Math question

[Novi]

Up to now I've seen a lot of most interesting formal and abstract explanations I will (try to) study soon. But what if you should explain it to your child? I mean why the hell multipling - 2 peaches by - 2 peaches you get (+) 4 peaches. There must be a conceptual way to illustrate the thing in pragmatic terms.

No you don't, you get (+) 4 peaches^2, 4 genetically modified monsters .

How about thinking of the '-' as a positional thing?

The peaches can be in two places: in your hands or in the fruit bowl.

So you start off with -2 peaches: there are 2 peaches in the fruit bowl.

Multiplying it by 2 means you now have 4 in the fruit bowl.

Multiplying it by -2 means you have still have 4, but they are now in your hands, because the '-' means moving them.
Magic!

Err, mathematics as casuistry ...

[Steve]

No you don't, you get (+) 4 peaches^2, 4 genetically modified monsters .

How about thinking of the '-' as a positional thing?

The peaches can be in two places: in your hands or in the fruit bowl.

So you start off with -2 peaches: there are 2 peaches in the fruit bowl.

Multiplying it by 2 means you now have 4 in the fruit bowl.

Multiplying it by -2 means you have still have 4, but they are now in your hands, because the '-' means moving them.
Magic!

Err, mathematics as casuistry ...

Very illuminating, Novitate 

For a virtually convincing proof, you would really have to go all the way back to the foundation of constants, themselves. I would suggest Frege's Principles of Mathematics.

[PSmith08]

At the risk of falling victim to an absurd idée fixe, let me put it like this. Basic math (the arithmetic, multiplication, and division with which you are familiar) happens in R, the field of real numbers. R is a commutative division ring: a set with two binary operations, multiplicative inverses, and commutativity. There are some other axioms that R has to fulfill as it is recognized as each, progressively more ordered, structure; it does, pretty much anyone with a course in abstract algebra could prove it to anyone's satisfaction, in fact to complete that course, you will have to prove it. A lot. One axiom of a field, because of the division ring bit, is this:

a*b=1 (and 1/=0), where b is called the inverse of a. So, in the case of R, a number multiplied by its inverse equals the multiplicative identity, 1 (s.t. a*1=a) - another rule of the game. These are rules of any field, and if it doesn't satisfy these rules, then it isn't a field. R is a field, and thus R satisfies these rules.

Let's do this, then: -1*b=1, where b is the inverse and 1 is - by the law of the structure - the inverse identity.[note: edit] Let's do some basic algebra, then,

-1*b=1
-1*b*(1/-1)=1*(1/-1)
b=1*(1/-1)=>b=1/-1
b=-1. We have seen, then, that the inverse of -1 is, shock and horror, -1. Because the field has associative and commutative properties for both operators, you can rearrange any situation to get to this point, i.e., where -1*-1=1. Why? It's trivially derived from the real numbers' status, a field, R.

[Novi]

At the risk of falling victim to an absurd idée fixe, let me put it like this. Basic math (the arithmetic, multiplication, and division with which you are familiar) happens in R, the field of real numbers. R is a commutative division ring: a set with two binary operations, multiplicative inverses, and commutativity. There are some other axioms that R has to fulfill as it is recognized as each, progressively more ordered, structure; it does, pretty much anyone with a course in abstract algebra could prove it to anyone's satisfaction, in fact to complete that course, you will have to prove it. A lot. One axiom of a field, because of the division ring bit, is this:

a*b=1 (and 1/=0), where b is called the inverse of a. So, in the case of R, a number multiplied by its inverse equals the multiplicative identity, 1 (s.t. a*1=a) - another rule of the game. These are rules of any field, and if it doesn't satisfy these rules, then it isn't a field. R is a field, and thus R satisfies these rules.

Let's do this, then: -1*b=1, where b is the inverse and 1 is - by the law of the structure - the inverse. Let's do some basic algebra, then,

-1*b=1
-1*b*(1/-1)=1*(1/-1)
b=1*(1/-1)=>b=1/-1
b=-1. We have seen, then, that the inverse of -1 is, shock and horror, -1. Because the field has associative and commutative properties for both operators, you can rearrange any situation to get to this point, i.e., where -1*-1=1. Why? It's trivially derived from the real numbers' status, a field, R.

But dammit, where are the peaches?

