Negative times negative makes positive

[joigus]

3 hours ago, studiot said:

It would actually be OK to define (-1) x (-1) = -1.

That would still form a hemi group (or semigroup if you wish) under multiplication.

However american practice requires there to be a unique non zero identity for multiplication, which would not be the case with -1.

 

It is notable that Birkhoff and MacLane (american definitions) start off on page 1 with this subject and reach the crux of it by page 6  (so a bit much to post here as an extract) with 'the integers following 8 postulates.

Definitely  a bit much for 12 year olds I feel.

 

Interesting. I wasn't aware of differences between us and them. Now that you mention it I seem to remember notes from some professor pointing out that one could either demand the existence of an non-zero identity or leave it out. And then rings with an identity would be 'ring with identity'. Like Z_p with p prime. (Please correct me if I'm wrong, I'm not checking anything I'm saying here).

The turn-around way that @Eise suggested, or the mirror way, or @StringJunky's yes-no, or @MigL's even/odd rule, or anything that's ultimately an involution (leaving fermions aside) would be perhaps the most intuitive way. Something like this:

Ok, kids. Multiplying by 1 is: Nothing changes

Multiplying by -1, on the other hand, is some kind of a 'switch'. When you switch the switching, it doesn't switch more. It just unswitches!

It may sound silly, but kids generally get it when you give them examples like these. 

Semigroups and the like are for geezers like myself. 

[Genady]

I have a question. Before they start with the multiplication, after they learn only addition and subtraction of integers, do they know how to deal with, say, 4-(-6)?

[Eise]

18 hours ago, Genady said:

... unless you're a fermion 🙂

Shouldn't the sum of all spins of the fermions in my body sometimes add to n + 1/2 so I am a fermion as a whole? Then by turning around 2 times I should look in the opposite direction. I try several times per day, but it never happened. Physics is wrong!

[Genady]

2 hours ago, Eise said:

Shouldn't the sum of all spins of the fermions in my body sometimes add to n + 1/2 so I am a fermion as a whole? Then by turning around 2 times I should look in the opposite direction. I try several times per day, but it never happened. Physics is wrong!

+1 (when the system lets me.)

Replace that remark with, "... if you're not an electron."

[Genady]

12 hours ago, joigus said:

Like Z_p 

Is 'Z_p' the same as 'Z/pZ'?

[joigus]

1 hour ago, Genady said:

Is 'Z_p' the same as 'Z/pZ'?

Yes, exactly. Those are the cyclic rings, or discrete... They go under several names.

[Genady]

37 minutes ago, joigus said:

Yes, exactly. Those are the cyclic rings, or discrete... They go under several names.

Then, I think, they always have a multiplicative identity, but when p is prime, they are fields, as then each element has a multiplicative inverse.

Edited Monday at 01:26 PM by Genady

[sethoflagos]

11 hours ago, Genady said:

I have a question. Before they start with the multiplication, after they learn only addition and subtraction of integers, do they know how to deal with, say, 4-(-6)?

How do you address the very different physical interpretations of:

a) I take six oranges from the box

b) I add minus six oranges to the box

There's a strong scent here of two different operations being conflated into one without explanation.

[Genady]

3 minutes ago, sethoflagos said:

How do you address the very different physical interpretations of:

a) I take six oranges from the box

b) I add minus six oranges to the box

There's a strong scent here of two different operations being conflated into one without explanation.

I don't understand. The example, 4-(-6), is neither a) nor b), I think.

[sethoflagos]

5 minutes ago, Genady said:

I don't understand. The example, 4-(-6), is neither a) nor b), I think.

I didn't say it was. I'm trying to highlight an intermediate step that's possibly being overlooked. 

[Genady]

4 minutes ago, sethoflagos said:

I didn't say it was. I'm trying to highlight an intermediate step that's possibly being overlooked. 

Ah, I see. IMO, adding and subtracting numbers do not reflect physical moving of objects but rather reflect ways of counting them.

[sethoflagos]

11 minutes ago, Genady said:

Ah, I see. IMO, adding and subtracting numbers do not reflect physical moving of objects but rather reflect ways of counting them.

To me, the very notion of negative numbers implies multiplication by -1. This suggests that prior introduction to multiplication of the natural numbers (not to mention the role of zero) is a more logical order of learning.

Edited Monday at 03:11 PM by sethoflagos
Clarification

[studiot]

16 hours ago, joigus said:

Multiplying by -1, on the other hand, is some kind of a 'switch'. When you switch the switching, it doesn't switch more. It just unswitches!

It may sound silly, but kids generally get it when you give them examples like these. 

Semigroups and the like are for geezers like myself. 

