-2 x -2 = 4.
2 x 2 = 4.
How can both of these equations be true?
I don't know. I've never known this. I've always wondered why this is so, since 4th grade when I pointed it out to the math teacher, who pretended it was no big deal. Next to a problem like this, the logical leaps of an Edward Lear seem ... like precision itself.
I got a book by Alberto Martinez called Negative Math (Princeton UP 2006), in which he states that this is perhaps the first book entirely about the history of negative numbers. And he states that lots and lots of people have believed that negative number rules don't add up. Here's the names of a few:
D'Alembert, Newton, Fermat, Descartes, Euler, Gauss, to name a few.
Some of these have argued that since geometry is always logical (you can see it, and seeing is believing), that we should therefore be able to translate all algebraic equations into geometrical and thus physical and visible positives, or else we are left talking nonsense.
Apparently the country where negative numbers have been most acceptable is in Germany. Kant thought they were useful in talking about areas of philosophy that had previously been too difficult to enter into without any precision!
The more empirical English had trouble with negatives. Even the French had some wariness with regard to negatives. Stendhal, for instance, argued that you shouldn't be able to subtract a larger quantity from a smaller quantity.
What about debt? In debt, you can be "in the hole." That is, you can owe ten thousand dollars more than you actually have, or have a mortgage and be a half million down. So that does make sense.
What about when you get the square root of negative one?
This is called an imaginary number. Euler said that we needed to know what we are talking about when we are talking about imaginary numbers (31).
Geometry is at least physical (I don't think there is such a thing as negative space), but algebra is just a roaring mess.
-4 + -4 = -8.
But -4 x -4 = 16.
So one moves to the left into the further reaches of the negatives. The other (if multiplication is supposedly a kind of addition) then we suddenly leap into the positives. I'm only half-way through the Martinez book. However, I'm happy to run into all the conundrums that have upset me since I was first introduced to negative numbers, and was infuriated by all the paradoxes, and seemingly arbitrary solutions.
Francis Maseres (a French mathematician) said that negative numbers and imaginary quantities (he gives many examples, such as x to the third power - bx = c, which has two solutions, one of which has x as a positive number, and one as a negative number, have been treated "with an uncommon degree of obscurity, and has been made the subject of much mysterious and fantastic reasoning, (or, perhaps, I should say, discoursing, since it deserves not to be called reasoning) concerning negative and impossible numbers)" (44).
The weirdest parts of philosophy are not nearly as arbitrary and impossible as the most basic notions in algebra, which are taught with a straight face by professors all the way down to elementary school teachers, seemingly blind to the paradoxes at the heart of their supposedly logical system of math. Math is as bizarre as myth, without even the slightest awareness of its mythical perplexities, which are presented to us as being the height of logic.
While most everyone can grasp geometry, I fear that only the unthinking can stomach algebra at present. It just doesn't make any sense, especially as soon as negative and imaginary numbers are introduced.
Sunday, May 17, 2009
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41 comments:
Kirby,
If you give me 20 dollars, you have -20 dollars. If you give me another 20 dollars (+ -20) you now have -40 dollars.
However,
If you take 20 dollars from me 20 times, you've now got $400.
Lower-division math always works with money.
g m
i would argue
that if kirby only had $20
and gives it to you
he has zero
he wouldn't have to get another
$20 to get even
it's like when they say
GM lost 20 billion dollars this year
did they lose it??
or did they simply fail to earn it
i mean
they didn't lose anything
i can see having $1000 dollars losing 20 on a horsebet
and looking at $980
no
kirby's right
there's something fishy going on
in math
logic is being held
with a knife to her throat
i guess we live in an age of identity theft
if my wallet is stolen and my credit card gets used
i am a negative number
minus 4
shit
this has got to stop
j
Kirby,
Honestly, -2 x -2 = 4 is the only thing that makes sense. It's the only thing that's consistent with the rest of multiplication. I'm sorry that it bothers you, but what do you want it to be?
If you answer that you want -2 x -2 = -4, then you have to have that, basically, -2 x -2 = 2 x -2 (or, in general, that -2 x A = 2 x A when A is negative, but not when A is positive). Is that really what you want?