(actually, your explanation brings back those algebra lectures ... )

[Steve]

At the risk of falling victim to an absurd idée fixe, let me put it like this. Basic math (the arithmetic, multiplication, and division with which you are familiar) happens in R, the field of real numbers. R is a commutative division ring: a set with two binary operations, multiplicative inverses, and commutativity. There are some other axioms that R has to fulfill as it is recognized as each, progressively more ordered, structure; it does, pretty much anyone with a course in abstract algebra could prove it to anyone's satisfaction, in fact to complete that course, you will have to prove it. A lot. One axiom of a field, because of the division ring bit, is this:

a*b=1 (and 1/=0), where b is called the inverse of a. So, in the case of R, a number multiplied by its inverse equals the multiplicative identity, 1 (s.t. a*1=a) - another rule of the game. These are rules of any field, and if it doesn't satisfy these rules, then it isn't a field. R is a field, and thus R satisfies these rules.

Let's do this, then: -1*b=1, where b is the inverse and 1 is - by the law of the structure - the inverse. Let's do some basic algebra, then,

-1*b=1
-1*b*(1/-1)=1*(1/-1)
b=1*(1/-1)=>b=1/-1
b=-1. We have seen, then, that the inverse of -1 is, shock and horror, -1. Because the field has associative and commutative properties for both operators, you can rearrange any situation to get to this point, i.e., where -1*-1=1. Why? It's trivially derived from the real numbers' status, a field, R.

You're already dabbling in Ring Theory as an Undergraduate? Very interesting...

[cx]

You're already dabbling in Ring Theory as an Undergraduate? Very interesting...

Isn't Algebra required for all undergrad math students?

[Scriptavolant]

When I was talking about a pragmatic explanation I was thinking of something like that.

Jack and John have two apple bags, and each day they match the number of apples in their bags.

They write down something like that on a paper.

day 1: Jack +1 John (Jack has one apple more than John)
day 2: Jack +1 John
day 3: Jack - 1 John (Jack has one apple less than John)
day 4: Jack -1 John
day 5: Jack +1 John

At the end they see that Jack had had one apple more than John, for one more time.
That equals: Jack had had one apple less, for one time less.

So: -1*-1= 1*+1= 1

Does it work?

[PSmith08]

You're already dabbling in Ring Theory as an Undergraduate? Very interesting...

The class started in permutation groups (i.e., S_n) and worked its way through groups proper to rings. The idea of the abstract sequence at my school is to get to rings and fields in the first semester and then to explore rings in depth in the second. It was just an opportunity to make some really tasteless Wagner jokes, like writing a quote from Das Rheingold into a couple proofs.

Isn't Algebra required for all undergrad math students?

My department, and I should note that I'm a minor, finally mandated abstract I for both pure and computational math students. It wasn't always that way.

[Steve]

Isn't Algebra required for all undergrad math students?

Not Abstract Algebra for Minors....

Certainly, Foundations of Algebra

[PSmith08]

Not Abstract Algebra for Minors....

Certainly, Foundations of Algebra

We really don't offer such a class. The basic sequence, for majors and minors, is Math 111 (Calc. I), Math 112 (Calc II), Math 223 (Linear), and Math 331 (Abstract). Majors have to find five more classes (usually in Multivariable, Real Anaysis, and some other classes), and minors get an elective.

[Steve]

We really don't offer such a class. The basic sequence, for majors and minors, is Math 111 (Calc. I), Math 112 (Calc II), Math 223 (Linear), and Math 331 (Abstract). Majors have to find five more classes (usually in Multivariable, Real Anaysis, and some other classes), and minors get an elective.

So you aren't required to take Ordinary Differential Equations or Statistics? That's flexible.

[PSmith08]

So you aren't required to take Ordinary Differential Equations or Statistics? That's flexible.

If you're in the track called "Algebraic Structures," then you probably won't come close to DiffEq. More likely, you'll take Calc. I, Calc. II, Linear Alg., Number Theory, Abstract I, Abstract II, Real I, Real II, and Seminar. If you're on the Analysis track, you'll probably take Calc I/II, Linear, Abstract I, Multivariable, Real I/II, DiffEq, and Seminar. Statistics are off in their own little pod, and depending on what you want out of a math major, you might need them. It is flexible, but only within a context of bigger choices. Once you've picked a track, and there are three I think (computational is the other one, we don't offer a stats program exclusively in Math), you're pretty well set.

[Mozart]

Real I, Real II

Yuck, these ones will gross me out.

[PSmith08]

Yuck, these ones will gross me out.

