Another Form for Complex Numbers (1)

To extend the generality of complex numbers and enhance their applicability, it's useful to write them in what's called "polar form".  This instalment and the next will deal with that treatment. A critical part is finding the angle shown, referred to as the argument. We can see from the diagram (Fig. 1) that the angle Θ may be found using:

arctan (y/x) = arctan(3/4) = 36.8 deg


Thus, Θ = 36.8 degrees is the argument

Now, any complex number (x + iy) may be written in polar form:

x + iy = r(cos (Θ) + isin(Θ))

And to get r:

r = [x² + y²]^1/2 =  [4² + 3²]^1/2 = [25]^1/2 = 5

Therefore we may write:   (x + iy) = 5(cos (36.8) + isin(36.8))

Note there is also the abbreviated function (based on the combo of sine and cosine):

cis (Θ) = cos (Θ) + isin(Θ)

so we can finally write:

C = r cis(Θ) = 5 cis (36.8)

Now we look at the vectors A and B, which we’ll henceforth call z1 and z2 to be consistent with complex notation. Our eventual goal will be to find the resultant, which will come in the next installment. In the meantime we will be working toward showing the multiplication and division of two complex forms, call them z1 and z2: 

e.g.  [z1 + z2]

From the diagram:

A= z1 = -2 + 2i

B = z2 = -2 -3i

So: z1 = x1 + iy1

And  arg(z1) = arctan(y1/x1) = arctan (-2/2) = arctan(-1)

So (Θ1) = -45 degrees = -π /4

Now find r1:

r₁ =[x₁² + y₁²]^1/2 =  [1 + 1]^1/2 = Ö2   Therefore:


z1 = Ö2 (cos(-45) + isin(-45)) = Ö2 cis(-45)

We now turn to the vector B which is:  z2 = x2 + iy2= -2 -3i

then: arg(z2) = arctan(y2/x2) = arctan (-3/-2) = arctan (3/2) = 56.3 deg

While:

r₂ = [x₂² + y₂²]^1/2 =  [(-2)² + (-3)²]^1/2 = [13]^1/2 =  3.6



Therefore:

z2 = 3.6(cos(56.3) + isin(56.3) = 3.6 cis(56.3)
Now, how do we obtain the complex product: [z1•z2]?

We have that:

[z1•z2] = (z1•z2) cis(arg(z1) – arg(z2))

But:

(z1•z2) = Ö2 (3.6) = 5.1

And:

arg(z1) – arg(z2) = (-45) – (56.3) = -101.3

so that:


[z1•z2] = 5.1 cis(-101.3) = 5.1 (cos (-101.3) + isin(-101.3))


[z1•z2] = 5.1((-0.195) + i(-0,98))


[z1•z2] = 0.99 + 0.98i

To get the resultant: z1 + z2 = z3:

A + B = z1 + z2 =[ (-2 + 2i) + (-2 – 3i)] = -4 –i

In any case:
x3 + iy3 = - 4 – i

Problems for the Math Maven:

1) Based on the resultant for (x3, y3) obtain arg(z₃)  and thence the polar form for the resultant. (Hint: Remember  arg(z₃)  =  Θ )

2) Now combine this resultant z₃ with that from C (Fig. 1) which we call z₄, to obtain z₅

Thence, find arg(z₅) and write in polar form.

3) Obtain the complex product for [z3• z4]

Why Is Breast Cancer Incidence Increasing for Younger Women?

A paper published in Wednesday's edition of the Journal of the American Medical Association., confirmed the suspicions of many oncologists who had noticed an uptick in patients younger than 40 with cancer that had spread to the bones, brain or lungs. In 1976, 1.53 out of every 100,000 American women 25 to 39 years old was diagnosed with advanced breast cancer, the study found. By 2009, the rate had almost doubled to 2.9 per 100,000 women in that age group — a difference too large to be a chance result.


According to City of Hope Cancer Center surgeon Dr. Benjamin Paz, who was not involved in the study. “Looking at a longer period of time, this study shows there's clearly been an increase. It's the first to do so."

