# Innumeracy

Oh FFS.  This simple question has apparently caused all sorts of mayhem among the innumerate:

The answer is of course “FALSE” — and to think otherwise is to be ignorant of two of the simplest definitions in mathematics, i.e.

• “A right angle is defined as two straight lines meeting at a 90-degree angle”, and
• “There are no straight lines in the circumference of a circle.”

And in the above picture, there’s only one straight line.

That anyone can even be fooled by the question means that math education has been completely screwed up.  I agree that it’s quite a tough question for a seven-year-old child (as posed in the article), but nobody with more than a seventh-grade education should be stumped by it, let alone a professor of mathematics.

By the way, ignore the red herring that a straight line consists of two right angles:  that’s only a partial definition of straight line.  (“The shortest linear distance between two points” contains only implied angles, not actual ones.)

And by the way:  the correct spelling is “two right angles”, no hyphen necessary.

I need another gin.

Update:  Oh FFS-squared.

For the above diagram to contain two right angles, one would have to add a third radius, thus:

Now the question “There are two right angles” has the answer “True”  (A0C, B0C).  If you were to answer “False”, giving “because there are four right angles” as your reasoning, you would (rightly) be given an “Incorrect” because there are only four right angles in the imaginary world (i.e. Thales’ Theorem et al.).  However, we are not in an imaginary world because we are not talking concepts, we are talking about an actual diagram.  And to cap it all, we are talking about a question posed to a seven-year-old child, for whom Thales has no existence.

As I explained to a Reader in an email on this very topic, it always pays to remember that mathematics has little basis in reality, e.g. where a line can have direction but no thickness and a point has a position but no size.  And I’m not even going to touch on division by zero… [eyecross]

1. The Ferguson Rifleman says:

Yeah, I have run across these too. What they boil down to is “poorly worded” math expressions that can be interpreted different ways. It’s like saying, “I saw the man with binoculars.” Did you see the man with your binoculars? Or did you see him carrying his own?

In this case, allow me to be the devil’s advocate: as geometric shapes get more and more sides (triangle, square, pentagon, hexagon, octagon, decagon, etc etc) – they begin to resemble a circle as the number of sides increases. In theory (so I’ve heard), all a circle is, is a polygon with an infinite number of sides. If that is true, then your diagram has two right angles.

I now take my leave amidst a hail of rotten vegetables, boos, hisses and death threats!💩👍

Have a great day, fellas.

1. Kim du Toit says:

“In theory (so I’ve heard), all a circle is, is a polygon with an infinite number of sides. If that is true, then your diagram has two right angles.”

And not even then. When a polygon becomes a circle, it ceases to have ANY straight lines.

1. bnations says:

Kim is right, but it’s not really intuitive as to why he is. To understand why, you have to consider the definitions of a tangent and a secant and their implications.

A tangent can only intersect a circle at a single point by definition. Were there two points on the circumference that could lie on the same line, then tangents wouldn’t be possible.

There is such a thing as a line that intersects a circle at two points — called a secant — but it by definition lies within the circle. Hence, there are no straight lines on a circle.

Math is fun; especially, when there are no numbers. 🙂

2. The Ferguson Rifleman says:

Pardon me, Kim – I misspoke – according to the tall foreheads I read… a circle is a polygon with an infinite number of “straight” sides. It kind holds merit. In a mathematical sense, in theory you can never have two points so close together that you can’t draw a straight line between them. Ergo, where the straight lines join the curved lines, on an infinitely small scale… you will have two right angles.

If you want to argue and say that there are no straight sides in a circle I am fine with it, but I don’t know if the tall foreheads would be. It would require a mathematical proof, and I left higher mathematics in the rear view 25 years ago.

1. ghostsniper says:

In AutoCAD those points are called “nodes” and while at normal viewing the circle does appear “round”. However, under magnification (zoom in real close) you can see the individual nodes. In the original AutoCAD set up the user specifies the number of nodes desired, up to 256 (I think) per any given circle or arc. Because I like my arcs and circles to appear properly curved I specified the highest number of nodes though it is a little memory resource demanding.