Here's another way to look at it. Consider the times table:
1 2 3 4 5 6
2 4 6 8 10 12
3 6 9 12 15 18
4 8 12 16 20 24
5 10 15 20 25 30
6 12 18 24 30 36
To find what A x B is, you find the column with A at the top, the row with B at the right, and look at the number where they intersect. Now, look at any individual row or column. In each of them, you basically count by some number. In the column with the 3 at the top, you can extend it indefinitely downward by continuing to count by threes, i.e., add 3 over and over again. Similarly with rows. So you can extend the table down and to the right as far as you want. You can also extend the table upward and to the left, by, instead of continually adding the number for that row or column, but subtracting the number. If you extend the columns up one more, and the rows to the left one more, you get:
0 0 0 0 0 0 0
0 1 2 3 4 5 6
0 2 4 6 8 10 12
0 3 6 9 12 15 18
0 4 8 12 16 20 24
0 5 10 15 20 25 30
0 6 12 18 24 30 36
This reflects that anything multiplied by 0 is again 0. Now, keep going up these columns, and to the left in these rows:
0 -2 -4 -6 -8-10-12
0 -1 -2 -3 -4 -5 -6
0 0 0 0 0 0 0 0 0
-2 -1 0 1 2 3 4 5 6
-4 -2 0 2 4 6 8 10 12
-6 -3 0 3 6 9 12 15 18
-8 -4 0 4 8 12 16 20 24
-10 -5 0 5 10 15 20 25 30
-12 -6 0 6 12 18 24 30 36
Finally, do the same for the upper left quadrant:
4 2 0 -2 -4 -6 -8-10-12
2 1 0 -1 -2 -3 -4 -5 -6
0 0 0 0 0 0 0 0 0
-2 -1 0 1 2 3 4 5 6
-4 -2 0 2 4 6 8 10 12
-6 -3 0 3 6 9 12 15 18
-8 -4 0 4 8 12 16 20 24
-10 -5 0 5 10 15 20 25 30
-12 -6 0 6 12 18 24 30 36
The numbers in the upper left quadrant are positive. It's the only thing that's consistent.
(To make this look decent, you'll need to copy and paste to something where you can get a monospaced font like Courier for formatting reasons.)
That's interesting. My thesis adviser wrote a book about negative or anti-geography. If I had known that was his plan I probably wouldn't have listened to him.
Kirby,
If you really want to get into the wackier parts of math, strike up a conversation about 0.9999... and 1 or how the number of infinite points in an infinite line is the same as the number of infinite plane. Those are good for starters.
George, it's not that I think that -2 x -2 should equal -4. I just don't think negative numbers have either rhyme or reason.
I suppose that they still must exist. In Alberto Martinez' book, he talks about great numbers of prominent mathematicians who believed that they really shouldn't exist, and that it was always possible to put a problem into positives, and proceed accordingly.
But I'm only an amateur in the outlying foothills of math. I keep getting into these quagmires on the way to getting up into the higher foothills, from which I may find that I have a larger perspective.
Thanks for your help, although I confess I couldn't grasp what you were getting at.
I haven't studied math since 4th grade when I first hit the negative numbers and decided it was all nonsense, but nowhere near as good as Edward Lear. I went through the motions through 12th grade so that I could pass the tests, but never thought it was truly accurate or meaningful.
I was briefly fascinated by geometry in 10th grade, but the clodhopper changed geometry into algebra after the first semester, since he felt more comfortable with it. I went back into a coma, from which I am now attempting to emerge.
I'm going to keep reading the Martinez book. Perhaps after I'm done I will have more to offer. Rather tied up this morning with kids, or else I would try to find some quotes from the book.
All numbers are inventions. They're place-holders for what we don't understand. Identity equals zero, since all identity is identical. Get it? 1 + 0 equals two places, one of which is empty. Right? Two stands for two ones, each of which is identical, but 2 can also stand for two unlikes, which aren't identical. Does this make any sense?
Infinity is beyond human consciousness. A line, or a plane, that is without boundaries, is beyond our comprehension.