The trend, which has yet to be explained, has raised real concerns about future efforts to treat the disease. Survival rates for young women with metastatic breast cancer are much lower than they are for older women, because the cancer tends to behave more aggressively in the young.  Meanwhile, the data from the U.S. Surveillance, Epidemiology and End Results, or SEER, database detected a significant increase among black and white women in both urban and non-urban areas, suggesting that the root cause or causes were widespread. This means whatever the cause or agent it can’t be a localized one.

It also (again) casts suspicion on the most likely culprits which are used across the nation: synthetic pesticides, weedicides (Most of which are banned in Canada), and bisphenol-A, a hormone imitator used in plastics. In her book  'The Body Toxic' , (pp. 148-49), author Nina Baker observes:

"Bio-monitoring studies of bisphenol A in human blood and tissue suggests that people are already exposed to levels that far surpass the current government reference. Humans quickly metabolize bisphenol A, so in order to account for the levels detected, people must be exposed to ten times more than the current acceptable intake level".


Risk benefit analyses, many done in Canada, show that even a billionth of a gram of bisphenol- A can engender toxic effects, such as causing human cells to mutate toward a cancerous form.

Unsatisfied with the usual health industry narrative that “people eat too many fats” – Baker and other women have actually undertaken “body burden analyses” to assess how many chemical toxins are in their blood. One RN cited, found 105 chemicals in her blood - all linked in animal studies to devastating health effects, including cancers of the liver, bone, pancreas, bladder and breast...not to mention disruption of the hormone system, and birth deformities (such as hypospadia in male infants, now occurring more and more in proportion to mothers' use of cosmetics).


Baker (p. 17) obtained her own body burden analysis performed by a lab in Manchester, England for $2,000. What she learned 8 weeks later floored her. An attached spread sheet disclosed her body contained at least three dozen highly toxic chemicals, including DDT and PCBs. These two chemicals and their metabolites (the products left after the original substance breaks down) are now routinely detected in the CDC's bio-monitoring program.

The U.S. Toxic Substances Control Act (TSCA) of 1976 was supposed to limit all this infiltration of chemicals, but it hasn't worked. Why not? Probably before its implementation it conferred a "blanket approval" (op. cit. p. 47) on some sixty two thousand chemicals used in commerce. As Baker observes (ibid.):


"No questions were asked. No hazard data were required. It's not suprising therefore that 99% (by volume) of chemicals used today are older substances that were grandfathered in under the toxics act, according to Inform, a New York based research organization."

Another obstacle to management and control are the permits for chemical manufacturers to deem "confidential" the most carcinogenic or debilitating of their chemicals - as also allowed under the TSCA. Baker notes that (ibid.):

"In 1998, for example, 40 percent of the substantial risk notices filed by manufacturers asserted the identity of the chemical was confidential".

This is serious, because it means that we the citizens don't know really how many serious poisons are being sucked into our bodies every day, or exactly how toxic these substances are! Even a simple lotion being used by a woman could have a devastating chemical in it, say PFOA -based, that causes monstrous birth or sexual defects in offspring or children around her.

Meanwhile, dedicated researchers at Tufts University produced groundbreaking work (p. 159) linking bisphenol A to pre-cancerous mammary tissue and breast cancers in mature rats, as well as incepting infantile endocrine system disruptions (DES syndrome) by virtue of the fact that a fetus can absorb bisphenol - A via its mother's blood.

Of course, many organs of the government-corporate nexus have seen fit to block these results from seeing the light of day. No surprise, since most Americans, cocooned as they are from real news and information, would literally shit bricks if they knew the extent to which they were being poisoned by their corporate Overseers. (Maybe to hasten their deaths so they can’t collect “entitlements”?)

These Overseers (mainly in the Chemical Industrial Complex) induce 548,000 cancers a year then make the bozos think it's their foods that are responsible. Don't eat that hot dog! It's got nitrites and you''ll get cancer! (Just move along, never mind that toxic chemical salesman trying to sell atrazine over there).