Nah. You'll be fine: it's exploring why functions on the real numbers work the way that they do. Easier said than done, but it could be worse.

[Steve]

If you're in the track called "Algebraic Structures," then you probably won't come close to DiffEq. More likely, you'll take Calc. I, Calc. II, Linear Alg., Number Theory, Abstract I, Abstract II, Real I, Real II, and Seminar. If you're on the Analysis track, you'll probably take Calc I/II, Linear, Abstract I, Multivariable, Real I/II, DiffEq, and Seminar. Statistics are off in their own little pod, and depending on what you want out of a math major, you might need them. It is flexible, but only within a context of bigger choices. Once you've picked a track, and there are three I think (computational is the other one, we don't offer a stats program exclusively in Math), you're pretty well set.

For Applied Math at Chicago, one would need to have:

Calc I, Calc II, Math Stats I (and/or II), O. D/Eq I (and/or 2), Numerical Analysis, Abstract, and Real, and at least one Advanced Calc, Single or Multivariable. And then pick from several electives... so they are pretty much the same

Are you planning on Graduate school, PSmith? In Classics or Math?

[Mozart]

Nah. You'll be fine: it's exploring why functions on the real numbers work the way that they do. Easier said than done, but it could be worse.

My math teacher told me when she took this class the whole thing was about a piece of string vibrating  That kind of scarred me.

[Steve]

My math teacher told me when she took this class the whole thing was about a piece of string vibrating  That kind of scarred me.

It's not a difficult class. Differential Equations, now there's a challenge...

Any wizzes on Maple around?

[Topaz]

Up to now I've seen a lot of most interesting formal and abstract explanations I will (try to) study soon. But what if you should explain it to your child? I mean why the hell multipling - 2 peaches by - 2 peaches you get (+) 4 peaches. There must be a conceptual way to illustrate the thing in pragmatic terms.

Ah, you didn't specify this at the beginning of this thread.  If you want an explanation a child might understand you might try the following, which I'm afraid is a bit longer but each step is simple:

1.  Think of a positive number (eg +5) as an asset, ie an amount you own.

2.  Think of a negative number (eg -2) as a debt, ie something you owe.

3.  Adding a positive number to another positive number means that your assets have increased, eg +4+ 3 = +7

4.  Adding a negative number (eg -3) to a positive number (eg 7) means reducing your assets by the amount of the debt to give a new value of your assets = 4

5.  This last step at No 4 above is the same as treating the debt (-3) as an asset (ie change the sign to give +3) and then subtracting it from the asset: 7-3 = 4

6.  Subtracting a positive number from a smaller positive number yields a negative result, ie an overall debt, eg +2-(+3) = -1

7.  Subtracting an asset from any debt gives a bigger debt, eg -3 - (+1) = -4.

8.  If you have a debt of −3 and then you get rid of part of it (-1) you still have a debt of -2, ie -3-(-1) = -2

9.  Multiplication of a positive number by a positive number gives a positive result.  This is because it's best to think of multiplication by a positive number as involving repeated addition, ie 2 x 4 = 2+2+2+2 = 8

10.  Likewise, think of multiplication of a positive number by a negative number as repeated subtraction, and thus gives a negative result for the same reason, ie 2 x (-4) = -2-2-2-2 = -8

11.  From the foregoing, multiplication of two negative numbers gives a positive result, eg  (-2)x(-4) = -(-2)-(-2)-(-2)-(-2) = 8

[Scriptavolant]

Ah, you didn't specify this at the beginning of this thread.  If you want an explanation a child might understand you might try the following, which I'm afraid is a bit longer but each step is simple:

I just wanted to collect some "practical" conceptual explanations, to add to the formal ones. Thank you for contributing twice, your first explanation is elegant as well, and I was able to get it (my mathematics is stuck with Russell's "introduction to mathematical philosophy" and little more).

[PSmith08]

For Applied Math at Chicago, one would need to have:

Calc I, Calc II, Math Stats I (and/or II), O. D/Eq I (and/or 2), Numerical Analysis, Abstract, and Real, and at least one Advanced Calc, Single or Multivariable. And then pick from several electives... so they are pretty much the same

Are you planning on Graduate school, PSmith? In Classics or Math?

Right now, I'm thinking about doing the law school thing (so much so that I'm taking the LSAT at the end of September), but I haven't ruled out grad school in Classics, in either archeology or ancient history.