The night before last, PBS had a special on the guy who invented fractal geometry. It turns out that most of the forms in nature are designed according to fractal formulas. Replicating relationships at increasingly smaller scales. Fascinating--and a fascinating man too.
does conceptual existence as in numbers indicate actual existence
the fact that i can say 1 or -3 is an abstraction
we cannot deal with reality
fractal or not
there are no numbers in nature
it is one thing to allow the mind to divorce itself from reality
it is another thing to impose the efforts of mental putation upon reality
the grid is grotesque
thanks euclid
mathematics is a distraction
a way of occupying the mind
OK so it gets a few people to the moon
and makes wall street tick like a heart juiced on methamphetamine
so what
numbers were invented as a way of keeping books
after first grade there's really not much use for them these days
machines do the books
betting is significant
better know the numbers there
truth to numbers
one is the loneliest number
two can be as bad as one
organizing life around negative potential is not good for people's self esteem
i'd like all streets to have names
numbering streets is a not so subtle form of manipulation civic manipulation
let's get back to names
real things real names
hardboiled egg street
needle-and-thread way
god is one
god is ayin
god is infinity
curtis
if the word infinity can be stated
and has a certain significance
can it be said to be beyond comprehension
at the very least the mind goes to the thought: space without limits is thinkable
like the ontological proof of god's being
that which nothing greater can be thought
even if pulling thought beyond adequate expression
is at least somewhat contained in thought
numbers are making us mad
negative numbers do little
to reduce the madness
grandmothers have no use for negative numbers
they simply want to know how many grandchildren there are
if all the grandchildren die does that mean grandma has
negative 10 grandchildren
no she has nothing
nobody
tragedy is rendered senseless by numbers
for tragedy is meaningful
we should never count up the dead in tragedy
it's an insult
algebra is an islamic conspiracy
and the jews have mastered it
you'd think they'd get along better
money is no longer money
money is numbers
how do i love thee
let me count the ways
how do i hate thee
shall i posit negative value
you are -6
your rejection sweetness
is a negative numerical concept
having no bearing in reality
but leaves me
well
destitute
damn
we are adrift in an incomprehensible sea of abstraction
and the oars are breaking
more numerous than the sands on the seashore
how many lord how many
maybe aborted humans are to be understood in negative quantities
instead of 1.2 million new humans every year we have
-1.2 million
yeah that's the ticket
j
"Number" isn't really much more of an abstraction than "animal" or "light" or "weight". We deal in abstractions all the time. The only way to make any sort of sense of the world around us is by abstraction and categorization, despite the inherent shortcomings.
Making statements like: "Infinity is beyond human consciousness. A line, or a plane, that is without boundaries, is beyond our comprehension." simply ignores that humans came up with the concepts of "infinity", "line", "plane". They are no more beyond our comprehension than "red", "tall", or "planet".
To type "after first grade there's really not much use for [numbers] these days" on a computer and send it over the internet shows an almost staggering ignorance or blindness.
And for "god is one
god is ayin
god is infinity", God is none of these things. God is God.
"numbers are making us mad
negative numbers do little
to reduce the madness"Baloney.
"algebra is an islamic conspiracy
and the jews have mastered it
you'd think they'd get along better"If this weren't so absurd, it'd be offensive.
"we are adrift in an incomprehensible sea of abstraction
and the oars are breaking"Piffle. Abstraction is wonderful. What is more abstract than love, faith, God?
"I just don't think negative numbers have either rhyme or reason."Negative numbers are rhyme and reason crystallized. They extend the patterns and structure of positive numbers in a consistent and extremely useful way. Honestly, you sound like a child saying, "Why should I have to learn to read? I don't see any use for it." And you won't see any use for it until you learn to do it. Without being comfortable with the language of number, you will be deaf to much of the conversation of the world, and be unable to understand a great deal of why things work the way they do. I hope you don't give that up.
