- Video I
- A negative times a negative is a ... ?
Mathologer | 24.VII.2015
- (... - 3)(... -4) = ... - ... - ... + 12.
But is this because -3 and -4 are numbers existing in their own right, with + 12 as their product per se, or is it because subracting from diverse dimensions (think of geometry for instance, squares, cubes) tends to make subtractions cancel out?
Your own "real life example" 4 "losses" of 3 € debts, would it be a brut and net gain of 12 in someone who owed nothing in the first place, or would it just cancel out debts insofar as they existed, like:
I owe 15 €, get rid of 12 € debt, and now owe 3€. We agree on this one.
I owe 2€, get rid of 12 € debt, and now owe nothing - but does whatever made me lose 12€ in debt automatically give me any positive 10 € if I had only two euros in debt?
If I have a prison sentence of 42 months and get rid of owing 14 of them (realistic example), if I am freed after the 28 months, do I get to keep someone in prison myself for the days or weeks I did too much? Or do I automatically get a monetary compensation for the days or weeks I spent too much in there? Or would I have to sue extra for possibly getting that too? Could I loose such a case?
- 2:14 Lots of ancient civilisations did very well without having recognised "zero" as a number.
Fibonacci told us how to use "zero" on the papers we put an algorithm on, old sense, like this is an algorithm*:
1 2 - 5 = 7
But he did not qualify "zero" as a number. He considered it a useful placeholder, to distinguish 10 and 100 from 1 or 105 and 1005 from 15.
- Better example with this placeholder:
1 0 5 - 7 6 = 2 9
- 6:29 Very nice proof in algebra that a negative times a negative is a positive.
However, algebra has everything to do with the "just do it" approach.
This does still not prove that "a negative times a negative" even exists in real arithmetic. You know, the kind where we get 2+2=4 from.
II+II = IIII (no zeros and no negative numbers in Roman numerals).
- 10:36 Now you have shown why algebraically a negative times a negative should make a positive.
You have STILL not shown there are per se negative numbers.
(50-2)2 = 2304 or even just (10-2)2 = 64, we certainly can use a formula which goes a2 - 2ab + b2.
But if you really lay out 100 pebbles, this would mean, if you take that as it stands, that you are first taking away 20 pebbles, then putting back four pebbles in one end of the taken away strip and then taking away another strip of 20 the other way.
What you would normally really do is, 100 - 20 - 16.
a2 - ab - (a-b)b.
Now, a2 - 2ab + b2 is not landing you with a wrong result, but that is only bc it is algebraically equivalent to a2 - ab - (a-b)b.
- 13:25 "it works" doesn't prove negative numbers actually exist as numbers in their own right ...
Now the reason I keep coming back to this, is, while (-a)2 = a2 in algebra, I'd not consider it a good theorem in arithmetic.
I definitely would consider the following as good theorems:
(x-a)2 = x2 - 2ax + a2 (as reformulation of x2 - ax - a(a-x) obviously)
(x-a)(x-b) = x2 - x(a+b) + ab
(x-a)(y-a) = xy - a(x+y) + a2
(x-a)(y-b) = xy + ab - ay - bx
But in each of these cases, I would argue, -a and when occurring -b also actually refer not to any "numbers lower than zero" but to subtraction.
Why am I saying this?
Bc, numbers begin from 1. This is philosophically important. You can potentially extend numbers forward as much as you like, but you can't extend them backwards even beyond 1. And this has implications for other fields where some kind of values are added one at a time.
YOU didn't come to stand here after an infinity of past causes leading to it.
YOU are where you are bc of a finite number of causes, including but not limited to own past choices - and so am I.
A finite number of causes implies a first cause - which according to St. Thomas Aquinas everyone calls "God".
Apparently this cultural generalisation is not valid for all types of modern men, some would consider "energy" the first cause or some would consider spacetime to be it, but let's not abuse mathematics to pretend there is no first!
- Video II
- Root 2 and the deadly Marching Squares
Mathologer | 15.V.2015
- 7 / 5 = 1.4
17 / 12 = 1.4166666666666667
41 / 29 = 1.4137931034482759
99 / 70 = 1.4142857142857143
239 / 169 = 1.4142011834319527
577 / 408 = 1.4142156862745098
1393 / 985 = 1.4142131979695431
3363 / 2378 = 1.4142136248948696
x2 = 2, x = 1.414213562373095?
Nah, that is also an approximation, just a better than the previous ones.
In other words, when we speak of "square root of two" as such, we speak of an irrational which is not a number.
When we write out a fraction to represent sqrt(2) we use an approximation, which is rational.
Either way, nothing irrational has been showed to be a number, and no number has been shown to be irrational.
Why am I into this?
Well, some people have argued Medievals did not understand numbers as well as we do, bc they thought all numbers were rational.
But they actually are. If you want a real exact sqrt of two, you take one square, and its diagonal is the side of a square twice as extensive, exactly. So, sqrt of two is diagonal : side (of a perfect square).
But diagonal : side is not one number, it's not even the ratio between two numbers, it's the ratio between two lengths.
And no, just bc length is often "counted" in numbers times a unit length doesn't mean length is an actual number.