Is 1 Negative? A Proof from Lang's 'A First Course in Calculus

[wifi]

This is from Lang's "A First Couse in Calculus".

In chapter 1 he makes two statements regarding positivity:

POS 1. If $a$, $b$ are positive, so is the product $ab$ and the sum $a+b$.

POS 2. If $a$ is a number, then either $a$ is positive, or $a=0$, or $-a$ is positive, and these possibilities are mutually exclusive.

("POS" meaning positive or postulate I guess.)

His proof that 1 is positive (using only the above statements) goes like this:

"By POS 2, we know that either 1 or -1 is positive. If 1 is not positive, -1 is positive. By POS 1, it must then follow that (-1)(-1) is positive. But this product is equal to 1. Consequently, it must be 1 which is positive, and not -1."

It seems like this proof hinges on that fact that (-1)(-1) is positive. However, since both (1)(1) and (-1)(-1) are positive, from this alone we really can't conclude that 1 or -1 are negative. Right? To me it'd make more sense to say that (-1)+(-1) is not positive, and by POS 1 -1 cannot be positive. By the same logic we can then claim that 1 is the positive one.

I know this is probably an elementary mistake on my part, but I'm lost.

 

[micromass]

This is from Lang's "A First Couse in Calculus".

In chapter 1 he makes two statements regarding positivity:

POS 1. If $a$, $b$ are positive, so is the product $ab$ and the sum $a+b$.

POS 2. If $a$ is a number, then either $a$ is positive, or $a=0$, or $-a$ is positive, and these possibilities are mutually exclusive.

("POS" meaning positive or postulate I guess.)

His proof that 1 is positive (using only the above statements) goes like this:

" By POS 2, we know that either 1 or -1 is positive. If 1 is not positive, -1 is positive. By POS 1, it must then follow that (-1)(-1) is positive. But this product is equal to 1. Consequently, it must be 1 which is positive, and not -1."

Do you understand this proof? Do you somehow disagree with it? It's not clear to me.

It seems like this proof hinges on that fact that (-1)(-1) is positive. However, since both (1)(1) and (-1)(-1) are positive, from this alone we really can't conclude that 1 or -1 are negative. Right?

Well, we know that $(-1)(-1)=1$. So if $(-1)(-1)$ is positive, then $1$ is positive.

To me it'd make more sense to say that (-1)+(-1) is not positive, and by POS 1 -1 cannot be positive. By the same logic we can then claim that 1 is the positive one.

How would you know that $(-1)+(-1)$ is not positive?

 

[wifi]

Do you understand this proof? Do you somehow disagree with it? It's not clear to me.

I don't understand the reasoning in the proof; I'm not questioning it's validity. I apologize if I was unclear.

How would you know that $(-1)+(-1)$ is not positive?

(-1)+(-1) is not positive, since it's < 0. Or, geometrically one can view it as going further left on a number line, further from 0 (ie. becoming more negative).

Well, we know that $(-1)(-1)=1$. So if $(-1)(-1)$ is positive, then $1$ is positive

I think this is what I don't understand. Wouldn't (-1)(-1)=1 imply that -1 is positive, since it's product is positive? .

 

[micromass]

(-1)+(-1) is not positive, since it's < 0. Or, geometrically one can view it as going further left on a number line, further from 0 (ie. becoming more negative).

Of course it's not positive. But my question is, how would you prove it using only POS 1 and POS 2?

I think this is what I don't understand. Wouldn't (-1)(-1)=1 imply that -1 is positive, since it's product is positive? .

The axiom says: "If a and b are positive, then ab is positive". You somehow seem to think that if ab is positive then a and b are positive. That is not true at all.
It's not because a product of two elements is positive, that the elements themselves are.

 

[wifi]

Of course it's not positive. But my question is, how would you prove it using only POS 1 and POS 2?

I really don't see how you could.

The axiom says: "If a and b are positive, then ab is positive". You somehow seem to think that if ab is positive then a and b are positive. That is not true at all.
It's not because a product of two elements is positive, that the elements themselves are.

I understand that the product of two numbers may be positive if both are positive, or both are negative. But from the manner in which POS 1 is defined, it seems like he's saying that, because the product ab is positive, then a and b are positive. Obviously this isn't true, but I don't understand this logic.

 

[micromass]

I really don't see how you could.

