word “auction” in it.

Engineer: Look, it’s simple. Total your items, add 8% tax, and 18% tip.

Mathematician: Sure. Does anybody know 12 plus 7?

Computer Scientist: You don’t?

Mathematician: What do I look like, a human calculator? Numbers are for children, half-wits, and bored cats.

The engineer looks at the cash they’ve gathered.

Engineer: Is everyone’s money in? It seems we’re a little short…

Physicist: How short?

Engineer: Well, the total was $104, not including tip… and so far we’ve got $31.07 and an old lottery ticket.

Physicist: Close enough, right? It’s a small error term.

Mathematician: Which of you idiots wasted your money on a lottery ticket?

Economist: I should mention that I’m not planning to eat here again. Are any of you?

Computer Scientist: What does that matter?

Economist: Well, in a non-iterated prisoner’s dilemma, the dominant strategy is to defect.

Engineer: Meaning?

Economist: We should be tipping 0%, since we’ll never see that waiter again.

Computer Scientist: That’s awful.

Physicist: Will the waiter really care – 0%, 20%? Let’s not split hairs. It’s a small error term.

The engineer looks up from a graphing calculator.

Engineer: All right. I’ve computed the precise amount each of us should pay, using double integrals and partial derivatives. I triple-checked my work.

Mathematician: Didn’t we all order the same thing? You could have just divided the total by five.

Engineer: I could? I mean… of course I could! Shut up! You think you’re so clever!

Economist: So, we’re all agreed on a 0% tip?

Computer Scientist: Well… the waiter did only bring two orders of fries for the table.

Physicist: We only ordered two.

Computer Scientist: Exactly. We got the 1st order, and the 2nd, but never the 0th.

Economist: I’ll be frank. At this point, my self-interest lies in not paying. And the economy prospers when we each pursue our individual self-interest. See you later!

The economist dashes off. The engineer and computer scientist glance at one another, then follow.

Mathematician: Looks like it’s just me and you, now.

Physicist: Good. The two-body problem will be easier to solve.

Mathematician: How?

Physicist: By reducing it to a one-body problem.

The physicist scampers away.

Mathematician: Wait! Come back here!

Waiter: I notice your friends have gone. Are you done with paying the check?

Mathematician: Well, I’ve got a proof that we can pay. But I warn you: it’s not constructive.

abhishek_sharma · Post by **abhishek_sharma** » 06 Dec 2013 11:01

I can't find a thread on Computer sci. so this goes here.

Celebration of John McCarthy's Accomplishments

Vayutuvan · Post by **Vayutuvan** » 14 Dec 2013 22:08

ramana garu:

ramana wrote:AmberG, Thanks. What physical phenomenon can we associate with fractional derivatives?

I was going through a site and came across this. It might help. Supposed to be collection of arXiv articles - survey and tutorial type on Fractional Calculus.

The Net Advance of Physics: FRACTIONAL CALCULUS

Vayutuvan · Post by **Vayutuvan** » 24 Dec 2013 07:12

Finally (not that it matters to one of the greatest thinkers of humanity, but still)

Enigma codebreaker Alan Turing receives royal pardon

SaiK · Post by **SaiK** » 28 Dec 2013 08:41

yes it matters, the great find of yours for math reference. thanks.

and I have nothing to offer on the matters that does not concern.

btw, from the quotes section: what is this?

6accdae13eff7i3l9n4o4qrr4s8t12ux.
Isaac Newton, 1676

Vayutuvan · Post by **Vayutuvan** » 28 Dec 2013 12:02

As I said in LGBT thread, here is a link to Dr. Andrew Hodges who works on Twistor Theory developed by Sir Penrose. Twistor theory material can be found at Penrose's website along with material written by Dr. Hodges (who, IIRC, was a student of Penrose's). I have read some work (collected papers of Turing) a few years back, but not the biography by A. Hodges.

Andrew Hodges (From Wikipedia, the free encyclopedia)

Andrew Hodges (born in London, 1949) is a mathematician, an author and an activist in the gay liberation movement of the 1970s.

Since the early 1970s, Hodges has worked on twistor theory which is the approach to the problems of fundamental physics pioneered by Roger Penrose.

Hodges is best known as the author of Alan Turing: The Enigma, the story of the British computer pioneer and codebreaker Alan Turing.[1] The book was chosen by Michael Holroyd as part of a list of 50 'essential' books (that were currently available in print) in The Guardian, 1 June 2002.[2] He is also the author of works that popularize science and mathematics.

He is a Tutorial Fellow in mathematics at Wadham College, Oxford University.[3] Having taught at Wadham since 1986, Hodges was elected a Fellow in 2007, and was appointed Dean from start of the 2011/2012 academic year.

His book, Alan Turing: The Enigma, is being made into a film starring Benedict Cumberbatch as Alan Turing which will be released in 2014.[4]

Books by Andrew Hodges[edit]

Alan Turing: The Enigma, Vintage edition 1992, first published by Burnett Books Ltd, 1983. ISBN 0-09-911641-3.
One to Nine: The Inner Life of Numbers, Short Books, London, 2007. ISBN 1-904977-75-8.
With downcast gays: Aspects of homosexual self-oppression, Pink Triangle Press, 1977. ISBN 0-920430-00-7.

