I've always had a doubt about why a negative multiplied by negative results in a positive. 

I could understand why two positives multiplied together result in a positive - you're duplicating an amount by a multiple so you have multiple or increase of that amount.  If you had lots and you multiply it by a factor, you are more 'lots'. 

The same logic applies to multiplying negative by a positive - if you had less than nothing or a debt and you multiply that debt you have more of nothing or more of that debt. 

So that explains why \(+ \times + = +\) or why \(- \times + = -\) but how do people accept that a negative multiplied by a negative is a positive?

I finally found how and why, and it's a testament to algebra's ability to reduce complexity that makes it easier to understand and accept. For example, using letters:

if \(a \times b = ab \tag{1} \) and \(-a \times b = -ab \tag{2} \) then \(-a \times -b\) must be negative or positive ab, i.e \( \pm ab\) - but which one? 

Well, intuitively its not easy to know which one (see below), but it can only be 1 or two possible signs, i.e \(-\) or \(+\) right? And if \(-ab\) is already made provision by \(-a \times b\), then it cannot be \(-\), so.... it must be \(+\)!

Simple!

This is purely by the elimination of choices for the signs that we have left, which is only \(+\). So purely because it cannot be anything other than \(+\) or \(-\) and \(-\) is already been made provision for, it has to be \(+\).

So we have no choice but to define \(-a \times -b\) as equalling +ab. 

Now, what I find interesting about this is that without algebra to simplify the numbers in terms of letters, and being able to make generalised rules, by those rules, you can determine something that's not intuitive otherwise. 

For example, if I intuitively try to reason what I've just determined through elimination, its not intuitive at all:

I did think about multiplying by a negative as successive subtractions instead of successive addition:

\(a \times -b\) as a reducing by itself by b times, so \(8 \times -2\) is:

8 reduce itself(8) by 8 is 0

0 reduce the second time by 8 is  -8 

this is wrong, so instead if instead of reducing 8 (itself) by itself (8), you reduce 0 by 8:

0 reduced by 8 is -8

-8 reduced by 8 is -16

This is correct, in the same way, that 8 times of -2 is 0 reduced by -2, 8 times. Essentially you start with 0 and perform 8 subtractions of 2. 

So incidentally, this shows that all expressions always are 0 to begin with, and then the numbers and operations apply to 0.

So \(-a \times -b\) is reducing -a by -b times. This doesn't intuitively make clear what would happen - its not intuitive. 

So \(-8 \times -2\) is

0 reduced by -8 once is 8 (this is not intuitive) 

8 reduced by -8 again is 16 (this is not intuitive) 

This seems to suggest that reducing by a negative (or a multiple of a negative, i.e. twice -8) is making a 0 less negative and doing so successively will eventually make it positive. In the same way, reducing 0 by a positive is making it 0 more negative and doing so successfully will eventually make it negative.

All of this is not intuitive by default, and I've tried to reason about it before (See Rules then Since then) however, the elimination of possibilities using Albegra at the beginning of this text makes it much more intuitive and acceptable.

Even if it doesn't explain why, it shows what it must be as there was no other choice. That's easier to accept than, "...a negative times a negative is a positive."

This is one pretty useful characteristics of Algebra - that it can reduce complexity so you can reason within the limits of rules that algebra puts in place.

I must admit, most of the time I read formal definitions of Mathematics and I wonder how anyone can actually learn from them. For example, when learning about p-values, this definition is used to explain what they are:

P-value is the probability of obtaining test results at least as extreme as the result actually observed.

Generally, the goal of Mathematics is to be general, abstract and all-encompassing, but this can lead to definitions that are verbose or unintuitive exactly because they try to describe the application of ideas to every possibility (generalization) into a sentence or words that represent that generalization or abstraction. This is not useful I feel for teaching or learning.

Apart from the age-old problem of not wanting to learn Mathematical concepts because learning it appears not immediately useful, which often is the case, eg. if you've got no interest in right-angled triangles and you are told you need to learn about Trigonometry, then you're less likely to learn it well.

Coupled with this is that when the Math concepts are taught, they are unintuitively explained, precisely because they start with the abstract/general concepts or the end-product of mathematics (like the definition above) instead of aiming to reach that abstraction point from a place of ignorance, which I'd argue most people who want to learn would agree they should start from. This harms Mathematics I feel.

It's almost like telling someone how to be a world-cup capable rugby player (or expecting them to be one) by giving them the trophy that won the rugby world cup. That's just silly.

So what about the p-value? Well, that is what I spent most of today trying to understand.

Ultimately my findings are that it is the likelihood that a claim (or conjecture or hypothesis) you make about something is actually wrong.

Weird you might think, why would you want to be so formal about that? The reason is that you want to provide a degree of confidence in your claim (the likelihood of being wrong being low) and you want to show that this likelihood is based on evidence of real underlying something, not just a hearsay claim about the something.

