-5=5

[echoVanguard]

I admit I forget what i*i ends up looking like.

i * i = -1. That's kind of the point of i.

echo

[shadzar]

I admit I forget what i*i ends up looking like.

i * i = -1. That's kind of the point of i.

echo

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

There are in fact two square roots of −1, namely i and − i, just as there are two square roots of every other real number, except zero, which has one double square root.

and here is where the square rooting of i becomes a problem, because i^2...

the imaginary unit's core property is that i^2 = −1. The term "imaginary" is used because there is no real number having a negative square.

[sabs]

Firstly
-1^2 and (-1)^2 are the exact same mathematical statement.

No, they're not the same thing. I know citing references doesn't do any good in this forum, but I'll mention that "Elementary Algebra for College Students" (Angel, 7th edition), notes on page 70 that

-4^2 = -16, and that (-4)^2 = 16.

Seriously, you could just use a calculator and see this with your own eyes.

In Arabic notation (i.e., the handwritten and book notation for writing arithmetic/algebra), it's acceptible to not write a zero in many cases, even when there could be one.

-4, really means "0 - 4", unlike (-4), which means "the additive inverse of 4". Computers don't actually use arabic notation, but instead a version of it suitable for computers, -4^2 might actually mean something different, depending on the computer language, even if in Arabic it's -16.

That's totally fucked up notation.
-1^2 should not default to reading as -(1^2).
That's completely different to how I was taught algebra, admittedly back in the stone age.

[Doom]

That's totally fucked up notation.
-1^2 should not default to reading as -(1^2).
That's completely different to how I was taught algebra, admittedly back in the stone age.

It's more likely you simply forgot the subtleties of the notation--it's an easy, easy, thing to forget, to judge by my students (or it's possible your teachers were of marginal competence, that happens often as well).

And you're correct, it's horrible notation, but it's the one we use nowadays. There are many other forms of mathematical notation, all of them have good parts and bad parts (one of the crazier ones is the barely understood textile-math of the MesoAmericas, which is probably similar to the Vedic method you can find on YouTube). I've even seen a case made that the Roman numeral system makes some types of calculation much easier.

Back to our miserable Arabic notation of the modern world, Arthur Clarke, a physicist and author, in one of his novels ("Songs of a Distant Earth", I think) has the physicists prove there's no energy in a vacuum. 50 years after the proof, the physicists realize everyone missed a minus sign, and there was actually an infinite amount of energy in a vacuum, and humans developed space travel shortly thereafter.

The point of the story? Even people with very advanced mathematical skills easily get messed up with the notation, especially as it relates to minus signs.

[shadzar]

The point of the story? Even people with very advanced mathematical skills easily get messed up with the notation, especially as it relates to minus signs.

that is why, and thankfully for me i did, it should be learned the computer notation so there isnt something missed and dont discard "obvious" things.

(-1)^2 as opposed to -(1^2)

dont discard the parenthesis.

notation should include it as would be said in words or it is read.

negative one squared, and the negative of one squared...respectively.

[fectin]

http://imgs.xkcd.com/comics/words_that_end_in_gry.png

[Maj]

Firstly
-1^2 and (-1)^2 are the exact same mathematical statement.
They both are short hand for -1 * -1 which /always/ equals 1.

I did not learn this is any math class I was ever in. I learned that:

-1^2 --> -(1^2) = 1
(-1)^2 = 1

Finally understanding my constant errors in notation changed my math grade from a C to an A overnight. Personally, I find it a lot easier to see when there's an x involved.

[sabs]

It's horrible notation to state that the default understanding of -1 is 0 minus 1, as opposed to Negative One)

That or you need a different notation for expressing a negative number, vs a substraction.

-x

is that 0 - x or is that the negative of x

It's possible I had horrible teachers, I learned math halfway in France, and then Halfway in the US in the New Jersey Public School System. So I had to relearn how to do division in 4th grade, but basically I didn't learn any new math at all until 8th grade, and it was confusing, because I did all my basic math in french, and all my algebra in english, mentally. Not to mention, I took algebra in 1984.. so it's been a while.