Yes that's not a bad explanation (but not proof) for beginners.

Consider a transformation T such that T² (-1) = 1

We could also compare a transformation T² (-1) = -1

 

The first transformation might actually be a reflection.

The second one will be equivalent to introducing i.

 

12 hours ago, Genady said:

I have a question. Before they start with the multiplication, after they learn only addition and subtraction of integers, do they know how to deal with, say, 4-(-6)?

I said I needed to dig out some old books from the depths.
 

Yes indeed htere are many approaches to this but we must remember two things.

Firstly who are 'they'  ?

Years ago only children at a grammar school would have learned anything about negative numbers.

 

In primary school great efforts were made to work only in the positive.

So for instance the subtraction of

2092 from 3513 would be carried out as follows

2 from 3 leaves 1

9 from 1 won't go so borrow 1 (ten) and find 9 from 11 leaves 2

1 from 5 leaves 4        (or alternatively reduce the 5 to 4 and take zero from it)

2 form 3 leaves 1

 

Answer 1421  All done in the positive.

 

These children would not have been introduced to graphs so there was no baclground for 'the number line'

 

Grammar School started at age 11 - 13  .

Most children did not go to grammar school.

Those that went to a non grammar secondary school were often apprentices and schooled in the practical.
So quite a number of introductory practical examples were developed for these.

Grammar school children were introduced usually by Hall and Knight - first pub 1895 and still going strong in the second part of the 20th century.
This has an algebraic development, less advanced than Birkhof and Maclean.

 

 

 

 

 

 

 

[Genady]

53 minutes ago, sethoflagos said:

To me, the very notion of negative numbers implies multiplication by -1. This suggests that prior introduction to multiplication of the natural numbers (not to mention the role of zero) is a more logical order of learning.

It is not so in the grown-up's algebra, but might be a good order for learning. However, there seems to be a logical gap there:

To introduce negative numbers it uses multiplication by -1, but where does the -1 come from if there are no negative numbers yet?

[sethoflagos]

21 minutes ago, Genady said:

It is not so in the grown-up's algebra, but might be a good order for learning.

Your OP is specific to the latter. I have clear memories of measuring and cutting strips of cardboard from old cereal packets, and arranging them as 'staircases' of varying gradients. Once we were familiar with ascending and descending the aboveground flights of stairs in various step sizes, we could have a look at what might call the first step down towards the cellar.

33 minutes ago, Genady said:

To introduce negative numbers it uses multiplication by -1, but where does the -1 come from if there are no negative numbers yet?

Following my own train of thought, it seems to start as simply a label for a descent of one step from ground zero. Giving the result the same name as the operation is an idea at least.

[Genady]

10 minutes ago, sethoflagos said:

Your OP is specific to the latter. I have clear memories of measuring and cutting strips of cardboard from old cereal packets, and arranging them as 'staircases' of varying gradients. Once we were familiar with ascending and descending the aboveground flights of stairs in various step sizes, we could have a look at what might call the first step down towards the cellar.

Following my own train of thought, it seems to start as simply a label for a descent of one step from ground zero. Giving the result the same name as the operation is an idea at least.

OK. We can use the "× -1" label to get from positives to negatives. How do we come to use the same label to get from negatives to positives?

[sethoflagos]

5 minutes ago, Genady said:

OK. We can use the "× -1" label to get from positives to negatives. How do we come to use the same label to get from negatives to positives?

Good question! Let me dwell on it a while!

(Though my label was simply '-1' without committing to multiplication or subtraction)

Edited Monday at 05:10 PM by sethoflagos

[studiot]

58 minutes ago, Genady said:

It is not so in the grown-up's algebra, but might be a good order for learning. However, there seems to be a logical gap there:

To introduce negative numbers it uses multiplication by -1, but where does the -1 come from if there are no negative numbers yet?

 

4 minutes ago, Genady said:

OK. We can use the "× -1" label to get from positives to negatives. How do we come to use the same label to get from negatives to positives?

Hello.

 

I am still assembling some of these other approaches, which contain cunning answers for you.

Two points.

Firstly the correct term is not 'negative numvers, but signed numbers.

The signs have different significances in different situations.

Secondly the cunning bit comes when you choose suitable combinations of situation so that you multiply two signed numbers.

 

[sethoflagos]

If we've learnt the operations of ascending-by-one, ascending-by-two, etc..

What happens when we ascend-by-zero?

What happens when we ascend-by-(-1)?

What happens when we descend-by-(-1)?

Do we see that ascending by 1, 2, 3... Is the same as descending by -1, -2, -3...?