Worse, they’re at the other end of the pipeline ALSO manufacturing the chemicals for chemotherapy to treat many of the cancers!

Given the sheer volume of synthetic pesticides and hormone disrupting chemicals has nearly tripled since 1976, it’s a wonder young women’s breast cancer rates have only doubled. Sadly, our women are the guinea pigs in a long running chemical experiment almost no one is aware of.

Fortunately, some places such as Boulder, CO, are declaring city—owned parks and other open space areas as “pesticide-free”. Rather than using these synthetic chemicals they’ve recruited hundreds of volunteers for manual management of weeds, pests etc. This has incited ‘turf wars’ in Colorado where other “green lawn” communities are terrified of being deprived of their chemical-laden green lawns by the manual management advocates (see e.g. ‘Chemicals Friend, Foe in Turf Wars’, Denver Post, February 10, p. 9B)
Alas, those communities may well preserve their green lawns, but at the cost of their young women burdened with increasingly aggressive breast cancers.

Meanwhile, since 2003 Quebec, Canada has enforced a full ban on lawn pesticides according to the David Suzuki Foundation (ibid.), and not surprisingly their breast cancer rates have not spiked like in the U.S.

Do we value the lives of our young women? If we do it’s time we let the truth be known about how our overuse of toxic chemicals is mauling their bodies!  How many breasts have been removed in mastectomy to rid younger women of cancers from these chemicals we will never know. How many younger women (ages 29-40) have died because of an aggressive breast cancer due to synthetic chemicals, is also something we may never know. But it's past time we found out!

The Austerity Barbarians At the Gate: And What's with the Chained CPI Still in Play?

John Boehner blubbered and huffed all through his interview with Scott Pelley yesterday. "We need the Senate to stop sitting on its ass and pass that bill!"

Huh? WHO was taking a vacation the past ten days, Mr. Speaker? Are you still in a parallel universe? Anyway, Mr. Boehner still demands more spending cuts in order to appease his austerity Barbarians, now salivating at the gate.

But I don’t get it! We’ve already had by proportion two dollars of spending cuts for every increase in revenue, despite the fact it was a revenue implosion (via unpaid for ‘wars’, tax cuts) that dug us into this mess. So why are even more spending cuts being demanded as a way to preserve the existing small growth economy?


Even this morning (MSNBC), Pat Toomey (R-PA) was bleating nonsense about closing loopholes to then be able to LOWER marginal tax rates and “unleash the power of the economy”. NO! NO! NO! You aren’t going to unleash anything but larger deficits and debt as you grind the economy to a halt. The evidence has already been manifestly shown that NO tax cuts of ANY kind work for enhancing growth!

Matt Miller showed this in spades in his ‘The Tyranny of Dead Ideas’ (tax cuts being a ‘dead idea’ still retained out of a long held supply side mythology). Meanwhile, The Financial Times in its issue of 9/15/10 showed the Bush tax cuts to be the frauds they are by not even having achieved the most modest growth objectives. Their main conclusion? "The 2000s were the weakest decade in U.S. postwar history for real, non-residential capital investment. Not only were the 2000s by far the weakest period, the tax cuts did not even curtail the secular slowdown in the growth of business structures. Rather the slowdown accelerated to a full decline".

So why do Repuke tax cut fetishists hold on to them? Because they are a convenient means of “starving the Beast” along with unpaid for wars. In this way they believe they can ensure cooperative Dems will cut domestic spending, when they ought to be increasing it!