[shadzar]

It's horrible notation to state that the default understanding of -1 is 0 minus 1, as opposed to Negative One)

That or you need a different notation for expressing a negative number, vs a substraction.

-x

is that 0 - x or is that the negative of x

order of operands comes into play here and explains why -x is assumed to be 0-x.

like in 10, zero is a placeholder same for 0.0001.

just because you dont HAVE to show the zero when talking about a number, you are subtracting x from something. so as not to change the value of x, you use 0.

-|5| = -5
|-5| = 5

(|-5|) + (-|5|) = 0

the zero was left out but still exists. to not NEED the zero to show a negative, the zero is replaced as above to, as someone else stated, add the additive inverse....aka subtract.

-5+5=0

we often write the above, to reduce the number of characters and space required to write it in the following notation.

5-5=0

same thing, just trying to reduce ink in books, but that need has ben removed so that you CAN, unless you are texting algebra, use full notation with computers as they are not bound by the cost of ink and paper.

this is why in the original thread i mentioned absolute value, as the minus sign or negative indicator, or additive inverse indicator just shows you moving the other direction on the number line.

[Maj]

I never learned it as 0 - x. I was taught that there's an invisible 1 there ("turned sideways so you can't see it") hanging out all ninja like.

So translated from my math teacher:

-x --> (-1)*(x)

If you think about it that way, -1^2 can be rewritten as (-1)*(1^2), and since you do exponents before multiplication, you'll end up with -1.

In high school I had a total hard ass for a math teacher - he was a very intimidating old-school kind of guy. I could frequently be counted on to get wrong answers - not because I failed to understand the material, but because I slipped up in notation while showing my work. My teacher spent hours helping me work through all the tiny details that I was overlooking until I finally could get it right consistently. It's because of him that I like math.

[Username17]

When you're writing it on a piece of paper, you are supposed to write negative one with a raised dash in front of it, and a subtract one operation with a mid-line dash in front of it. On a computer, there is no easy way to put a raised dash before anything because there isn't a keystroke to do that. However, when I see a dash in front of a solitary number, I assume it to be the negative sign because there is no fucking reason to have a freestanding "minus one" and lots of reason to have a freestanding negative integer.

In accounting, a negative number is either bracketed by parentheses or written in red. So negative one would be "(1)" or "1". Forum discussion is generally not in bank notation, so the usually assumption of a parenthetical one would be that it was a positive one that was first in the order of operations for whatever reason.

But yeah, if someone says "-1^2=-1" then you are perfectly within your rights to punch them in the mouth. Because while the computer text does literally map to the pen and paper text that says that, in actuality that computer text is supposed to stand in for the pen and paper text where a positive one should be on the right side of the equals sign.

-Username17

[Doom]

I never learned it as 0 - x. I was taught that there's an invisible 1 there ("turned sideways so you can't see it") hanging out all ninja like.

So translated from my math teacher:

-x --> (-1)*(x)

If you think about it that way, -1^2 can be rewritten as (-1)*(1^2), and since you do exponents before multiplication, you'll end up with -1.

This way is just as valid. Implied 1 also exists in Arabic, much like implied 0.

In Arabic notation, a "1" can always be implied as part of a product, so I can write "1x" or "x" when referencing a variable term.

While many remedial/poor math students have, well, politically incorrect reasons for doing poorly in math, about 10% do poorly simply because they've never actually been told the 'real' rules for Arabic notation. These types of students typically end up whizzing through remedial courses once they learn how to really read the symbols. My best was a student I took from pre-remedial through several other courses, only to have him come back to me after a few years for help in a Differential Equations course he was taking at LSU...a 20 year old can go from remedial to DE in a few years, because understanding the language is just that important to understanding the mathematics of the language.

[shadzar]

This way is just as valid. Implied 1 also exists in Arabic, much like implied 0s.

In Arabic notation, a "1" can always be implied as part of a product, so I can write "1x" or "x" when referencing a variable term.

the cost of ink and paper is probably why it is "implied" rather than written properly, but that needs to be discard, and like you say, stop taking shortcuts and write 1x or -1x, or just include parenthesis and teach better what they stand for.

