-5=5

[RadiantPhoenix]

Because it's really not related to the ability score thread...

He is also trying to prove that in algebra, and not 2e, -5=5.

I think I might be able to do this one.

[Kaelik]

From 2-3 is definitely wrong, and I think 4-5 is also wrong.

After having done some head maths, switch the definitely and think.

[RadiantPhoenix]

Yeah, I'm pretty sure the problem is between 4 and 5 too; 2 -> 3 is legit, though.

[Doom]

Between steps 3 and 4 is wrong because you can't use square root properties ("square root of a ratio is a ratio of square roots") when an argument is negative.

Steps 2 to 3 is also fine (just substituting symbols which are equal). Step 4 to 5 is ok, just the notation is improper (you're supposed to factor out the -1, then write it as a complex number), but you'll get the same answer formally or informally.

[Leress]

To be fair he did say it was the absolute value of -5, but it had fuck-all to do with the subject in the thread.

[Josh_Kablack]

Well if we're doing this sort of thing there are at least 2 other such "proofs" out there.

Charles Lutwidge Dodgson has one using variables to obscure a division by zero error in one of his letters. He started by letting A=3 and then in one of the steps of the proof he used A-3 as a divisor, which therefore let him go on to "prove" that 3==4

And my 9th grade algebra text had another that used subtly incorrect expansion and factoring of a difference of squares binomial. Sadly I don't recall the particulars, but it was something like expanding (a - b) ^2 to ( a^2 - b^2 ) which looks reasonable at a glance, despite being incorrect.


On a related tangent, another board I frequent uses a meme of -1^2 == -1 as a way to mock the mathematically ignorant.

Because if you strictly follow order of operations, exponentiation is performed before multiplication, even multiplication by a negative coefficient.

So -1^2 should most properly be read as (-)(1^2), which is indeed -1. However orthography has led to many people writing -1^2 when what is truly meant is (-1)^2, which works out to a positive 1.

[shadzar]

it was something like expanding (a - b) ^2 to ( a^2 - b^2 ) which looks reasonable at a glance, despite being incorrect.

to anyone understanding order of operands it does NOT look reasonable, because braces and parenthesis are done before multiplication and division.

1. [(
2. */
3. -+

those are the groupings for the simplest operands, or ones used most often, and you do them in that order for a reason.

since (a-b) has been reduce to simplest terms already in order to exponential it (a-b)(a-b) = (a-b)^2 you cant always try to break out of the parenthetical to convert it.

(a-b)^2 is already the simplest notation.

on this note, i asked MANY people yesterday to see how well algebra was maintained by average people...it was about 20-30 in different fields of work and such, so a small set.

the results were the same for the majority.

a-b=c
solve for b

most people said just by looking at it:
b=a+c

some didnt know how to figure it out, while others had to right it down in steps to come to:
b=a-c

of course a=b+c

so it seems in all reality, that those not using algebra on a daily basis, soon lose it as it, like other information and skills, atrophy from non-use.

which may explain, when coming into 2nd and the number of existing players no longer in school, that didnt like to figure THAC0 properly, because they were out of school, and figured they had no need for that stuff, and no longer COULD convert a formula to solve for each variable, but get stuck with the "what Zeb said to do" and use the wrong form of the formula, such as the above (a-b)^2

but everyone remembers aa+bb=cc, even if they dont know it is the Pythagorean theorem, or can tell you verbally "the square of the hypotenuse is the sum of the square of the 2 sides."

@Leress: it does but people dont see it, the ability scores are reflected in ALL other system that use them. so a change in those other systems contribute to how ability scores changed.

moving from THAC0 to BAB, shifted how the ability scores interacted with the attack resolution system, and thus the ability scores had to change in order to work with the new system. then the new save system could be built off the new BAB and ability scores.

but nobody paid attention tot hat in the original thread, just saw the term THAC0 and wanted to attack it.

[Juton]

Step 4 is invalid because it should reduce to:

Code: Select all

2.24i   2.24
----- = -----
2.24    2.24i     or i = 1/i

Which is not equal, therefore the proof fails.

EDIT: simplification

[Doom]

Exactly, the property of radicals being used from step 3 to step 4 is invalid, hence the numbers are not equal in step 4.

[fectin]

-1 is a number, not an operation. Order of operations is inapplicable.
Claiming that -1^2 = -1 is like claiming 12^2 = 14.