when it comes to numbers i prefer ignorance
it's a highly overused
quasidiscipline numbers is
we could do without most math
words are not abstraction insofar as they are articulated by children
and children doing numbers is sort of cute
but after that nobody should take them numbers very seriously
poems require more mental acuity than
any math equation
if you get a poem wrong it's tragic
if you get a math problem wrong
it's just a red mark on the page
so poets
try to get the poems right
the world depends on that
wall street has people hypnotized
with numbers that are empty of significance
gone is the knowledge of the dollar in the pocket
i know some of the things i write sound silly
but think about it
who invented algebra
who runs wall street
suzi orman
she's the new profitess
of numbers and love
how much can you afford not to lose
that's the thought we should all be going to bed with
milton friedman lives
keynes squirms in his grave
i'm a little insulted that you seem so resistant to acknowledging the insanity of numbers theory and it's extended impression on our culture george
we're being fed this psychotic schmaltz drama that says we're all tied into this thing called the economy and the indicators are shown to us every night like a monitor on a minotaur heart patient and they've gotten most people to believe it because most people do fairly well in math through highschool
but let's be real
time is a wastin'
you gots to spend money to make money
so sue me
numbers schmumbers
words and things that's what it's all about
do the hokey pokey
it's all blood transfusion now
that's what the bailouts are
trying to keep the ficticious monster hungry and wired is a game highstakes numbers gambling
the guys at the table are enjoying themselves
numerical abstraction tends to propogate a form of ignorance so subtle that people actually think they are thinking
ah well
put no numbers on my grave stone
nature needs no numbers
she's what she is when she is how she is
and will be without numbers
although
some say she is getting hotter
by degrees
be well all manner of thing be well
slurp slurp
take all you have and give to the poor
and don't count the cost
numbers bring out the worst in people
this computer is killing me
ahhhgggh!
j
Curtis: The math logician and axiologist Hilbert and other deep math people seemed to think otherwise--that number was some quasi-Platonic or Pythagorean fundamental quality inhering in things. Perhaps George can comment on this, but the Kuhn craze took place mostly among parrot-like ("Bertie, stop parroting"--and Bertie dutifully responds: "Parroting!?") "paradigm-shifters" and lit-critters who were the first scammed in the delicious Sokal Hoax that made English dept math/physics- challenged post-human wise-asses like Stanley Fish into figures of fun and ridicule.
Still, George, have you a suggestion for a book on well, . . . nothing . . .i.e., the concept of zero and its aftermath. In the immortal "Vanity Fair," there is the chapter about Rawdon Crawley and Becky: "How to live on Nothing a Year," or WS's Lear, who tells Cordelia that "nothing comes from nothing . . .,
or,
the great negative of created non-creation of the deliciously wicked 17th c. rake and eminently distinguished degenerate, John Wilmot, Earl of Rochester, one of the King's close comapanions. From his masterpiece, "Upon Nothing":
[ . . .]
Great Negative, how vainly would the wise
Inquire, define, distinguish, teach, devise?
Didst thou not stand to point their dull philosophies.
Is, or is not, the two great ends of Fate,
And true or false, the subject of debate,
That perfects, or destroys, the vast designs of Fate,
When they have racked the politician's breast,
Within thy bosom most securely rest,
And, when reduced to thee, are least unsafe and best.
But Nothing, why does Something still permit
That sacred monarchs should at council sit
With persons highly thought at best for nothing fit?
While weighty Something modestly abstains
From princes' coffers, and from statesmen's brains,
And Nothing there like stately Nothing reigns,
Nothing, who dwellest with fools in grave disguise,
For whom they reverend shapes and forms devise,
Lawn sleeves, and furs, and gowns, when they like thee look wise.
French truth, Dutch prowess, British policy,
Hibernian learning, Scotch civility,
Spaniard's dispatch, Dane's wit are mainly seen in thee.
The great man's gratitude to his best friend,
King's promises, whore's vows, towards thee they bend,
Flow swiftly to thee, and in thee never end.
I remember one prissy Protestant Presbyterian in our English dept at Portland State fond of abusing Catholics (I'd respond to Black Jack C-----, as I called him, with "Cavaliers--Right and Romantic! Roundheads? Of course, BJ,--wrong and repulsive!" Vivent the Stuarts forever! And after good Queen Bess II passes on, kick Charles and the rest of that royal German rabble back to Hanover where they belong! At any rate, when a Inspector Morse mystery concerned the great reprobate himself, and the "Mystery!" was introed by the another ol' queen, Vincent Price, and he "quoted" Rochester as concluding one of his most distinguished poetic obscenities with "And then to love [sic] again," not knowing--and this priggish Protestant pilchard was a Yale-trained 17th c. "expert"--Rochester's actual injunction was "And then to c**t again!--what girlish preciosity, BJ--how very charming!
Bonus: from memory: "This Be the Verse"--by Philip Larkin (I'd slightly bowdlerize this gem of "optimism" a bit for my freshman only). I know you all know--or should know, it:
They fuck you up, your mum and dad,/ They may not mean to--but they do,/They fill you with the faults they had/And add some extra--just for you/
But they were fucked up in their turn/By fools in old-style hats and coats/Whohalf the time were sloppy-stern/And half at one another's throats/
Man hands on misery to Man/It deepens like a coastal shelf/Get out as early as you can/And don't have any kids yourself.