Edited Monday at 05:35 PM by sethoflagos
Continuation

[TheVat]

It might be easier to explain to a child that negation in math is like an inversion or flipping the number line.  Multiplication is simply making an inversion one or several times.  So, one inversion of minus three is -1x-3.  Invert the number line and you flip minus three to three.  So -3x-3 is inversion of minus three three times, and each time adds a flipped segment 3 units long, getting you to 9.  This shows that positive and negative are really somewhat different in character - one just extends, the other flips.  

[Genady]

25 minutes ago, TheVat said:

It might be easier to explain to a child that negation in math is like an inversion or flipping the number line.  Multiplication is simply making an inversion one or several times.  So, one inversion of minus three is -1x-3.  Invert the number line and you flip minus three to three.  So -3x-3 is inversion of minus three three times, and each time adds a flipped segment 3 units long, getting you to 9.  This shows that positive and negative are really somewhat different in character - one just extends, the other flips.  

This is one of the recurring themes in this thread, e.g., 

 

[KJW]

2 hours ago, studiot said:

Years ago only children at a grammar school would have learned anything about negative numbers.

When I was in Year 2 at a regular public school, we were taught matrices, including matrix multiplication. They didn't seem at all useful to a seven-year-old. I didn't encounter them again until senior high school, when their use was revealed.

I do not remember when I learnt about negative numbers.

 

[Genady]

2 hours ago, sethoflagos said:

If we've learnt the operations of ascending-by-one, ascending-by-two, etc..

What happens when we ascend-by-zero?

What happens when we ascend-by-(-1)?

What happens when we descend-by-(-1)?

Do we see that ascending by 1, 2, 3... Is the same as descending by -1, -2, -3...?

I'd say that the basic operations are ascend/descend-by-one. Then to ascend/descend-by-two you just ascend/descend-by-one twice, to ascend/descend-by-three you ascend/descend-by-one three times, etc. To ascend/descend-by-zero you then ascend/descend-by-one zero times, i.e., you don't move. I think it is not difficult to swallow that to ascend/descend-by-(-1) you descend/ascend-by-one, to ascend/descend-by-(-2) you descend/ascend-by-two, etc.

[studiot]

28 minutes ago, Genady said:

I'd say that the basic operations are ascend/descend-by-one. Then to ascend/descend-by-two you just ascend/descend-by-one twice, to ascend/descend-by-three you ascend/descend-by-one three times, etc. To ascend/descend-by-zero you then ascend/descend-by-one zero times, i.e., you don't move. I think it is not difficult to swallow that to ascend/descend-by-(-1) you descend/ascend-by-one, to ascend/descend-by-(-2) you descend/ascend-by-two, etc.

Here is an example of what I mean

 

Let us consider some deals on the stock exchange.

Define

Buying 3 shares as -3 shares

Selling 3 Shares as +3 shares

 

Selling a share at $5 above par as +$5  or $5 profit

Selling a share at $5 below par as -$5  or $5 loss

par is the same as face value.

Now let us say consider 4 different deals assume that as soon as each deal is done the company redeems the share at face value.

 

Deal 1)

Sell 3 shares at $5 above face value

That is +3   x  +5   =  +15 or $15 profit

Deal 2)

Sell 3 shares at $5 below par

That is +3  x  -5  =  -15  or $15 loss

Deal 3)

Buy 3 shares at $5 above par

That is  -3  x  +5  =  -15 or $15 loss

Deal 4)

Buy 3 shares at $5 below par

That is -3  x  -5  =  +15  or $15 profit

 

Deal 4 is obviously the case of negative times negative makes positive.

 

So it depends upon how you assign the plus and minus.

Which is why mathematically they are called directed or signed numbers.

[Genady]

6 minutes ago, studiot said:

Here is an example of what I mean

 

Let us consider some deals on the stock exchange.

Define

Buying 3 shares as -3 shares

Selling 3 Shares as +3 shares

 

Selling a share at $5 above par as +$5  or $5 profit

Selling a share at $5 below par as -$5  or $5 loss

par is the same as face value.

Now let us say consider 4 different deals assume that as soon as each deal is done the company redeems the share at face value.

 

Deal 1)

Sell 3 shares at $5 above face value

That is +3   x  +5   =  +15 or $15 profit

Deal 2)

Sell 3 shares at $5 below par

That is +3  x  -5  =  -15  or $15 loss

Deal 3)

Buy 3 shares at $5 above par

That is  -3  x  +5  =  -15 or $15 loss

Deal 4)

Buy 3 shares at $5 below par

That is -3  x  -5  =  +15  or $15 profit

 

Deal 4 is obviously the case of negative times negative makes positive.

 

So it depends upon how you assign the plus and minus.

Which is why mathematically they are called directed or signed numbers.

Got it. +1