[Juton]

Radiant, did you change the proof? Because now it starts with -1=-1 where before it started with 1=-1.

In this proof there is an error in line 5 as well, sqrt(-5^2) = 5, not -5.

EDIT: Brackets

[Doom]

-1 is a number, not an operation. Order of operations is inapplicable.
Claiming that -1^2 = -1 is like claiming 12^2 = 14.

Actually, no.
A) (-1)^2 = 1
B) -1^2 = -1.

Order of operations:

Parenthesis
Exponents (and Radicals, though often this isn't listed)
Multiplication and Division
Addition and Subtraction.

Useful mnemonic: Psychotic Enemies (Really) Must Die Agonizingly Slowly.

In case A), you're multiplying -1 by -1, to get 1. In case B), you have to do the exponent first, and then put a negative sign on whatever you get after you exponentiate..."-" doesn't really mean subtraction so much as "additive inverse", but that is probably a bit too technical for here.


And I'm sure in the original proof, sqrt(-5)^2 is just poorly written, and is meant to be ( sqrt(-5) )^2...doesn't much matter, of course, since the bogus line invalidating the proof comes a few steps earlier.

[fectin]

You're conflating subtraction and negation. They are not the same, even though they use the same symbol.

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

[RadiantPhoenix]

And I'm sure in the original proof, sqrt(-5)^2 is just poorly written, and is meant to be ( sqrt(-5) )^2

This is true. My program for writing formulae has some bugs.

[Manxome]

Between steps 3 and 4 is wrong because you can't use square root properties ("square root of a ratio is a ratio of square roots") when an argument is negative.

You are mistaken. That's just distributing exponentiation over multiplication:

(a*b)^c = a^c * b^c

That's a completely general rule, and is true even when a is negative and c = 1/2.

Step 2 -> 3 is also entirely valid.

Here's an alternate version I saw in college that requires no division:

1
= sqrt[ 1 ]
= sqrt[ (-1)*(-1) ]
= sqrt[ -1 ] * sqrt[ -1 ]
= i * i
= i^2
= -1

Here's the actual error:

Two lines are wrong, but it's the same error in both places. If you pay attention to the first step, you'll notice that what this proof is effectively arguing is that 1 = -1 because they are both the square root of 1. It is entirely true that both of them are square roots of 1--however, every number has two square roots! So the fact that two numbers are both square roots of the same number does not imply they are equal.

In actual fact, sqrt[1] = +/- 1 and sqrt[-1] = +/- i. I have simply ignored a different square root on each side of the equals sign; if you carry the +/- through, you get +/- 1 = +/- 1, which is not a very interesting proof.

-1^2 == -1
Because if you strictly follow order of operations, exponentiation is performed before multiplication, even multiplication by a negative coefficient.

First of all, the same order of operations is not observed in all contexts. For example, mathematicians usually give multiplication higher precedence than division, but in computer programming they are usually given equal precedence and simply read left-to-right. So an expression like "6 / 3 * 2" actually has a different answer depending upon which set of rules you use.

But more importantly, that is not a multiplication in any notational scheme I have ever heard of. I believe most mathematicians would simply say that "-1" is a number, but in computer land we would say that that is the number "1" preceded by the unary negation operator.

If you take a look at this table of C++ operator precedence, you will notice that addition and subtraction have a precedence of 6, multiplication and division are at 5, but unary negation is all the way up at 3. There is no exponentiation operator in C++, but the important thing is to notice that negation is in a separate class unrelated to any of the arithmetic operators and having a precedence higher than any of them. No reasonable person is going to give exponentiation a higher precedence than that.

Now, it is true that

(-1) * 1 ^ 2

would evaluate to -1, under either mathematical or computer programming rules. But "-1" is not the same expression as "(-1) * 1".

[Doom]

You are mistaken. That's just distributing exponentiation over multiplication:

(a*b)^c = a^c * b^c

That's a completely general rule, and is true even when a is negative and c = 1/2.

Uh, no.

Try a = -1, b = -1, and c = 1/2, and see for yourself it doesn't hold as a "general" rule. Honest, go and look in an algebra textbook, it specifically says you can't use certain radical properties if the arguments are negative. It is exactly because square root means the principal root (i.e., the positive root), that you cannot use radical properties for negative numbers.