Great Tory, Larkin. Great minor poet.
PS: How 'bout a Romantic walk on the dark side--may clouds of Emmy's wormwood absinthe carry you all to you rest! Cheers,
Jacques,
I've read three books on nothing, as you put it:
John Barrow's "The Book of Nothing: Vacuums, Voids, and the Latest Ideas about the Origins of the Universe". This is about half physics, half history about zero. It is definitely my favorite of the three, although your mileage may vary.
Robert Kaplan's "The Nothing That is: A Natural History of Zero". Sometimes interesting, but the author seemed too married to his thesaurus, as it were.
Charles Seife's "Zero: The Biography of a Dangerous Idea". I found this book too meandering and at times unconvincing.
You might see what you can find at the library.
when it comes to numbers i prefer ignorance
it's a highly overused
quasidiscipline numbers is
we could do without most mathSays the self-proclaimed ignorant man.
poems require more mental acuity than
any math equationSo you say, in your ignorance.
if you get a poem wrong it's tragic
if you get a math problem wrong
it's just a red mark on the page
so poets
try to get the poems right
the world depends on thatGetting a poem wrong (whatever that might mean) is not tragic. If you get a math problem wrong, that's not tragic, either. If you are ignorant of poetry, that's tragic. If you are ignorant of mathematics, that's tragic. The world depends on neither.
jh just likes to hear himself 'talk' a lot of the time.
It's usually at least somewhat entertaining, because the way he talks is pretty fun.
But you can't take him seriously.
I think he should give up, eternally and permanently, all technology that uses math that goes above a 1st grade level.
Naked and shelterless would seem to fit him quite well---
Oh, and a bad calculation in poetry could never hurt anyone - but a bad calculation in, say, space flight, or house-building, or car-manufacturing, well that can be quite deadly.
GG + Brett:
You're coming down pretty hard on JH.
I think what JH was getting at was that poetry and literature requires more "humanity" than pure math does.
But Math can be beautiful!
I had a Hawaiian math teacher in high school and he showed me the math of the sunflower, of the nautilus shell...he also played the ukelele (SP?) That doesn't mean I learned algebra. I stink at algebra, but was somewhat better at geometry (like Kirby!)
But in defense of JH, math can be a bit impersonal, if not downright cold.
One death is a tragedy, a million deaths is a statistic.
people are dying for lack of what poetry has for them and they do not even know it (para. WCW)
i will concede that astronomy should use math
but only that
but stick with the distance between
stars and planets
give me my cave
a rosary
and a bible
i'll be fine
maybe a little water
if god would be pleased
maybe a crumb of bread
math forms automatons
poetry forms human souls
we have to find better ways to tell lies
did jesus ever do any math problems
now that's an interesting question
he liked 12
and then 72
3 works in there somehow
i think his death reversed
all negatives
they manipulate value and worth
compromising both
and we pay and pay
the days of heartbreak squared
lost
the GPS is broken
no recourse to numbers
fate and death
twitter
this could be the end
nature does not count her deaths
j
Mathematics is a language, no more, no less. The power of an equation is no greater than a sentence which states a fact.
All languages are handy tools with which to organize our thought and apply it to empirical problems in the world.
But language, like the mind, has limits. Since we live in three dimensions, in a world of matter, we have built an edifice of abstraction out of our observation and experimentation which is broadly corroborated (confirmed). But modern physics has gone beyond materialism to confront the macro and micro phenomena which our senses cannot directly apprehend.
The world of the increasingly small, and the increasingly large, test our limits of understanding. And we have yet to unlock most of the true mysteries: What is light? What is gravity? What is matter? How did life begin? What is time? How did the universe "begin"? How big is it? Is there an inherent plan in the structure of the universe?
Language enables us to confront all these issues, but it provides no answers. Mathematics (a language) is a way of describing what we don't know.
Mathematics won't buy your groceries for you. It will tell you how much you can afford to buy, and can help you fix the car, and whether your blood sugar is high or low.
Language and mathematics aren't "inside" things. They're mental constructions which are intended to describe phenomena, both through naming and descriptions of relationships of mass and energy.