@Frank, journal and ledger entries arent exactly something everyone can handle, because sadly accounting is needed by everyone to be able to do their own taxes, but isnt really taught as part of a school curriculum.

Lotus has an option, as does that Microsoft spreadsheet, and forums CAN include BBCode that will write negative numbers in accounting format, but not having learned it in schools, wont help people much.

also the problem in schools is that red ink is for teachers to grade with...failure!

but at least MANY people discussing RPGs know the accounting terms "bleeding bottom line" and "falling into/getting out of the red", etc as pertains to being in debt, which is having assets equal to a negative balance.

c'est la vie.

[Doom]

the cost of ink and paper is probably why it is "implied" rather than written properly,

Actually, the fundamental rule of mathematical notation/thought is to do things as quickly as possible, to maximize time for drinking beer. Ok, now that's a joke, but the point is still, the notation should be quick.

I mean, should I really write 0 + 2 + 4 instead of 2 + 4?

If so, why not write 0 + 0 + 2 + 4? So, yeah, let's just leave the implied zeroes out, and put them back when they're needed.

Similarly, why not write 1*2 + 1*4?

I'm sure there was a case or two where the notation was changed to save ink, but, really, most of the time it's done simply to speed up calculation. People are generally pretty good at adding 0 or multiplying by 1, including such notation does not help clarity.

Bottom line, people need to learn the 'silly' rules for reading/writing mathematics the same why they need to learn the silly rules for reading/writing any other language.

And it turns out, Frank does have a point--calculators, for example, give different answers for 0 - 4^2, as opposed to the squaring of negative 4. Granted, I don't think punching people in the face is much of a solution as this is rather problematic, considering Frank proposes this as a resolution for online communication; it's also problematic since not every computer language reads -4^2 the same way. On the other hand, the same Arabic notation is taught to everyone.

[RadiantPhoenix]

On the other hand, the same Arabic notation is supposed to be taught to everyone.

FTFY

[shadzar]

the cost of ink and paper is probably why it is "implied" rather than written properly,

Actually, the fundamental rule of mathematical notation/thought is to do things as quickly as possible, to maximize time for drinking beer. Ok, now that's a joke, but the point is still, the notation should be quick.

I mean, should I really write 0 + 2 + 4 instead of 2 + 4?

right, that is why i said it SHOULD be taught right in schools, rather than skip steps to include confusion for people.

that is why i said to use a number line, your starting point IS 0. to do the problem 2+4, you first must move two spaces to the right which is 0+2, then once you are at the correct starting point you can add four to it to get to 6.

it shows the relationship that 0 exists and is implied in the cases, rather than just skipping the step and trying to mentally got from 2 to 6. that way in further math you dont mess up by skipping the step because you understand the step exists, and can look for it WHEN needed.

in the interest of quicker learning these days, many places dont even use a number line to teach, they just teach memorization of counting and multiplication tables, and the basic math principles get lost.

so in the initial discussion of THAC0, zero isnt hidden, but right there, and even character sheets had the number line on it for you.



and this is a D&D character sheet with THAC0 on it...cant currently find an AD&D one online with the THAC0 chart, but will keep looking.

[jadagul]

Good lord, you people are going to make me actually post on this unbelievably idiotic topic. Is there anyone other than Frank who isn't trolling?

Doom, I feel sorry for your math students. Really. This is the kind of fetishistic obsession with arbitrary trivia that gives us a bad name. As an actual mathematician, if someone sent me an email with "-1^2" written in it my response would be to send an email back asking for clarification because it's ambiguous. Which I've actually done when students email me. Because otherwise I have no idea what they think they're saying.

And actually, I'd read

Code: Select all

a/b*c = (a/b)*c = ca/b
a/bc = a/(bc)

although again in the second case I'd incline to ask for clarification. (Actually, in the first case I'd pretty much always write the third option, and in the second case I'd usually go with the second option but not always). Order of operations is really only useful for writing polynomials, because that's the only place it's really clear.

Also, "-1" totally uses the symbol "-" as a unary negation operation, not a binary subtraction operation. Think about it like a group inverse, which it is.