Other than every single algebra textbook ever written, Wikipedia also specifies you can only use non-negative arguments. see for yourself, it's down a bit on the article.

Because of the discontinuous nature of the square root function in the complex plane, the law √zw = √z√w is in general not true.

Here's an alternate version I saw in college that requires no division:

1
= sqrt[ 1 ]
= sqrt[ (-1)*(-1) ]

And the very next step is the mistake. "The square root of a product is a product of square roots", only applies if the arguments are positive; much as in your 'general rule' above, it doesn't work if both arguments are negative.

First of all, the same order of operations is not observed in all contexts. For example, mathematicians usually give multiplication higher precedence than division, but in computer programming they are usually given equal precedence and simply read left-to-right. So an expression like "6 / 3 * 2" actually has a different answer depending upon which set of rules you use.

Well, it has one answer according the proper rules. There is only one order of operations. Honest, mathematicians give multiplication the same precedence as division, reading left to right (again, go and pick up an algebra textbook and see for yourself). Every mathematician will give the same value to that calculation, 4. Every. Single. One. There is only one answer, as per order of operations.

But more importantly, that is not a multiplication in any notational scheme I have ever heard of. I believe most mathematicians would simply say that "-1" is a number, but in computer land we would say that that is the number "1" preceded by the unary negation operator.

And, yes, in formal notation, "-1" means "the additive inverse of 1". When a mathematician writes " 5 - 1", what he formally means is "5 + (-1)", honest.

[Manxome]

Uh, no.

Try a = -1, b = -1, and c = 1/2, and see for yourself it doesn't hold as a "general" rule. Honest, go and look in an algebra textbook, it specifically says you can't use certain radical properties if the arguments are negative. It is exactly because square root means the principal root (i.e., the positive root), that you cannot use radical properties for negative numbers.

OK, you are discussing principal roots and I am discussing general roots. I'm sure there exists some restricted set of transformations that are allowed on principal root operations that preserve equality, but I can't recall ever learning or needing to know exactly what that set is, despite taking math classes up through differential equations and multivariable calculus.

If you allow negative roots, I'm pretty confident it holds for negative numbers.

Well, it has one answer according the proper rules. There is only one order of operations. Honest, mathematicians give multiplication the same precedence as division, reading left to right (again, go and pick up an algebra textbook and see for yourself). Every mathematician will give the same value to that calculation, 4. Every. Single. One. There is only one answer, as per order of operations.

I, and at least 3 other people I talked to when determining the grammar for a parser I was writing, were all taught in school that multiplication binds tighter than division.

If you're reading a math textbook as you see the expression "a / bc", you would interpret that as "(a/b)*c"?

[Doom]

OK, you are discussing principal roots and I am discussing general roots...but I can't recall ever learning or needing to know exactly what that set is, despite taking math classes up through differential equations and multivariable calculus.

I'm discussing 'square root' as it means in the context of the original proof, and nothing else. There's not much I can do but again ask you to look it up in a textbook if you just don't believe me. You might have gotten away with not knowing the difference in the classes you've taken, but I teach it that way in every math class, up through differential equations and multivariable calculus, and complex analysis, for that manner.

Alternatively, you can read Wikipedia if you don't have any of your textbooks, much like I specifically quoted. The square root of a product is not, in general, a product of square roots (fwiw, Wikipedia does have a 'what if' discussion if you change the definition of square root to mean something that it doesn't normally mean). Keep in mind, I've already shown your 'general rule' to be false in a specific case, which should go a long way to suggesting your general rule is not, in fact, general.

I, and at least 3 other people I talked to when determining the grammar for a parser I was writing, were all taught in school that multiplication binds tighter than division.

Again, consult a mathematics textbook and see with your own eyes if you don't believe me. I just don't know what else to tell you. Division means "multiply by the reciprocal", it's just multiplication by a different name, it simply doesn't make sense to claim that multiplication 'binds tighter' than multiplication.

If you're reading a math textbook as you see the expression "a / bc", you would interpret that as "(a/b)*c"?

As a slash, yes, a/bc is read as (a/b)*c (note how I use parenthesis to be perfectly clear, rather than use ambiguous spacing). There's an important 'special rule' that distinguishes between division with a slash, and a straight line.