Exactly, this is why your proof doesn't work.

I understand that the product of two numbers may be positive if both are positive, or both are negative. But from the manner in which POS 1 is defined, it seems like he's saying that, because the product ab is positive, then a and b are positive. Obviously this isn't true, but I don't understand.

POS 1 doesn't say that at all. I don't really understand why you think that it says that. POS 1 clearly says that if a and b are positive, then ab is positive. It says nothing about when ab is positive.

 

[wifi]

Exactly, this is why your proof doesn't work.

I wasn't claiming to prove that in terms of only POS 1 and POS 2.

POS 1 doesn't say that at all. I don't really understand why you think that it says that. POS 1 clearly says that if a and b are positive, then ab is positive. It says nothing about when ab is positive.

I feel like I'm angering you. I'm just going to go think some more about it; I'm slow at these things.

 

[micromass]

I wasn't claiming to prove that in terms of only POS 1 and POS 2.



I feel like I'm angering you. I'm just going to go think some more about it; I'm slow at these things.

You're not angering me at all! My apologies if I gave you that impression.
I simply don't understand what exactly your problem is, so I can't really help until I understand.

 

[lurflurf]

The argument is
suppose -1 is positive
then by pos 1
(-1)(-1)=1 is positive
this violates pos 2
thus -1 must not be positive

another approach would be
suppose either -1 or 1 is positive
that is 1 is not 0 (pos 2)
(-1)(-1)=1
(1)(1)=1
by pos 1 we conclude
1 is positive

 

[wifi]

The argument is
suppose -1 is positive
then by pos 1
(-1)(-1)=1 is positive
this violates pos 2
thus -1 must not be positive

How does (-1)(-1) being positive violate POS 2?

 

[D H]

How does (-1)(-1) being positive violate POS 2?

Contradiction. The assumption that -1 is positive means that 1 is positive. A number and its additive inverse cannot both be positive by POS2.

 

[SteamKing]

This is from Lang's "A First Couse in Calculus".

In chapter 1 he makes two statements regarding positivity:

POS 1. If $a$, $b$ are positive, so is the product $ab$ and the sum $a+b$.

POS 2. If $a$ is a number, then either $a$ is positive, or $a=0$, or $-a$ is positive, and these possibilities are mutually exclusive.

("POS" meaning positive or postulate I guess.)

His proof that 1 is positive (using only the above statements) goes like this:

" By POS 2, we know that either 1 or -1 is positive. If 1 is not positive, -1 is positive. By POS 1, it must then follow that (-1)(-1) is positive. But this product is equal to 1. Consequently, it must be 1 which is positive, and not -1."

It seems like this proof hinges on that fact that (-1)(-1) is positive. However, since both (1)(1) and (-1)(-1) are positive, from this alone we really can't conclude that 1 or -1 are negative. Right? To me it'd make more sense to say that (-1)+(-1) is not positive, and by POS 1 -1 cannot be positive. By the same logic we can then claim that 1 is the positive one.

I know this is probably an elementary mistake on my part, but I'm lost.

If a = -1 and b = -1, then by Postulate 1, in order to conclude that a and b are both positive, we must calculate the product ab and the sum a+b.

Now ab = (-1)(-1) = 1 : so far so good.
But a + b = -1 + -1 = -2 : there is a different result here which is not positive

We can conclude thus that -1 is not positive. It's the word 'and' in the postulate which is key, where it specifies that the product AND the sum of a and b must be positive.

 

[HallsofIvy]

This is from Lang's "A First Couse in Calculus".

In chapter 1 he makes two statements regarding positivity:

POS 1. If $a$, $b$ are positive, so is the product $ab$ and the sum $a+b$.

POS 2. If $a$ is a number, then either $a$ is positive, or $a=0$, or $-a$ is positive, and these possibilities are mutually exclusive.

("POS" meaning positive or postulate I guess.)

I suspect that it stands for "positive" since these are axioms or postulates describing "positive" numbers.

His proof that 1 is positive (using only the above statements) goes like this:

" By POS 2, we know that either 1 or -1 is positive. If 1 is not positive, -1 is positive. By POS 1, it must then follow that (-1)(-1) is positive. But this product is equal to 1. Consequently, it must be 1 which is positive, and not -1."