Flash forward to the current situation and the coming “sequester”. Barring an unlikely last minute deal, here's a short list of some of the massive, national bi-partisan-created austerity cuts arriving by Friday, according to the New York Times:


-600,000 food stamp recipients will be cut from the program

-Massive education cuts


The first will be murderously brutal given 1 in 3 kids in marginal income families ($24,000/yr for a family of 4) is already going to bed hungry and the mean benefit is only $3 /day with food stamps running out for most families within 3 weeks. A new documentary film, ‘A Place At the Table’ will highlight this abomination in the alleged “richest nation” on Earth. The sad thing is now these people will be in worse straits, all because the Repukes don’t want to even close one tax loophole for the richest hedge fund managers, or stock brokers, traders. See my earlier blog ( http://www.brane-space.blogspot.com/2013/02/scarborough-bwwaaaahaha-we-cant-have.html) on why the Rs are wrong, dead wrong on not increasing taxes!


Meanwhile, we learned this morning (‘Morning Joe’)that  most college students don’t even want to hear the word “sequester” and they flee from any subject tag for a blog, or tweet bearing it. Well, I hope not! The fact is they will be adversely affected in terms of being able to secure further (government-based)  college loans if this thing goes through. When your future interests are at stake you do pay attention, and you try to learn how your own welfare is threatened. In addition, thousands of teachers will be laid off as the federal support purse strings dry up, including for ‘Head Start’ and ‘Race to the Top’.

Other impending cuts:

-12 billion in Medicare cuts (more to come after 2013)

-Millions receiving unemployment will see their checks cut by 11% (an average of $132 a month)


So don’t tell me that the amount of the cuts ($85b) is ‘trivial’ or regurgitate the Repuke line that “Obama is using scare tactics”.

Note this is just for 2013. The current plan for the austerity "sequester" cuts is $100 billion of federal cuts every year for ten years, equaling massive cuts to jobs, Medicare, education, and completely destroying federally funded social programs.

We will be laid waste as never before, and left as a marginal Third World nation, perhaps even worse than Barbados.

Now, WHO is to blame? Despite the finger pointing in Washington, D.C. and the sequester being described as “stupid” (e.g. no targeted cuts) in many quarters, in reality both Democrats and Republicans created these "sequester" cuts (back in 2011 with the really stupid debt ceiling increase and Obama unwilling to invoke the 14th amendment instead) , and they can just as easily undo them with a snap of the finger. But both parties are choosing not to delete the cuts. They just don't want political responsibility for the fallout, which many economists have predicted will push the U.S. economy over the edge into official recession.


Obama could also just as easily appeal to the American public —over the heads of congressmen — to demand that the cuts be shelved forever. Instead, he's still proposing a "grand bargain” deal that he knows the Republicans won't go for.  What's in Obama's grand bargain deal? According to the White House website :


-$130 billion in "savings" [cuts] to Social Security, by implementing a ‘superlative CPI'
-$35 billion in "savings" [cuts] to the retirement of federal employees.

- $400 billion in health care "savings" [cuts], much of it Medicare cuts.


Obama (cynically, in the minds of some) fails to mention the words Social Security or Medicare in the above plan, choosing instead to write in code "superlative Consumer Price Index". But there’s nothing ‘superlative’ about prolonged, corrosive cat food cuts that demeans seniors, especially those who do not have the wealth to sustain lowered adjustments to inflation over time. Already Bernanke and his Beltway Club of “serious people” (e.g. assholes) are pushing the BS there’s “no inflation”, despite fuel, food, and medical costs soaring. Seniors even with moderate incomes therefore will be shafted as the chained CPI kicks in and woe betide them if a medical catastrophe strikes – especially with Medicare cuts on a $400 b scale.

Most real progressives’ jaws dropped last week when, incredibly, White House spokesman Jay Carney  told a press conference that Obama is a  "macho man". He told the reporters that President Obama is still willing to cut Social Security benefits by using the Chained CPI as the basis for the annual cost-of-living adjustment (COLA).  Believe me, Mr. Carney, there is nothing "macho" about sending 45 million seniors into cat food diets, and  forcing them to grovel in dumpsters for remains of kibbles when their meds end up breaking the bank!

This willingness to cut the benefits of retirees is presumed to establish President Obama as a “serious person” in elite (read: NEOLIBERAL)  Washington circles. But in progressive circles it defines him as a wuss and enabler of the Repukes. Moreover, it will defame the Dem brand in perpetuity if he goes through with this BS, marking him as the destroyer of the primary principles first put forward by FDR, JFK etc. Is THAT really worth a “legacy”?


