Renamed “A bridge too far”.
And they call them “real” numbers!
The Cantor Set is constructed in the following way:
Start with the interval [0,1]. Next, remove the open middle third interval, which gives you two line segments [0,1/3] and [2/3,1]. Again, remove the middle third for each remaining interval, which leaves you now with 4 intervals. Repeat this final step ad infinitum.
The points in [0,1] that do not eventually get removed in the procedure form the Cantor set.
How many points are there in the Cantor Set?
Consider the diagram below:
An interval from each step has been coloured in red, and each red interval (apart from the top one) lies underneath another red interval. This nested sequence shrinks down to a point, which is contained in every one of the red intervals, and hence is a member of the Cantor set. In fact, each point in the Cantor set corresponds to a unique infinite sequence of nested intervals.
View original post 222 more words
Trump, Watergate, Nixon, Rockefeller: the real lesson —Two very different men, two very different presidents, Trump and Nixon; but the real reasons for attacking them are the same— by J…
(Copied from elsewhere; author unknown but reprinted with thanks to whoever it was. Also thanks to my friend Kathryn Wadsworth who put this out on FB!) A DAY IN THE LIFE OF JOE REPUBLICAN AMERICAN …
Now for more exiting and less mathematical stuff.
I don’t know if this is “the way forward” or “just bullshit”, but McKinsey knows (something).
So on with “Digital” in education:
This is the introduction –
Companies today are rushing headlong to become more digital. But what does digital really mean?
For some executives, it’s about technology. For others, digital is a new way of engaging with customers. And for others still, it represents an entirely new way of doing business. None of these definitions is necessarily incorrect. But such diverse perspectives often trip up leadership teams because they reflect a lack of alignment and common vision about where the business needs to go. This often results in piecemeal initiatives or misguided efforts that lead to missed opportunities, sluggish performance, or false starts.
And the rest of it is here
Have fun. The words are easy.
The row operation matrix inversion method is so neat and ingenious, and it has the same operations for all dimensions of matrix.
Here is a step by step approach, where firstly the dimension is chosen, then the first of the buttons is selected (Start). After which the buttons are selected in order. The states of the left and right matrices are displayed at each stage, and finally the identity matrix appears on the left, and the inverse of the original matrix appears on the right.
The first display shows the original matrix on the left. Nothing has been done yet.
The second one is the 2 x 2 matrix inverted.
The third is the 4 x 4 matrix inverted.
This is the standard way of inverting a matrix. I came across this first in 1960. Oh so long ago! Goodbye Cramer’s Rule.
My dad figured this out years ago.
The string method work for all matrices, and it is at least ten times quicker to “do” than to “write about”.
Optional, yet optimal…..
Dear Assistant Superintendent MacGuffin:
Thank you for your interest in partnering with aggravatED, the education consultant-management education consulting firm, and for so thoughtfully completing our preliminary readiness screening tool, the Partner, Initiative, & Stakeholder Survey (PISS™).
Our consultant-management consultants looked with great interest at your PISS™, and they concluded definitively that your district would benefit from our innovative, strategic, and accountability-focused guidance.
As you may remember from our informational materials, our initial PISS™ analyses categorize participating schools/districts into four categories describing overall focus and consistency: Laser, Inconsistent, Diffuse, and Without Tangible Focus (WTF).
According to this scale, our initial analysis — which included a study of historic district performance and engagement data — identified your district’s PISS™ as just at the division between ‘Diffuse’ and ‘WTF’. Below are a few noteworthy — and seemingly competing — initiatives/partners that helped us reach these conclusions:
- Partnership with disruptED, ed-tech firm facilitating district’s…
View original post 240 more words
I was pondering the reality of negative numbers and after figuring out that a sequence of dots on a line can be extended in each of the two directions, and then arbitrarily selecting one dot as “the zero”. The line can be further labelled as 1, 2, 3, … to one side and -1, -2, -3, … on the other side.
(better to label the 1, 2, 3, … as +1, +2, +3, … and consider the lot as “signed numbers”)
Soon proceeding towards arithmetic I concluded that 7-3 is 4, and also 8-4 is 4, and therefore 13-9 is 4, and then 3-7 is -4, and -2-2 is -4. It was then observed that if a-b=c then a-y-(b-y) is also equal to c, regardless of the signs of the specific numbers involved.
This of course is stunningly obvious when looking at the signed difference of the first and the second number as an extended number line diagram.
The outcome of all this was an arithmetic for 0, 1, 2 modulo 3, and the signed difference x-y is a binary operation diff(x,y) with table:
…x … 0 1 2
0 0 1 2
1 2 0 1
2 1 2 0
Example: 1-2 is -1, which is 2 modulo 3
So a non abelian, non associative algebra with a not quite identity satisfies the conditions, where A=1, B=2 and C=0
There are three objects and an operation called “doesn’t have a name”.
Two are similar, and the third is a bit different
They are paired to yield a single object as follows:
AA = BB = CC = C
AB = BC = CA = B
AC = CB = BA = A
Notice that BC and CB are different, so non-abelian.
Worse is that (AC)A = C and A(CA) = B are different, so non-associative.
And consequently A and B and C are different.
Interestingly, and maybe separately, the minus sign behaves very differently from the plus sign:
a-(-b) is a+b, but there is no way of writing a-b using only addition.
This means that all expressions can be written with “minus” alone.