Kirby,
I believe that the idea of -2 X -2 = 4 is a simple idea of turning negative things into postitives in our lives.
When life gives you lemons, turn the lemons into grape juice and leave the world wondering how the hell you did it.
Since mathmatics incorperates much truth it would realize you are better off turning negative things into positive things.
But then again I'm not a math major so take that with a very large grain of salt.
math forms automatons
poetry forms human souls
You wouldn't say this if you spent much time around university math departments and English departments.
did jesus ever do any math problems
Did Jesus ever write a poem?
Emmy Bee,
I think what JH was getting at was that poetry and literature requires more "humanity" than pure math does.
And I say that this is not true. Both literature and mathematics are human endeavors, attempts to better understand God's creation. They complement each other, and both are important. They both are part of our "humanity".
But Math can be beautiful!
Indeed it can. And it doesn't even need to be beautiful in regards to something like sunflowers, pine cones, or nautilus shells. Mathematics can be beautiful on its own terms. I think, for example, that Gauss's theorem on quadratic reciprocity, or his Theorema Egregium, or the fundamental theorem of calculus are beautiful in their own right.
But in defense of JH, math can be a bit impersonal, if not downright cold.
I could make the same claim about a lot of poetry and literature.
One death is a tragedy, a million deaths is a statistic.
Which has nothing to do with mathematics, insofar as it is even true. Numbers are not mathematics, any more than words are literature.
In any case, I'm replying in this thread as a public service to all the young Lutheran surrealists out there reading it, to try to get them not to ignore mathematics, a major important and beautiful edifice of human thought, tool of communication, and aspect of God's creation.
"Numbers are not mathematics, any more than words are literature."
You might want to re-think those assertions.
This thread has run a while, and I have had to reflect, because I am dealing with an actual mathematician, and I am not on very firm ground, since I don't know mathematics very well.
But for openers, I think that addition and subtraction are no-brainers with negative numbers. So I think that the first half of GM Palmer's first comment, doesn't give me too much trouble (except that he subtracted forty, when he should have subtracted twenty).
My problem is when you start multiplying and dividing negative numbers.
If I am four dollars in the hole, that would be -4.
Let's say I then double the debt. Suddenly, I have 16 dollars. How the heck did that happen?
I don't think that negative numbers work when it comes to multiplication and division. I think it should in fact be banned.
We already have a precedent when we ban dividing by zero. It's meaningless.
Zero is a kind of black hole when it comes to multiplication and division, pulling everything into its warped universe.
Since negative numbers are even less than zero, I think they are even stranger when we get to multiplication and division.
George Grady, Lutheran Surrealist Mathematician, has argued that negatives are INTERNALLY consistent within algebra. That's probably true.
Jungian psychology is also INTERNALLY consistent.
But do either of them really measure or describe REALITY consistently.
The answer is flatly no in terms of Jungian psychology (Jung, for instance, argued that everything in a dream is about the dreamer, and therefore if you go to NYC and dream about walking down the sidewalk that night, your unconscious is sending you a message about yourself). Jungian psychology is not really falsifiable, but it seems logically goofed up. The whole notion of archetypes -- that Mary with Jesus is the same as another image out of Egypt with some batty goddess holding some demented little infidel -- are the same -- is just another matter of how Jung's INTERNAL consistency has no relationship to EXTERNAL adequacy.
Heck, even MARXISM is INTERNALLY consistent. (Which doesn't mean that Marxists aren't virtually HANDICAPPED by their mental mapping of the world -- or shall I say -- DIFFERENTLY ENABLED).
Lacanians, Deleuzians: ditto.
Mathematics claims to be accurate, and to be the basis of the sciences.
So let's say I'm going four miles an hour in reverse or -4 mph. Now let's say I go twice as fast in reverse.
Suddenly I'm going 16 mph?
What the - ?
I'm not arguing that negative math isn't INTERNALLY consistent, I'm only arguing that in terms of multiplication and division, it just doesn't seem adequate to the real world.
And I am proposing a ban on multiplication and division with negative numbers, since they don't seem to work. And I am showing that the number ZERO already has a similar ban on at least division, since it doesn't seem to yield anything productive.
as Lutherans, we must try to be productive and realistic, even if the surrealist half of the equation urges us on otherwise.