Likely most of the D.C. Neoliberal insiders (e.g. Fareed Zakaria, Bob Woodward, Joe Klein etc.) probably don't understand the Chained CPI. However, everyone else should recognize that this technical fix amounts to a serious cut, never mind the odious PR to disguise it. It reduces benefits compared with the current schedule by 0.3 percent annually. This adds up over time. After someone has been getting benefits for 10 years, the cut in annual benefits is 3 percent, or equivalent to one whole month of Social Security. After 20 years, people would be seeing a benefit that is 6 percent lower, and after 30 years their benefit would be reduced by 9 percent. (For those interested, AARP has a nice calculator that shows how much retirees can expect to lose from the Chained CPI.)

Now, one can certainly debate whether the Chained CPI should be viewed as a "large cut," but there is no debate that Chained CPI cut imposes a bigger hit to the typical retiree than the ending of the Bush tax cuts were to the typical high-end one percent earner. Social Security provides more than half of the income for almost 70 percent of retirees. This means that the 3 percent cut in Social Security benefits amounts to a reduction in their income of more than 1.5 percent.

Let us sincerely hope that Obama isn’t gaming us, and trying to disguise an underlying Neoliberal instinct with progressive rhetoric. We didn’t go “all in” with him last November just to put a Neoliberal plant into office, when voters chose him because the alternative, “Mittens”, planned to do just what Carney proclaims Obama is “macho” enough to do now.

We deserve better. It’s time for Obama to show he is a fighter and for the Middle Class, including not enabling cuts to cripple seniors in their most vulnerable years.

Mr. President, take DOWN that CPI proposal from the White House site! Do NOT, I repeat do not, seek any "grand bargain" with the Republican ratfuckers!

Complex Numbers Solutions

The Problems again:


1) Find the sum of (1 –i) and (6 – 6i)

Solution:

(1 - i) + (6 - 6i) = (1 + 6) + (-i - 6i) = 7 - 7i


2) Find the sum of (0- i) and (15 – 0i)

Solution:

(0 - i) + (15 - 0i) = (0 + 15) + (-i - 0) = 15 -i



3) Subtract: (3 + 3i) from 10 – 4i

Solution:  

(10 - 4i)  - (3 + 3i) =  (10 - 3) + (-4i - (3i)) = 7 - 7i

4) Find: (1 –i) + (11 – 2i) – (4 – 4i)

Solution:  Write:

[(1 - i) + (11 - 2i)] - (4 - 4i)

Then: [(1 - i) + (11 - 2i)] = 12 - 3i

Suibtracting:

(12 - 3i) - (4 - 4i) = (12 - 4) + (-3i) - (-4i) = 8 + i

5) Represent the two parts of (1) as vectors on an Argand diagram, and show their resultant.

Solution: The vector construction procedure will be analogous to that depicted in Fig. 1 of the Complex numbers blog, except in this case the vectors are added end to end, starting with the first (ending at 1,-1) then the next, culminating with the resultant vector terminating at (7, -7) in the lower right quadrant of the Argand diagram. The overall resultant is at 45 degrees with respect to each axis (x, y), which is the same orientation as each of the component vectors. (How do you confirm this?)

6) Using the Argand diagram of Fig. 1 and complex addition, find the resultant of the vector sum B + C.

B = (2 - 3i) and C = (4 + 3i)

So vector addition will yield a resultant for a parallelogram (you need to complete the other two sides, e.g. B' and C') extending from the point (2,-3) and ending at the point (4,3) which is 6 units in length. That is, conforming to:

(2 + 4) + (-3i + 3i) =  6

Why is Downton Abbey So Addictive? (Spoiler Alert!)