Keep in mind, a straight line, as in

A
-------
BC


is interpreted as (A) / (BC), this is because there's a caveat that a straight line in this context is interpreted as including parenthesis for whatever is above, and below, the straight line.
Thus,

A + B
----------
C + D



means (A + B) / (C + D), and not A + B/C + D....I have to use parenthesis for the slash notation, since I don't have a caveat there.

The 'straight line' rule of implied parenthesis also applies for the straight lines in absolute value, and the straight line that is the top of the radical sign in 'square root'. For example, |a- b| really means |(a-b)|, and if I could draw a radical symbol, I could show that it's understood that anything in the radical is implied to be in parenthesis, even though such parenthesis are seldom written.

Hope this helps, I find students often miss learning these types of caveats, which causes all sorts of problems when it's time to read a formula.

[Manxome]

I'm discussing 'square root' as it means in the context of the original proof, and nothing else.

Sorry. I was also discussing "square root" as it means in the context of the original proof, but I was under the impression that a radical meant "square root" and that you put a sign in front of it if you wanted only one of the roots, so that the notation used in the original proof referred to square roots. Wikipedia apparently disagrees with this usage, claiming that a radical always and only means the principal root. I apologize for the confusion.

However, that said...

Keep in mind, I've already shown your 'general rule' to be false in a specific case, which should go a long way to suggesting your general rule is not, in fact, general.

It would go a long way, if you had actually done that. But I was most emphatically clear that I believed it applied to square roots (as in, the inverse of the square function) and NOT to principal square roots, and your example (a = -1, b = -1, c = 1/2) does not constitute a counter-example in that context:

sqrt[ (-1)*(-1) ] = +/- 1
sqrt(-1) * sqrt(-1) = (+/- i)*(+/- i) = +/- 1

No contradiction that I can see.

there's a caveat that a straight line in this context is interpreted as including parenthesis for whatever is above, and below, the straight line.

Yes.

it's understood that anything in the radical is implied to be in parenthesis, even though such parenthesis are seldom written.

Yes.

|a- b| really means |(a-b)|

Absolute value involves putting symbols on both sides of the expression it's applied to, I can't think of anything else it could even theoretically mean. How would you even evaluate |a-b| if that were not true?

yes, a/bc is read as (a/b)*c

But I am utterly surprised by this one. I would have bet money that you would agree that was a/(b*c).

No, I don't have algebra textbooks lying around my house to compare against.

note how I use parenthesis to be perfectly clear, rather than use ambiguous spacing

I thought it was fairly obvious that I was using ambiguous spacing specifically so that I could ask you how you'd interpret the ambiguity. I thought that was a fairly reasonable way to use it.

Division means "multiply by the reciprocal", it's just multiplication by a different name, it simply doesn't make sense to claim that multiplication 'binds tighter' than multiplication.

And here you're just being deliberately obtuse. This makes exactly as much sense as claiming that multiplication is just looped addition so multiplication and addition can't possibly have different precedence.

Symbolic notation can self-evidently have any arbitrary precedence rules we make up, up to and including multiple operators with precisely the same meaning except for having different precedence. I could make up my own notation where the # symbol means multiplication with operator precedence higher than addition but the @ symbol means multiplication with operator precedence lower than addition, and any person even passably good at math should have no trouble evaluating expressions written with this notation after having it explained.

The least stupid way I can possibly read your statement here is as meaning that you would personally find it counter-intuitive if the / and * operators had different precedence, which is at best tangential to the discussion.

[Username17]

Multiplication and division have equal precedence and are read left to right. So "a/bc" is indeed (a/b)*c. But -1 is still the way you write "negative one". It happens to also be the way you write "subtract 1", allowing people to be douchebags by using one when it is situationally implied that you should be using the other, but negative one is still a number and "-1" doesn't have an order of operations place at all.

None of that has fuck all to do with the fact that if you remove any Square Root you lose the equals sign because a Square Root has two solutions.

-Username17

[Doom]

Wikipedia apparently disagrees with this usage, claiming that a radical always and only means the principal root.