Kirby,
If you have a 4 dollar debt (-4), and you double it, you don't multiply by -4, you multiply by 2. That's what doubling is: multiplying by 2. Multiplying a number by itself is usually called squaring, and it's not at all the same. Similar comment to your speed example.
Curtis,
You consider a dictionary literature? Neither is a list of statistics mathematics.
George, yes, this makes better sense. So if I'm going negative four and the speed is doubled, I'm going negative eight.
I don't know what it would mean to square a speed, but if I did THAT, I would then be going 16.
And if my debt was negative 4 and was doubled, my debt would then be negative eight.
But if I square my debt, then it would be 16.
Maybe therefore it's squaring negatives that I would like to ban.
well
none of this "adds up"!
so how to feed 4 people on
3 apples
3x+4y-1n= apple sauce .
that way we can feed the hungry in Nigeria and Pennsylvania and tax the rich Republicans to pay the freight charges
Kirby,
You also have to consider units, too. When you add dollars to dollars, you still have dollars. When you multiply dollars by dollars, you have dollars squared, whatever that would be. You can only add things which have the same units (for example, you can't add 20 dollars to 10 cents to get 30 somethings; rather you have to convert them so they have the same units: 20 dollars plus 0.10 dollars is 20.10 dollars.)
But, you can multiply things regardless of their units: their units also multiply, but you can't necessarily interpret the result in any meaningful sense. For example, if you travel 60 miles per hour for 2 hours, you've traveled 60 * 2 = 120, and the units are miles per hour times hours. Since "per" is a synonym for "divided by", the hours units cancel, and we just have 120 miles.
If you interpret positive distances as distance in one direction and negative distance as distance in the opposite direction; positive speed as speed in the positive direction and negative speed as speed in the negative direction; and positive time as time in the future and negative time as time in the past; then multiplying distance = speed * time with the standard sign rules for multiplication gives you the right answers.
Squaring can often make sense, even for negative numbers. For example, the kinetic energy of, for example, a car, is one-half the mass of the car times the square of the speed. The direction (i.e., sign) of the speed ends up going away here, since we're multiplying the speed by itself. But this gives you the right answer. For example, hitting a tree at 60 miles per hour hits thirty-six times as hard as hitting a tree at 10 miles per hour; a speed 6 times faster hits 6*6=36 times harder.
Another place that squaring takes place is dropping (throwing is more complicated) a rock off a cliff. If you see how fast the rock is going after it's fallen 30 meters, you'll find it's going 9 times as fast as it was after it had fallen 10 meters, since 9 = 3*3. Galileo basically did this experiment, although he rolled cylinders down ramps to make the speeds small enough to measure accurately.
Kant believed that algebra had to do with intuitions about time, whereas geometry had to do with intuitions about space.
(Martinez 84).
Apparently Einstein combined the two, using rather idiosyncratic math, according to Martinez (105).
Ok, now it's off to church. My daughter is the cruxifer tonight.
Ed:
Remember when the Reagan Administration called catsup a "vegetable" so they could add it to the nutritious free (Federally funded) lunches for American school kids?
Okay, Grady, numbers comprise mathematics--they are its symbols. Just as letters and words are the "stuff" of literature.
Literature actually precedes dictionaries, as you may well know. Dictionaries are typically compendiums of accepted usage and spelling.
Literature is what you call it. I'm sure there are many kinds of literature which you would exclude.
I, like many people, consider dictionaries to be a form of literature. And they certainly are. Though most of them, these days, are so uninspired as not to be. Sam Johnson's definitions are certainly inspired literature.
Think it over.
when jesus came to the defense of the woman caught in adultery
he wrote a little poem in the sand with a stick
i think it went something like:
father
men with laws can be such insufferable assholes
i love this delightful woman
now shamed by all these
assholes
the beatitudes
were probably spoken
over and over by jesus
locked into the consciousness
of at least one eventual writer
jesus was an oral not a scribal poet
"plentious the harvest
pitiful the workers"
some really funny stuff here
statics at least indicate
math has happened
distionaries indicate at least
poetry has taken place
plenty of found poems to be found
in the pages of lexicons
i think the negative value of speed would only add up if you were falling ass first into a black hole
stay away from those things (are they things?) kirby
this is all a gas
thanks to everyone
this made my day
george i honestly have
nothing but respect for your mathematical knowledge
i hope you can accept my surrealist
efforts here
this is a place where i can be a little nuts in the blogosphere
i should hope i'm taken with a few grains of salt and pepper and some paprika to boot
my blog is a place where i do things
in a little more serious tone
you're welcome over there
peace
ciao
j
Curtis,
If literature is what I call it, I don't call dictionaries literature, Samuel Johnson notwithstanding, although I suspect I actually have a rather broad category of literature. I also have a broad idea of what mathematics is. But it isn't mere numbers, anymore than literature is mere words.