In the latest issue of ‘New Republic’ (Feb. 25), in an interview(p. 34) with Aussie film critic Clive James, the topic of ‘Downton Abbey’ came up and he’s asked why Americans are so into it. First, he dismisses the series as “a dime a dozen” comparing it with previous Master-servant drama series, and ‘Upstairs-Downstairs’. He then adds, referencing Yankees’ addiction:

“It’s a reflection on America. They’d like to have that kind of aristocratic structure. Very soon Julian Fellowes will be in America writing the American equivalent.”


Having watched the first two seasons of the series with wifey, and started the third, I can’t agree. This makes me wonder if Clive James himself has watched it through, as opposed to cherry-picking a couple episodes and forming a presumptive opinion. The thing is, you have to watch it continuously and consecutively, you can’t do a ‘cafeteria-style’ pick and choose then arrive at any valid conclusion.

What we’ve found is the series is absolutely riveting, as much or more so than ‘Babylon 5’ the gripping scifi series produced and written by J. Michael Straczynski, which we’d just finished viewing for the 2nd time before commencing Downton. No, there are no Vorlons, TEEPs or ‘Shadow Wars’ in Downton Abbey, but none are needed. The intrigues with the servants downstairs, and the subtle collisions with their flawed ‘masters’ are every bit as engrossing as when Sheridan and his alliance takes up arms against Earth, and the Vorlons are found to have been the real masters of the Earthers and lesser races in the galaxy. (After the Shadow war).


The ‘Shadow war’ in Downton, meanwhile, is fought at multiple levels, between the footmen, butlers, maids, and cooks and between them and the aristocratic Crawleys. Just as Babylon had it’s n’er do wells, so does Downton, what with the scheming footman (now valet) ‘Thomas’ and Lady Crawley’s servant, O’Brien.


But Downton is not scifi, so why are hard-core science fiction aficionados like wifey and myself so attached to a post-Edwardian period drama set in old England? Maybe even addicted to it? This is hard to say, but perhaps boils down to the spectacle and drama of an isolated and privileged human enclave or mini-society (based at Downton) passing through one of the most turbulent eras of human history, taking in the sinking of the Titanic, the First World War, then the aftermath and the Spanish Flu epidemic (which killed 100 million) and the turbulent 1920s with the rise of fascism and formation of the Irish free state.


But do I think of aping the entitled British aristocrats while watching it? Hell no! How could I? Not with what they have to themselves endure. (And the one scene from season 2 when the chicken falls on the floor while being cooked, then brushed off and served to the Crawleys and friends, is choice.) Then there was O’Brien, believing Lady Crawley (Cora) had it in for her, i.e. to replace her, leaving a bar of soap near her tub and thereby inducing a miscarriage when Cora slips on stepping out of the tub. Terrorist servants anyone?

The image that occurred to me wasn’t lording it over the ‘help’ or grinding them under, but realizing that when a whole class of humans is demeaned and yes, exploited (though the servants believe they’re being cared for) it is likely they will strike back at their oppressors, and yes, become de facto terrorists. After all, the modern phenomenon of terrorism itself is a product of the weak being preyed upon by dominators of more powerful nations, then finally using the only means they know….terror, to strike back.


In Downton, one can even see a budding (IRA)  terrorist lurking in the Crawley household in the person of the chauffeur Branson, an Irish rabble rouser and sympathizer of Irish nationalism and ‘hurling off the yoke of English oppression”. The weird thing is that he actually marries one of the Crawley’s daughters (Sybil). She is determined to break away from the entitled clan anyway, and her announcement of her impending choice of fiancée nearly brings the Crawley household to eruption of its own.

So no, the attraction of Downton isn’t viewers envisaging the adoption of an American aristocracy, but rather seeing human struggles played out in a miniature cosmos, in a controlled setting that gradually sees their lives unraveled owing to outside pressures, and class wars.