It seems like this proof hinges on that fact that (-1)(-1) is positive. However, since both (1)(1) and (-1)(-1) are positive, from this alone we really can't conclude that 1 or -1 are negative. Right?

Yes, both (1)(1) and (-1)(-1) are positive. However, we can from "Pos 2" (the "trichotomy" axiom) conclude that (since 1 is not the additive identity it is NOT 0) 1 is either positive or negative. If 1 is positive, we are done. So assume that 1 is negative and derive a contradiction. That is, assume that -1 is positive.

To me it'd make more sense to say that (-1)+(-1) is not positive,

HOW do you say that? Of course, (-1)+ (-1)= -2 but how do you know that is not positive?

and by POS 1 -1 cannot be positive. By the same logic we can then claim that 1 is the positive one.

I know this is probably an elementary mistake on my part, but I'm lost.

 

[Seydlitz]

The argument is
suppose -1 is positive
then by pos 1
(-1)(-1)=1 is positive
this violates pos 2
thus -1 must not be positive

another approach would be
suppose either -1 or 1 is positive
that is 1 is not 0 (pos 2)
(-1)(-1)=1
(1)(1)=1
by pos 1 we conclude
1 is positive

I'm also beginner like wifi, but maybe I can rephrase this informally. (Please correct me also in case of fallacy)

I want to know whether -1 or 1 is positive. The clear thing is 1 is not 0 because of the existence of multiplicative identity. Any numbers squared $a^2$ will always be positive. If we multiply (-1)(-1), then it will be positive. If we multiply (1)(1) it will also be positive. But what is (1)(1)?

We also know for certain that (1)(1) = 1 because of the identity axiom, not only because of pos1 and pos2. (a)(1) = (a)

If -1 is positive then (-1)(-1) should be (-1), and consequently (1)(1) = -1. Surely this is not right.

The important point I think is the special case that $a^2$ will always be positive, where a is not 0. I think if we only know that (1)(1) = (1) and the special case of number squared \, it's already enough to prove that 1 > 0. (Is this right?)

I also think it might be possible to agree on the fundamental definition to switch sign of positive and negative. So we treat 1 as negative and -1 as positive.

It seems like this proof hinges on that fact that (-1)(-1) is positive. However, since both (1)(1) and (-1)(-1) are positive, from this alone we really can't conclude that 1 or -1 are negative. Right?

I know (1)(1) = 1. Still, I'm not sure whether it's positive or negative but I know $a^2$ must be positive for any non-zero number. In that case 1 is positive, and so (1)(1). If it's positive it's not negative. Hopefully this is not a circular argument and you can understand it.

 

[D H]

If a = -1 and b = -1, then by Postulate 1, in order to conclude that a and b are both positive, we must calculate the product ab and the sum a+b.

You do not have to calculate a+b. Calculating ab is all that is needed.

There is two items left out of the proof, one of which is showing that (-1)(-1)=1. The supplied proof merely says "But this product is equal to 1." Proving that (-1)(-1) is indeed 1 is trivial.

If -1 is positive, then so is 1 by POS1 and the fact that (-1)(-1) is 1. But this contradicts POS2. That's all that is needed to show that -1 is not positive.

What about 1? Assuming 1 is positive does not lead to a contradiction. Lack of a contradiction is not sufficient, because POS1 says nothing about the possibility of 1 being zero. Dealing with that possibility is the other item left omitted from the proof.

 

[wifi]

If a = -1 and b = -1, then by Postulate 1, in order to conclude that a and b are both positive, we must calculate the product ab and the sum a+b.

Now ab = (-1)(-1) = 1 : so far so good.
But a + b = -1 + -1 = -2 : there is a different result here which is not positive

We can conclude thus that -1 is not positive. It's the word 'and' in the postulate which is key, where it specifies that the product AND the sum of a and b must be positive.

This is exactly how I see it! How is this not correct?

 

[D H]

This is exactly how I see it! How is this not correct?

You don't know whether -2 is positive or negative. Using the sum doesn't accomplish a thing.

 

[wifi]

You do not have to calculate a+b. Calculating ab is all that is needed.

There is two items left out of the proof, one of which is showing that (-1)(-1)=1. The supplied proof merely says "But this product is equal to 1." Proving that (-1)(-1) is indeed 1 is trivial.

If -1 is positive, then so is 1 by POS1 and the fact that (-1)(-1) is 1. But this contradicts POS2. That's all that is needed to show that -1 is not positive.

