Take charge young man and reckon your soul,
With math it's possible, so why contest?
Why close the door to wonders untold?
Dimensions unseen, or places blessed?

Did you ever wonder about the things you could do,
Simply with numbers? You can do miracles—it's true!
The whole numbers, for example. They never end,
Yet contain an infinitude of primes, while seeming to blend,
Composites together, perfect squares and like kin,
In such perfect harmony within a land of no sin.
And as infinite as they are, still they cannot contend,
With the more numerous reals, which further extend,
To infinities beyond what we normally comprehend.
When Cantor first discovered this, he blushed with fear,
At the import and magnitude of this extraordinary idea.
Oh what wonder, oh what a day!
But how could this be you say?
In fact, pay heed and listen, it's true,
There are more real numbers between 0 and 1,
Then all the counting numbers which forever grow,
And whether we move along the line fast or slow,
No matter. The reals always win and beat them out,
They outnumber and defeat, exercising their clout,
Oh what a wonder, oh what a day!
But how could this be you say?
Even more amazing and I will put this forth,
Is that we can shrink the interval between 0 and 1,
To anything you like. Now wouldn't that be fun?
For example, take 0.1. I assert—and I do this with care,
That there are more reals between 0 and 0.1,
Then all the counting numbers, be patient don't fear.
For I will get to the proof, but give me some time,
As I lay out some facts, and work on my rhyme.
Oh such wonders, what a glorious day!
But how could this be you say?
By extension you see, once the proof is laid bare,
You will realize strange things, very strange things indeed.
The intervals can shrink, forever vanishing,
To virtually nothing, yet listen, take heed.
The reals will still beat, within this small world,
The counting numbers, though they continue forever.
A very strange world, I am not this clever,
To ponder, to believe such, to muse to endeavor,
To enter this realm of abstract thought.
Yet do it I must, for it's a marvelous day!
But how could this be you say?
To begin with take the counting numbers which begin 1,2,3...
They topple like dominoes going on to the next,
We show how we pair each one with a mate,
A number from the interval between 0 and 1,
And we do this in such a way that every one has a date.
It's so simple you see, this is sure to amaze,
How this proof rings so clear, requiring only a gaze,
Of thought pure and true, let there be no delays.
Let us put forth its case, presenting it here,
For its mere truth will assuage any latent fear,
Mind-boggling, astounding, let us bless this day,
But how could this be you say?
This pairing we give a very special name,
A one-to-one correspondence, and this serves to tame,
The pairing of numbers, each real to each natural,
In such a way that the proof comes to light.
Thus we proceed, commencing with 1,
We pair it with 0.1 and we're done,
Onto natural 2, and for this we make haste,
To pair it with 0.11, not to waste,
Any room in the middle for unwanted space,
Thus onward we go, adding 1 to the next place.
In this manner we take 3, and find him quite right,
Cavorting with 0.111 in the night.
We continue in like manner, forever and apace,
In order to finish at last this grand race.
Never realized such dealings, before this great day,
But how could this be you say?
Thus all numbers paired, and enjoying their dates,
When something would happen to alter their fates.
To unsuspecting minds, this would never occur,
But happen it did and left quite a stir.
Suddenly from afar came these numbers in tow,
Reals—whose decimal extensions would not go,
With any of the naturals because their sequence defied,
What was already established, in the correspondence we tried.
For example, no date for 0.22 could be found,
Nor for 0.55 or even 0.1333.
The reason was simple, all the naturals were bound,
To the reals already mentioned, as you can clearly see.
Thus these reals, feeling left out, decided the party to crash,
And bump off a natural, and start quite a clash.
When the naturals espied what was forming in place,
They ran with their dates in hope to save face.
Thus the infinitude of naturals just barely got away,
From the fierce unpaired reals who came to ruin their day.
What a bizarre occurrence, on such a very strange day,
But how could this be you say?
So you see there are more reals embedded in 0 to 1,
Then all the naturals though they continue to run.
The story just told relates this most curious fact,
Of how this could be, now no need to retract.
And what this implies without any reason to doubt,
Is that many infinities exist, and for this we should shout.
Thus new truths come to light in a most interesting way,
A very important lesson on this most fascinating day,
Something to ponder and to another relay,
So that progress in learning continues away.
What a most unusual tale told on this most phenomenal day,
Now at last we can say why these things are this way! QED


Keep those Modifiers from Dangling, George!
If you want to write well, one of the stylistic elements that you need to pay heed are those nasty dangling modifiers, whether in participial, gerund, or phrasal form. Although sometimes insidious to spot, these grammatical faux-pas will tinge your writing and confuse your readers' train of thought. Much like a poorly segued musical transition, dangling modifiers impinge somewhat strangely on the ear and often lend unintended yet grotesque humor to your writing in general.
In the following sentence, see whether you can detect the dangling modifier:

It was early morning. George rose from his bed taking his furry slippers with him and began his march downstairs where breakfast awaited him.