In this way it bears close similarity to Babylon 5. The latter saw its own servant class fighting to stay alive in "down below" (Brown sector) while the principals in the higher levels engaged in their own conspiracies, bickering, and infighting,  in a space station housing 250,000.   Downton, meanwhile, is set in the confines of a mammoth home housing 25, masters and servants - all involved in their own conspiracies and struggles to get ahead. Terrorists, conspirators and loyalists stand out in Babylon and also in Downton. Both show that beneath the veneer of superficial physical setting we all face the same fears, basic goals, e.g. of security and a better life for our offspring. The human saga is thus both timeless and unlimited by geography, location.
This is perhaps a lesson that Clive James and other Downton critics need to learn.

Downton has been recognized by Guinness World Records as the most critically acclaimed English-language television series of 2011, and became the first international television series to receive the largest number of nominations in the history of the Primetime Emmy Awards, with twenty-seven in total. It was the most watched television series on both ITV and PBS, and subsequently became the most successful British costume drama series since the 1981 television serial of Brideshead Revisited.By the third year, it had become one of the most widely watched television shows in the world.

People in the U.S. could do lots worse than to invest time in Downton Abbey!

What are Complex Numbers?

Fortunately, mathematicians devised complex numbers long before the practical need for them arose. In the case of the latter, complex numbers are relevant to everything from the quantum wave function to oscillating-alternating circuits and wave forms. For example, the representation of monochromatic plane waves in free space:
E = E(o) exp[i(wt – kz)]

Or the voltage at the end of a transmission line:

V(x=0) = A exp iwt[l + K]

where w = 2πf  and k = 2π/ l

where f is the frequency, and  K is the voltage reflection coefficient.

It was inevitable that after the invention of counting numbers, i.e. integers, a new breed would be invented which made use of the imaginary number i. This is found when one solves the quadratic:

x² + 1 = 0

To get in the first step:

x ² = -1

Then solving for x:   x = (-1)^½

  Or the square root of minus 1. This is defined and referred to as i.   Thus: i×i= -1

Placement of numbers follows a similar analogy to the placement of real numbers on the Cartesian x-y axes. In this case, one uses an Argand diagram with axes:

iy

^
!
!

!

!

!
!---------------------------------------------------> x


Thus, the basis is laid to place complex numbers, say of the form: a +bi, on the above graph, where a is located by using the ordinate (x-axis) and b by using the iy (or imaginary ) axes. (Note: in most cases the i is dispensed with in the diagram)

For example, to find: 2 + 3i, you would mark off 2 on the x axis, and 3 on the iy axis.

Addition and subtraction of complex numbers follows simple rules and is straightforward. The main rule is to keep imaginary and real parts separate when performing the operations. For example:

1 + 2i + 3 – 3i = (1 + 3) + (2i – 3i) = 4 - i

And:

7 + 8i - (3 – 3i) = (7 – 3) + (8i – (-3i)) = 4 + 11i

An Argand diagram in “action” is shown in Fig.1. Here, several vectors are represented. We see that the vector A = -2 + 2i, B = -2 – 3i, and C = 4 + 3i.

How would we combine the vectors A + B? We can use the simple addition process:

A + B = (-2 + 2i) + (-2 – 3i) = -4 –i

Using Fig. 1 to complete the parallelogram formed by the vector should lead the reader to see the resultant terminates at the coordinate (-4, -1) (Actually, -4, -i)

What about adding the vectors A + C?

A + C = (-2 + 2i) + (4 + 3i) = 2 + 5i


Completing the parallelogram, the reader should be able to satisfy himself that the resultant terminates at
 (2, 5) or just beyond the upper limit of the graph.

Problems:

1) Find the sum of (1 –i) and (6 – 6i)

2) Find the sum of (0- i) and (15 – 0i)

3) Subtract: (3 + 3i) from 10 – 4i

4) Find: (1 –i) + (11 – 2i) – (4 – 4i)

5) Represent the two parts of (1) as vectors on an Argand diagram, and show their resultant.

6) Using the Argand diagram of Fig. 1 and complex addition, find the resultant of the vector sum B + C.

Worried About My Next Trip to Vegas? Hell NO!

Caesar's Palace casino in Las Vegas- is terrific to visit, but be sure not to miss the many shows as well - including some that are free.