What about 1? Assuming 1 is positive does not lead to a contradiction. Lack of a contradiction is not sufficient, because POS1 says nothing about the possibility of 1 being zero. Dealing with that possibility is the other item left omitted from the proof.

Hm. I think I'm getting closer to understanding this.

So, at a certain point in the proof we've basically proved that -1 and 1 are both positive. This certainly isn't the case, because by POS 2 we know only one of them can be positive. But how we do we know that it's 1 that's positive?

 

[D H]

You know that -1 is not positive, so by POS2 -1 can only be zero or negative. Thus 1 is either zero or positive. All that's left is to show that 1 is not zero.

 

[Seydlitz]

You know that -1 is not positive, so by POS2 -1 can only be zero or negative. Thus 1 is either zero or positive. All that's left is to show that 1 is not zero.

I wonder if it would be possible to leave the contradiction argument entirely by deciding with certainty that 1 is in fact positive and not 0, i.e knowing (1)(1) = 1 and squaring any non-zero number will produce a positive number.

 

[lurflurf]

^It is true that a^2 is positive when a is not zero
We cannot use that result here as we have not yet proven it.

The way we decide with certainty that 1 is in fact positive is that
(-1)(-1)=(1)(1)=1
if -1 is positive 1 is positive
if 1 is positive 1 is positive

since either -1 of 1 is positive this is equivalent to
1 is positive

 

[D H]

The way we decide with certainty that 1 is in fact positive is that
(-1)(-1)=(1)(1)=1

That doesn't work because (-0)(-0) = (0)(0) = 0. You need something else such as the definition of what "0" and "1" are, to distinguish 0 and 1.

 

[Seydlitz]

I found Spivak's approach to this problem quite elucidating and straightforward for beginner. (no prove by contradiction yet) He proved first 1 is positive using $a^2>0$

Then it is evident from Trichtomy Law that if 1 is positive it can't be 0 at the same time.

 

[lurflurf]

^That is not really better.
The proof in Lang is not by contradiction, though it is easy to make into one if desired.
We start with 1 is not zero
so either -1 or 1 is positive
we conclude 1 is positive

Lang does not give enough structure and definitions to show 1 is not zero (the proof is on page 4)
presumably that is to be assumed
One could require
x+0=x
if 1=0
so do all integers
integers are very uninteresting if that is the case.

 

[bahamagreen]

wifi,
Lots of math confounds normal thinking by using familiar symbols and signs in assigned and assumed roles within an argument where their properties are different, or the opposite. Part of that stems from the use of logical forms that take the reverse orientation, and some of it stems from math folks being rather proud of their ability to abstract and hold symbols apart from their common face meanings while manipulating them conceptually.

This means the ones that are good at this may not see why you are confused...

The test assumes that "-1" is positive;

the test equation results in "1", which you are used to thinking of as positive.

But since "-1" has been assumed as positive, that means "1" is assumed to be negative.

The test equation results in "1", which is assumed negative, which means the test fails because it is seeking a positive result for the test.

This is really just a case where the substitution of numbers for the variables in the postulates tends to make it confusing - because we are used to knowing 1 is positive and -1 is negative... but the proof assumes and asigns the reverse properties, so the test result must be evaluated as a "reverse" as well. So "1" as a result is considered a negative number.

 

[wifi]

wifi,
Lots of math confounds normal thinking by using familiar symbols and signs in assigned and assumed roles within an argument where their properties are different, or the opposite. Part of that stems from the use of logical forms that take the reverse orientation, and some of it stems from math folks being rather proud of their ability to abstract and hold symbols apart from their common face meanings while manipulating them conceptually.

This means the ones that are good at this may not see why you are confused...

The test assumes that "-1" is positive;

the test equation results in "1", which you are used to thinking of as positive.

But since "-1" has been assumed as positive, that means "1" is assumed to be negative.

The test equation results in "1", which is assumed negative, which means the test fails because it is seeking a positive result for the test.

This is really just a case where the substitution of numbers for the variables in the postulates tends to make it confusing - because we are used to knowing 1 is positive and -1 is negative... but the proof assumes and asigns the reverse properties, so the test result must be evaluated as a "reverse" as well. So "1" as a result is considered a negative number.

Thanks so much bahamagreen, I get it now!