------------

SaiK, did not get the reference tro Newton. Could you please elaborate further?

SaiK · Post by **SaiK** » 28 Dec 2013 22:59

http://mathpages.com/home/quotes.htm

Amber G. · Post by **Amber G.** » 29 Dec 2013 00:08

Editorial in Boston Globe about Alan Turing.
Britain: Reclaiming the hero it maligned

BTW, there is a movie coming out where Benedict Cumberbatch is playing Alan Turing in " Alan Turing: The Enigma" A still shot from ..

Amber G. · Post by **Amber G.** » 29 Dec 2013 00:24

SaiK wrote: btw, from the quotes section: what is this?
6accdae13eff7i3l9n4o4qrr4s8t12ux.
Isaac Newton, 1676

The quote, the legend has, comes from a letter he wrote to Leibniz (or Oldenburg ?). I am quoting it:

The foundations of these operations is evident enough, in fact; but because I cannot proceed with the explanation of it now, I have preferred to conceal it thus: 6accdae13eff7i3l9n4o4qrr4s8t12ux. On this foundation I have also tried to simplify the theories which concern the squaring of curves, and I have arrived at certain general Theorems.

The anagram expresses, in Newton's terminology, the fundamental theorem of the calculus: (If you break the code written in Latin) "Given an equation involving any number of fluent quantities to find the fluxions, and vice versa."

If you are curious, arrange the letters of above ..("6a" becomes "aaaaaa" and "13e" becomes "eeeeeeeeeeeee" etc..(aaaaaaccdaeeeeeeeeeeeeeeff... etc) and one gets
"Data aequatione quotcunque fluentes quantitates involvente, fluxiones invenire; et vice versa"

(Some other ciphers around the same time... by Hooke:)
Hooke, A Description of Helioscopes and some other Instruments, Royal Society, 1676

cdei(2)n(2)o(2)ps(3)t(2)u(2) ut pondus sic tensio

(As the weight, so the extension - First statement of Hooke's Law of elasticity.

or
cei(3)nos(3)t(2)uv ut tensio, sic vis
("As the extension, so the force" - Restatement of Hooke's Law of elasticity)

Vayutuvan · Post by **Vayutuvan** » 29 Dec 2013 00:52

AmberG +1. Nice.

SaiK · Post by **SaiK** » 29 Dec 2013 01:29

boy, thanks.. that could potentially be an info hiding algo for aam brains like me.

Amber G. · Post by **Amber G.** » 29 Dec 2013 02:34

^^^ You may enjoy A Passion for Mathematics: Numbers, Puzzles, Madness, Religion,

The newton's quote is on pp 30, there are some other interesting pieces eg Namgiri writing equations on Ramanujan's tongue etc..

Eric Demopheles · Post by **Eric Demopheles** » 20 Jan 2014 21:49

An achievement of a Chinese-born mathematician in remarkable circumstances:

https://www.simonsfoundation.org/quanta ... prime-gap/

he had found it difficult to get an academic job, working for several years as an accountant and even in a Subway sandwich shop.

Edit: Ooh, sorry, had visited this thread after a long long time, and didn't see that kasturi already reported it.

Eric Demopheles · Post by **Eric Demopheles** » 20 Jan 2014 22:27

Nilesh Oak wrote: Do we any evidence for existence, usage or inkling of complex numbers in ancient Indian mathematics/astronomy/trigonometry..etc., anytime before say 1500 CE ?

Inkling. Ah. Good word.

Yes, there is. Indians had already developed quadratic equations. For example:

http://en.wikipedia.org/wiki/Brahmagupta

This is a rather subtle point, via considering math as an abstract manipulation of *stuff*. Here examples of stuff include numbers(of various sorts, natural numbers, real numbers, etc.), geometric objects(lines, points, parabolas, etc.) and more abstract things like polynomials, differential equations, etc..

This is in the same sense as: When you already have natural numbers and addition and subtraction, you already have an *inkling* of negative numbers, which are nothing but a formal setup added to ensure that you can always subtract. Here the stumbling block is that first you need to arrive at zero; but once you have the notion of zero, it is easy to teach someone about negative numbers.

When you consider the positive numbers and quadratic equations based on them, you have all the background for complex numbers ready and at this point a skilled instructor can already introduce complex numbers. Nothing more is required. It is about just formal manipulation of "stuff needed to denote solutions of polynomial equations." In fact the standard way of introducing it is just this: Consider the solutions of the equation x^2 + 1 =0, and call them i and -i, and then it so happens that all the solutions of all the polynomial equations can be expressed in terms of the real numbers and this extra "number" -- this statement is called the fundamental theorem of algebra. Actually, from a philosophical viewpoint, the passage from real numbers to complex numbers is an easier leap(assuming the learner is well-versed in algebra) than the passage from natural numbers to negative numbers, because for the latter, the enormously important(philosophically important) notion of zero is first necessary. That step, although very simple, took a long long time for civilization to arrive at.