The shootout and fiery early morning crash last Thursday on Las Vegas’ famed Strip (intersection of Flamingo Road and Las Vegas Blvd.) has many previous visitors turning into ‘nervous Nellies’ regarding future trips. One theory bruited in yesterday’s paper (Denver Post, p. 14A) is past location recall and identification. Basically, as the news story flashed on, some fraction of the 40 million yearly visitors will look and say: “I was THERE! I walked that part of the Strip! It could have been ME in the middle of it!”


Errrr…..not quite! The shootout and crash actually occurred at 4.20 a.m. and most good little do-be’s that visit Sin City are bedded down in their respective hotels and firmly in the Land of Nod, not out skylarking on the Strip. (As one long time Vegas’ hobo put it when interviewed, “Nothing good happens at that hour of the day!”)

But that’s true in ANY city, from San Francisco to Colorado Springs, to Milwaukee. Check their police logs and the greatest number of bar fights, stabbings, assaults and shootings occur after midnight. It’s when basically all the losers and n’er do wells come out and well,…..start trouble.

The interesting aspect here is that Vegas’ crime stats are actually falling, this despite several violent incidents so far this year. As the Post article notes:

“Violent crime, which includes murder, rape, robbery and assault in the city’s main tourist hub fell 13 percent in 2012, from 256 to 223 incidents, and is down 11 percent for the first part of 2013.”

In other words, the perception is worse than the reality.

Hence, it is clear that the inflation of one or several incidents (like the Thursday morning shootout and crash) have the potential to inflate the psychological identification and derivative perception, but this is wrong. The chances of anyone being shot or blown up in Vegas are likely many times lower than any other major U.S. city, precisely because Vegas has an investment in protecting its image as an adult “Disneyworld”.

The recent incident, we now know, was a case of two rappers head-butting. The altercation actually started at the nearby Valet service of the Aria Hotel and spilled over into city streets. (Jeebus, some rappers, like certain "pastors", have a real pathological ego problem!). Like the Tupac Shakur incident back in 1996 it was basically a one-off, certainly for another long while.  Would -be visitors need to put these sporadic, oddball bursts of violence behind them.

Having visited Vegas three times in the past eight years, the most disturbing incident wifey and I beheld was the collapse of a sixty -something guy near the exit of the Excalibur casino back in 2005. He’d evidently dropped dead from a heart attack. The most upsetting event was a very pathetic buffet breakfast taken at the Wynn Resort in 2009, with all the eggs and ham, bacon etc. at lukewarm temperatures. When my wife complained, the chef gave her a fast pass ‘voucher’ to the next day’s breakfast. She tore it up. Thereafter, though we were guests at the neighboring Encore, we took our breakfast buffet at the superb Bellagio. (All primary breakfast foods, such as meats, eggs, regularly have their temperatures taken.)

But by and large, all our experiences in Vegas have been fantastic, from enjoying the Bellagio’s musical fountains, to the sexy Sirens of Treasure Island, to seeing Cirque de Soleil shows, to enjoying the Fremont Street attractions (including the light shows) and seeing the whole city at night from the overlook at the top of the Paris Hotel. Oh, we also thoroughly enjoyed playing the slots – penny slots! – where we won most of our money (an average of $100 net gain on each of our visits). The last time we even faced off at the Encore against twenty other guests in a slot competition, to see who could ring up the most points in ten minutes.


What’s your biggest physical danger when visiting Las Vegas? Easy! It’s gorging on rich foods at too many buffets and clogging your arteries! The food at any of the hotel casinos is generally fantastic, and there’s plenty of it: whether you go to the Luxor’s breakfast buffet (or the Bellagio’s), or the Bally’s lunch buffet. The temptation always is to “get your money’s worth” although truth be told, the prices are already rock bottom.

Bottom line, if you’re planning a trip to Vegas, don’t let the recent bad press deter you. All you need to do is use a grain of sense, as you would visiting any other place.