Well, WitCH 2, WitCH 3 and Tweel’s Mathematical Puzzle are still there to ponder. A reminder, it’s up to you, Dear Readers, to identify the crap. There’s so much crap, however, and so little time. So, it’s onwards and downwards we go.
Our new WitCH, courtesy of New Century Mathematics, Year 10 (2014), is inspired by the Evil Mathologer‘s latest video. The video and the accompanying articles took the Evil Mathologer (and his evil sidekick) hundreds of hours to complete. By comparison, one can ponder how many minutes were spent on the following diagram:
OK, Dear Readers, time to get to work. Grab yourself a coffee and see if you can itemise all that is wrong with the above.
Well done, craphunters. Here’s a summary, with a couple craps not raised in the comments below:
- In the ratio a/b, the nature of a and b is left unspecified.
- The disconnected bubbles within the diagram misleadingly suggest the existence of other, unspecified real numbers.
- The rational bubbles overlap, since any integer can also be represented as a terminating decimal and as a recurring decimal. For example, 1 = 1.0 = 0.999… (See here and here and here for semi-standard definitions.) Similarly, any terminating decimal can also be represented as a recurring decimal.
- A percentage need not be terminating, or even rational. For example, π% is a perfectly fine percentage.
- Whatever “surd” means, the listed examples suggest way too restrictive a definition. Even if surd is intended to refer to “all rooty things”, this will not include all algebraic numbers, which is what is required here.
- The expression “have no pattern and are non-recurring” is largely meaningless. To the extent it is meaningful it should be attached to all irrational numbers, not just transcendentals.
- The decimal examples of transcendentals are meaningless.
15 Replies to “WitCH 4”
Ok, just so there’s enough crap to go around for everyone, I’ll point out just the one specimen: The “no pattern” crap stated in the definition of a transcendental number.
A simple connter-example to the given definition: Liouville’s constant (historically, one of the first decimal examples of a transcendental number given) has a very definite pattern ….
Thanks, John. Of course the “no pattern” is merely the tip of a massive crap iceberg. But nice of you to leave some crap for others.
The number sometimes credited to Euler also has quite a distinct patter, as does Phi if you are allowed to define pattern in a way that doesn’t refer to decimal strings…
But before even seeing the crap you can smell it with this one: why does a student of the level presumed here need to know the phrase “transcendental”???
Thanks, Number 8. Two very good points. (For others looking, there is *plenty* of less subtle crap to identify: don’t let your eyes glaze over, and look at what is actually suggested by or explicitly written in the diagram.) Your first point is excellent, that one is taught that irrationals are “no pattern” things, but this is pretty much just because decimals suck as a way to understand irrationals (or anything). And yes, what indeed is the point of introducing here the notion of “transcendental” whatsoever? Sure, if one explored what it actually meant, which could be done in Year 10, then yes, that’s great. But without decent exploration/explanation, it is incomprehensible.
Bad diagrams like this one seem to be common in maths teaching. Google images ‘real numbers venn diagram’ and you’ll see many as bad or worse than this.
To add some crap:
Confusing placement of bubbles, e.g. it seems that integers, recurring decimals and terminal decimals make up only some of the rationals.
Their definition of transcendental would imply sqrt(2) is transcendental.
Not obvious what is meant by ‘surds’. It seems here to mean a*sqrt(b). What about other ‘rooty’ expressions involving cube roots etc? A student could reason they must be transcendental.
Thanks very much, Dan. Poorly thought out Venn diagrams are indeed common, though I’d say the diagram above is gold medal material. As to the specific crap you’ve noted, you’re spot on: yes, the disconnected bubbles inside the larger bubbles suggest that there are other rationals and irrationals around; yes, whatever “Have no pattern and are non-recurring” is supposed to mean, it presumably applies to all irrationals, not just transcendentals; and yes, whatever “surd” is supposed to mean, the accompanying list suggests it refers only to irrational square roots of rationals, with the zillions of remaining algebraic numbers left without a home.
For the other hunters, there’s still (at least) two major pieces of crap to note …
The line a/b where a, b are not defined is always problematic, but even when some books do try by defining a, b as *whole numbers* this seems to raise a whole new level of WITCH (to borrow an acronym) which I seem to recall was discussed in an earlier post (or in a newspaper column).
A valid criticism, though nitpicky.
Thinking over the diagram for about as long as it takes to make a coffee I found the use of symbol … To be inconsistent . I recall seeing lines above repeating sequences of numbers in my o level texts of the last Millenia
Also I am uncertain to the transcendental nature of some of the numbers listed in bottom right hand corner
e^( i * pi) = -1 even though both e and Pi are transcendental and i complex (Euler)
Thanks, Steve. Definitely the use of … in the bottom right of the diagram is crap.
If … is used it must be clear what the dots are replacing. So, for 0.999… and 3.1415…, for example, there is no real question what the dots are representing (even if some consider an overbar more standard for repeating/recurring decimals.) For 1.257308… and 0.0097542…, however, it is impossible to know what the dots mean.
With no definition/restriction for a and b, you can just write anything as x/1. And the sqrt(11)/3 is just so misleading with no restrictions given, I mean it’s literally there in the form of a/b.
If I had no idea what irrational meant, I’d be equally clueless after looking at that page, if no other explanation was given… And could say the same for “Surds” and transcedental numbers.
Thanks, Spiros. Yep, the unqualified a/b is definitely crap. Of course one wants to avoid making such diagrams overly noisy or pedantic, and we can all guess the intended meaning, but obviously *some* explicit definition is appropriate for the central definition.
A percentage doesn’t necessarily have to be a positive integer divided by 100. For example, you could cut a circular hole out of a square and find the percentage of area removed from the square. The exact answer wouldn’t be a rational number. So their example using 16% = 0.16 as a rational number suggests percentages are always rational.
Ah, I missed that one. Very good point and very nice example.
Well, posters have found no shortage of crap, including some I had overlooked. Nice work. In the next day or so I’ll update the post with a summary of all the crap (including one piece as yet unmentioned).