jh,
when jesus came to the defense of the woman caught in adultery
he wrote a little poem in the sand with a stick
I'm pretty sure he was drawing the diagram of the Pons Asinorum.
i hope you can accept my surrealist
efforts here
I think (and hope) it's all in good humor.
just for the sake of argument
in websters 3rd new international dictionary -- unabridged
literature --literatura --- writing, grammar, learning
it would seem the dictionary indispesable to that
john henry cardinal newman is quoted
"literature stands related to man as science stands to nature"
after all it's not the
words or the numbers
it's what we presume to do with them
and malt does more than milton can
to justify god's ways to man
(o...housman or...somebody like that)
dictionaries are to literature
what slide rules are to calculus
(no that ain't right
but it sounds good)
literature is unthinkable without dictionaries
math can get along just fine without numbers
(no i didn't quite mean that)
i suppose where the quatrain the sestet the couplet the sonnet are concerned i suppose where meter is discerned hexameter dactyls and the like i suppose there's a bit of math in it the poetry certainly william gaddis took math into his computations known as novels
o certain poetry of the making in the orally rehearsed and delivered numeric exercises at the blackboards of kids doing math and i would suppose where a professor goes on in the explication of theorems and convoluted however logical or ill equations there is a tad of poetry
when i was young they referred to
marijuana cigarettes as numbers
that seemed poetry and math perfectly wed
i must to bed
go
well
"they" recently changed the spelling!
can no longer find catsup anywhere
so I looked it up in my Funk and Wagg-it
it is now spelled Katchup!
my guess is them little skool kiddies eating their salty/sugary "vegetable
squeezed out of a little plastic bag onto their swine-flu hot dog
might think that they were really eating bloody ground up cats! and instead of hot-dogs real dogs? eating
nutritious H1N1
all of this redefining of nouns/verbs/adjectives/etc just a ploy to sell more Lulu (POD)produced dictionaries!
PBS now showing Michaelangelo Revisited 2009
worth a watch...
George, the idea that in math things have to add up within the same kind of units was a big help. It just dawned on me this morning. I have a kind of time delay at points when learning math.
I thought it was Byron who favored malt over Milton.
George: Thanks for your suggestions on reading about "nothing." Em n' me are in transit much these days, but know that my old colleague has agreed (now back from foreign shores) to contribute to this site at some near future time--his mathematical speciality is orthomodular lattices and his political preferences are for Michael Savage. He's a military vet like me (though not a war vet as I am), and should add something to the mix on Kirby's site.
When we were colleagues at a small college in Virginia, we once drove over forty miles to attend synagogue (my first time as a monarchist right-wing RC) in North Carolina, and I remember the female rabbi read prayers from her own "degenderized" prayer book while the faithful read prayers from their own "genderized" versions (my colleague was not scandalized by this, but not particularly pleased either). I'm not sure if these details are pertinent, but he's an animal lover and plays a mean game of: tennis, ping-pong, pool, and cribbage (BTW, apparently the invention of the Cavalier poet, Sir John Suckling). He also shares my passion for Scrabble, though perhaps not for poker. I hope your exchanges will be profitable when they occur (and I do so appreciate your contributions on this site), though they may be limited becuase he's working so many hours these days. As I've said before, we taught at a 99% African-American college--and our band of new hires at the college--now all elsewhere--included a Protestant (business), a Jew (mathematics), a Muslim (criminal justice), and me (English) in 1999--all of us over forty, and all all military vets. Now what's the chance of that phenomenon occurring at a much larger school, however much these bigger schools try to vaunt their "diversity"?
Kirby -- dija ever realize that zero was invented by A-rabs and that all a-rabs is terr-rists?
Hmmm
I asked a math wizard from Ohio State how -4 squared could equal positive 16, and he said, in outer space, things like that are easier to imagine.
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