Not, not just Wikipedia, but every single mathematics textbook defines the radical in that fashion. It's a definition, Wikipedia 'disagrees' with your usage because your usage violates the definition, and is thus wrong. Doyou have a calculator? If so, press the radical (for square root) of, say, 9, and see if the answer is "3" or "+/- 3". The reason why your calculator says 3 is because it's using the same definition for radical (and note, "radical", as opposed to "square root") as everyone else but, apparently, you.

square roots (as in, the inverse of the square function) and NOT to principal square roots, and your example (a = -1, b = -1, c = 1/2) does not constitute a counter-example in that context:

sqrt[ (-1)*(-1) ] = +/- 1
sqrt(-1) * sqrt(-1) = (+/- i)*(+/- i) = +/- 1

No contradiction that I can see.

Yes, if you take squre root to mean something besides what it actually means, there is no contradiction, but generally, making up your own meanings for words is bad form, particularly in mathematics. Square root (in this context, as a radical) has a particular definition, as per Wikipedia and every single mathematics textbook...you can't change the definition to something else at a whim.

Thus, in the original proof, when they use a radical property when the arguments are negative, the proof becomes invalid because that's not how the radical symbol is defined.

And here you're just being deliberately obtuse. This makes exactly as much sense as claiming that multiplication is just looped addition so multiplication and addition can't possibly have different precedence.

Again, I will reference every single mathematics textbook ever written. There's just not much more I can do about it...division is defined as the inverse of multiplication, and generally is done by multiplying by the reciprocal (note the reciprocal is the multiplicative inverse of a number), so it has the same precedence as multiplication.

("Basic College Mathematics" by Miller, O'Neill, Hyde defines multiplication in this way on page 134, although Wikipedia's definition is a bit arcane, says the same thing...again, I just dont' know what else to tell you but to look it up yourself it you don't believe me.)

Subtraction is defined as "add the additive inverse", so it has the same precedence as addition.

Seriously, I'm just quoting the basic definitions here, I'm not being obtuse.

The least stupid way I can possibly read your statement here is as meaning that you would personally find it counter-intuitive if the / and * operators had different precedence, which is at best tangential to the discussion.

I was trying to help by quoting the most basic definitions, I'm sorry if the basic definitions look stupid to you. Again, please consult a basic math textbook and read it with your own eyes...I just don't know what else to tell you.

[fectin]

Could someone give me an example where order of operations makes a difference between multiplication and division, or between addition and subtraction? I'm pretty sure you can evaluate them however you want, as long as you get innermost parentheticals first, exponents next, and all division and multiplication before any addition or subtraction.

And no, it's not even ambiguous. There is no "interpretation" in which -1^2 = -1, because there is no situation where subtraction can involve only one term. If you were a tool, you could say it was 0 ('not true and not true', Boolean logic), but that's both dumb and an uncommon notation.

If it were 0-1^2 = -1, that would be correct. That generalizes real well though: "if you had said something different, it might have been correct."

[sabs]

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

secondly.
sqrt[(-1)*(-1)] = 1
sqrt[-1] * sqrt[-1]= i * i

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

you always do multiplication/division in pairs, starting from left working right.

abcd/efg always gets read as: (((ab)*(c)*(d))/(e))*(f)*(g)

[shadzar]

Could someone give me an example where order of operations makes a difference between multiplication and division, or between addition and subtraction? I'm pretty sure you can evaluate them however you want, as long as you get innermost parentheticals first, exponents next

you just gave an example...

an exponent is the act of multiplying something by itself...and that comes BEFORE multiplication or division....

x^2 = x*x
x^4=x*x*x*x

subtraction, as was said is just addition.

the order really matters in cases such as others have said with computer, you have to tell the computer what comes first otherwise it reads right to left.

in the case of the previous a/bc, a computer could NOT handle that as 3 variables, but bc being the name of a single variable.

a computer would require a/b*c. then it would do a divided by b times c

originally and since computer cannot represent fractions as they still mostly print text on a single line unless you format 2 or more lines to do it, a/bc is representation in typed form of a/(b*c) as a computer would need it, or:

a
---
bc

as humans cam write with pen or pencil on paper.

a clearer example of why you do multiplication first how about hand-written form as opposed to printer form.

ab/c+d

computer form: (a*b)/(c+d)

hand written form:

ab
----
c+d

before you can divide the product of a and b by the sum of c and d, you have to work on both sides of the dividing line. that is the basic principle why multiplication comes first, in addition to the exponents.

you cannot divide something by something else when you have nothing to tell you what you are dividing by. you have to solve both sides before you can do the division.

a*b=X
c+d=Y

then we get x/y and NOW you can divide.

[Doom]

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.