This likelihood of being wrong is based on testing your claim against the source of observations, instead of just merely making the claim and having no testing of the observations to back your claim up. This idea now seems easier to grasp but can also appear a weird thing/concept to want to understand.

You want the likelihood of your claim being wrong to be low, so a low p-value is good. So how do you calculate this likelihood of being wrong?

You need to construct a test that checks that each actual observation matches your claim, and run all the observations through this test, and if all the test results show that your claim is correct for each observation, then the likelihood of your claim being false overall is 0%, i.e the p-value is 0. 

This might seem strange but its basically a way to substantiate your claim by showing that you based it on testing your claim on the actual data.

Here is an example:

If I have collected 5 apples and from those apples, I've somehow determined and claimed that they are [all ok to eat] then that's fine, but if you can say that they are all ok to eat because I tested every single one and that's how I came up with that claim, then the claim is now based on testing the observations as a means to show evidence that my claim is unlikely to be wrong, then this is better than just making the claim based on nothing. 

I can derive a p-value that indicates this likelihood that my claim is wrong, based on actual testing:

My actual testing on the observed apples: 

• Apple 1 has no discolouration and smells ok, so its [OK to eat], and it actually is [OK to eat]
• Apple 2 has no discolouration and smells ok, so its [OK to eat], and it actually is [OK to eat]
• Apple 3 has no discolouration and smells ok, so its [OK to eat], and it actually is [OK to eat]
• Apple 4 has no discolouration and smells ok, so its [OK to eat], and it actually is [OK to eat]
• Apple 5 has no discolouration and smells ok, so its [OK to eat], and it actually is [OK to eat]

That's 5/5 of the tests that show all the apples are OK to eat, meaning that it's 0% likely that my claim that all the apples are [OK to eat] is wrong.  My p-value is 0.  Furthermore, because my test which is [if the apple has no discolouration and smells ok then it means it is OK to eat] was actually correct about the apple being OK to eat, it's also a good test because it matches the actual reality of being OK to eat.

However, if my testing results were different:

• Apple 1 has no discolouration and smells ok, so its [OK to eat], and it actually is [OK to eat]
• Apple 2 has no discolouration and smells ok, so its [OK to eat], and it actually is [OK to eat]
• Apple 3 has no discolouration and smells ok, so its [OK to eat], and it NOT is [OK to eat]
• Apple 4 has no discolouration and smells ok, so its [OK to eat], and it actually is [OK to eat]
• Apple 5 has no discolouration and smells ok, so its [OK to eat], and it NOT is [OK to eat]

That's 3/5 of the tests that show my claim to be correct while 2/5 show my claim to be wrong, meaning it's a 40% chance that my overall claim is wrong, based on testing. My p-value is 0.4 (40%) and that test of mine doesn't seem to be that good. Either way, my claim is now being based on testing the data, not just a claim based on no evidence.

So we now have a level of confidence that our claim we made that all apples are [OK to eat] is correct. That confidence level is p = 0.4 in the last example, meaning we aren't 100% confident based on the testing of that data that claim is universal across all the apples - because it wasn't in our testing.

So ultimately we have a claim (all apples are OK to eat), and we have a level of confidence in that claim (p-value), derived from testing the apples/data. This is really it.

That p-value can also be seen as the likelihood of our tests producing false positives when testing our claim to be true. Above there were 2 false positives (Apple 3 and Apple 5). The p-value can also be seen as the rate of failure e.g 2/5 that the claim was correct against all the data tested.

I've just finished watching Midnight in Paris for, what seems, like the umpteenth time. I'm always left feeling pleased and remarkably content after watching it because the main character represents an unassuming authenticity which I wish was more common.

The main character is honest, unpretentious and, in many ways, a naive main character who has no underlying agenda or malice towards anyone - a peaceful, and seemingly likeable existence. Yet, he is now gradually realizing that the price he is paying for this, is a mediocre life.

Through the film, he slowly realized that he has deprived himself of his underlying ambitions in order to take the path of least resistance in life. He has a pretentious fiance who chides and cheats on him and is ultimately at odds with him.

He mysteriously time travels into the past - seemingly induced by a quiet desperation or longing to be in the past where he can fulfil his true ambitions of being a writer without the reality of his present.

However, in finding solace by escaping to the past, he soon realises that the past can't solve his problems and that he is merely kicking the can further down the road, so to speak, and that to fix his problems he needs to take charge of his circumstance in the present which happens to mean taking a stand to fulfil his ambitions, and leaving his fiance.

It's quite an entertaining watch as you get to meet Hemmingway, Picaso, Dali, Gertrude Stein, Scott and Zelda Fitzgerald and other influential figures of the past.

Its a Woody Allen film and stars Owen Wilson and Rachel McAdams.

Will watch it again.