In this complex sentence—as grammarians would call it—complex because the sentence contains the independent clause, "George rose from his bed taking his...downstairs" and the subordinate clause, "where breakfast awaited him," we have a "not-so-obvious" dangling participial phrase. The reason that the "dangler" is not so obvious is because the intent of the writer is obvious from the context; we know who is taking the slippers. Yet the way the sentence is structured, the bed, by its juxtaposition to the participial "taking," is the one doing the taking. One could argue that this is splitting hairs and being overly pedantic, yet this example, by the very insidiousness of the "dangler," demonstrates very well that if we are not careful with these grammatical structure razers in cases such as these, we would become very sloppy in more serious cases, in which the meaning becomes grotesquely distorted.

Though the meaning be clear in the previous example, this does not make the infraction pardonable. A good writer needs to be aware of these "danglers" and has to conform his writing so as to minimize their occurrence. We are all guilty of these infractions and should not stop writing because our writing is not always perfect. Since the dangling modifier is such a common error—even among good writers—we need to be ever aware of its stealthy manner of sneaking into our prose. We should rewrite the above by making any of the following changes:
Rising from his bed, George took his furry slippers with him and began his march downstairs where breakfast awaited him.
Or,
Taking his furry slippers with him, George rose from his bed and began his march downstairs where breakfast awaited him.
If you want to write well, beware the "dangler." And go get a good book on grammar and learn about the language you wish to express yourself in. This will be well worth your while.
More on this in another article. Stay tuned...


Teach Your Kids Arithmetic - The Quick-Add - Part II
In continuation of Part I, we now plunge more deeply into the Quick-Add Method and show how this makes doing addition quite easy. This procedure hinges on two key ideas: 1) the method of complements; and 2) the Quick-Add Conversion. To refresh your memory (also see "Teach Your Kids Arithmetic - The Quick-Add - Part I), complements of a number are those numbers, which when added to the given number, yield a sum of 10, or some multiple of 10. For example, the 10-complement of 8 is 2, since 8 + 2 = 10. The 10-complement of 4 is 6, since 4 + 6 = 10. The Quick-Add conversion is simply the way in which we convert our given addition problem into a "quick-add;" for once done, the problem becomes—well, what the method says: a quick-add. That is, the addition can be done quickly and easily. As mentioned previously, the Quick-Add works as follows: in analyzing 10 + 7, we rewrite this example as 10 + 07. We insert a 0 in front of the 7 as a placeholder for the empty "tens column," and to bring the numbers into parallel structure. The brain performs 1 + 0 in the "tens column" and 0 + 7 in the "ones column," thus capitalizing on the "Additive Identity Property" of 0.

Whenever we are confronted by an addition problem, we are going to convert it to a "quick-add." For example, take the addition of 7 + 5. This is 12, but some children might not get this straight away. Ask them what 10 + 2 is, however, and the answer is for the most part immediate. Nobody struggles with the latter addition problem because it is in "quick-add format." Now to get the problem into this format, we simply do the "Quick-Add Conversion," and this is when the idea of complements comes in to play. We always work with the bigger number, which in this example is 7. We take the 10-complement of 7, which is 3. We reduce the smaller number, 5, by 3 to become 2. Now we have the converted example: 7 goes to 10, and using its complement 3 to reduce 5, 5 goes to 2. We now have the "quick-add" 10 + 2 = 12. Let's look at another example: 8 + 9. In this case, the 10-complement of 9 is 1; thus 8 is reduced by 1 to 7, and we have the "quick-add" 10 + 7 = 17. A snap! If both numbers are the same, no problem. Look at 6 + 6. The 10-complement of 6 is 4, thus the other 6 gets reduced by 4 to 2. We now have the "quick-add" 10 + 2, which is 12.

This method can be extended to larger and larger numbers, using the idea of 100-complements, 1000-complements, and so on. For now, I will examine just another two examples, using additions with numbers bigger than 10. Take 18 + 8. We break down 18 into 10 + 8, and observe that the 10-complement of 8 is 2; 18 then becomes rounded to 20, the next 10 up from 18, and 8 becomes reduced by the 2 to 6. Thus we have 20 + 6 = 26. For the example of 19 + 17, we have 19 is 10 + 9 and 17 is 10 + 7. The 10-complement of 9 is 1, so 19 goes to 20, and 17 is reduced 1 to 16. So the converted example is 20 + 16, which can be further broken down to 20 + (10 + 6) = 20 + 10 + 6 = 30 + 6 = 36. In the last example, I was using some forgotten rules of arithmetic, such as the Associative Property of Addition, and breaking down the example quite extensively; however, I think the point is made and the procedure is now established.

Try looking at addition problems from this perspective by using the idea of complements and "Quick-Add" conversions. I don't think you or your kids will be having trouble with addition anymore. Stay tuned for more arithmetic magic in my future series of articles on this most important topic.