Manxome: You're right that every number "has two square roots" except for 0. But when we look at square root as a function, by definition it has to give a unique answer. We set that up easily on the positive (non-negative, really) reals by saying the square root function sends positive reals to positive reals. We can extend that to the negative reals by arbitrarily picking a square root of -1 and then scaling everything else linearly. You can't actually extend the function to the rest of the complexes in any coherent way without throwing continuity on some ray totally out the window, so we don't usually think about that unless we're complex analysts, and really who wants to be a complex analyst?

Fectin: the easiest example where order of operations would make a difference is in the expression

Code: Select all

1 - 1 + 1

which under usual conventions (and the way I'd probably read it) equals one. You can, however, rewrite it with parentheses

Code: Select all

1 - (1 + 1)

to get negative one instead.

Frank: I've never seen anyone who can actually distinguish between a minus sign and a negation sign in handwritten math. Certainly no one whose math I'd be comfortable interpreting based on the height and length of the penstroke. This is why god invented parentheses.

[Lago PARANOIA]

http://rationalwiki.org/wiki/Mathematic ... ite_note-0

[Doom]

Jada, damn, that's hysterical, but I don't think anyone will fall for it.

[jadagul]

Jada, damn, that's hysterical, but I don't think anyone will fall for it.

Fall for what? That's how I and basically every mathematician I work with handles things.

[DSMatticus]

The - symbol means three different things. It is a binary subtraction operator (2-1), unary negation operator (-2), and the negative number symbol (a raised -, otherwise identical to -2). -1^2 is definitely not the binary operator, and the existence of the raised dash as separate from unary negation is completely debatable. Either way, computers are killing the raised dash; that, and what Jadagul said about trusting a mathematician's handwriting.

But whether or not you think the raised dash is a thing of its own, the unary negation operator totally exists. We're very well agreed on that. But we are not at all agreed about the rules for handling it. Wikipedia doesn't even agree with itself:

As unary operations have only one operand they are evaluated before other operations containing them.
[http://en.wikipedia.org/wiki/Unary_operation]

There exist differing conventions concerning the unary operator − (usually read "minus"). In written or printed mathematics, the expression -3^2 is interpreted to mean -(3^2) = -9
[http://en.wikipedia.org/wiki/Order_of_operations]

And you want to know the kicker? Neither of those wikipedia articles has a citation that supports their statements about precedence. They're the equivalent of "some guy on the internet said."

Calculators do something really weird with unary negation: my calculator has a button just for it, but it puts it below exponentiation and above addition/subtraction.
-1^2 = -(1^2) = -1
-1+2 = (-1)+2 = 1
1+-1^2 = 1+-(1^2) = 0
2+-1 = 2+(-1) = 1
2+-1+1 = 2+(-1)+1 = 1

Excel puts unary operations above all binary operations, so -1^2 = 1, and 1+-1^2 = 1+(-1)^2 = 0, rest being identical.

tl;dr there is no convention for -1^2 beyond "use parentheses, fuckwit."

[Doom]

I'm glad I finally got the idea across that wikipedia should not be taken as an ultimate authority. You can also consult any mathematics textbook that covers the completely accepted conventions of basic order of operations in Arabic notation, or, heck, ask Dr. Math, among many other sources.

And, sure, when dealing with folks that don't really know the subtleties, it makes all sorts of sense to be absolutely, positively, clear, typically via parenthesis.

[DSMatticus]

I'm glad I finally got the idea across that wikipedia should not be taken as an ultimate authority.

Wikipedia has something called a citation system. You should look into these so-called citation systems; they will greatly assist you in overcoming your usual inability to distinguish between information that is and isn't credible.

Like Dr. Math; that's an loltastic source. The problem with you continually appealling to order of operations is that the unary negation operator is an operation that is explicitly not contained in the very order of operations you described earlier in this thread. So you have exactly nothing to appeal to.

[Count Arioch the 28th]

I've found that believing the exact opposite of whatever Doom says is usually the correct answer. The fact that he's in charge of keeping poor people from getting a college education with his stupidity is not a surprise, I've had too many teachers just like him.

[Doom]

Heh, you trolls are always good for laughs. Every single mathematics textbook explains it the same way, because the rules really are as I and others have given.