If, using “AutoSnap” to intersect a polyline into any given radius the polyline will indeed seek out the most appropriate node, implying that indeed the spaces between nodes are straight and thus a right angle is achieved. Keep in mind that all of what I wrote is only within the AutoCAD program and no reflection on real reality. FWIW, in my 28 year history with AutoCAD I have found several instances where the program developers made decisions that were not exactly correct, however, for all my uses it as always been very accurate. You know what they say, trash in, trash out. AutoCAD is like the perfect employee and will do EXACTLY what you tell it to do, so the user must be very careful what it tells it.

2. OldTexan says:

Is that the reason why you tend to fall off of the world when you reach the edge? That is a dumb shit question about two right angles when a straight line bisects a circle and unfortunately it is a normal ‘new math’ question that in some class rooms does not have correct answer and a good way to screw up a grade school kid’s mind. My wife works for my daughter with her math tutoring business and it is a challenge teaching math, especially to children who have no idea about fractions, percentages or making change since they seldom see people actually spending cash.

A close friend of mine is a retired grade school principal and I asked him years ago why math had become so complicated for students. His explanation was that for years the people who develop text books work very hard to make a four year old text book obsolete so they come up with new shit that will require the school systems to replace all of the old books with ‘up to date’ educational materials. Same old crap from kindergarten through college. Lots of money from publishing houses to wine and dine those who make decisions on books.

1. Don Curton says:

“His explanation was that for years the people who develop text books work very hard to make a four year old text book obsolete so they come up with new shit that will require the school systems to replace all of the old books with ‘up to date’ educational materials.”

I think there’s a lot of truth in that statement. When I was in school back in the 70’s, every text book had a lined column on the inside of the front cover for kids to put their name and room #. Every book I had would have at least 8 or 10 names before mine. Fifteen years ago when my kids were in middle school, it was common for them to not have books for half the fall as there was always a book shortage!! And I lived in one of the better school districts in the state. Then they started handing out laptop computers and/or tablets with the books saved in digital form, but as fast as they put porn filters on them the kids found a way around them. I think the last several years of high school for my kids (roughly 7-8 years ago) everything was “online” and the kids simply logged in from home. No homework during the bus ride anymore, you had to be at a computer with internet.

I don’t know how the print industry makes money anymore, but the whole textbook thing is a total scam.

2. JQ says:

Old Texan,
Your friend is absolutely right. Many college professors get involved with writing and editing textbooks and make more money doing that than they do teaching. They need to have the teaching position and credentials to edit the book so they keep their professorship. Some athletes are the same way. They make more from endorsement deals than they do playing sports but I digress.

A professor I knew in college made more money from his chemistry text book than teaching. The publishers keep updating the edition in order to prevent people from using perfectly good textbooks that are only a few years old.

Another assault on mathematics is the demand to include social justice in them. If you look at the indices of math text books 20 years ago, theyd be filled with math terms and types of problems to do. Now, there is far more stuff in them like racism etc. In a mathbook that’s just no appropriate

JQ

3. libertyman says:

I wonder when the professor of “maths” as they say thinks the circle becomes a straight line? Because you draw a line through it?
Of course, the shortest distance between two points isn’t always a straight line, if one is traveling across a sphere, like planet earth.

1. Kim du Toit says:

Hence the word “linear” in the definition.

2. Mark D says:

The shortest distance between two points on the surface of a sphere is still a straight line, THROUGH the sphere. The shortest distance between two points on the surface of a sphere, while staying on that surface, is Great Circle distance, i.e. the arc described by extending the two points of interest to the center of the sphere and connecting them.

4. Fred Z says:

The curve comes asymptotically close to a straight line, which is the argument used to support the “two right angles”.

But asymptotically close still leaves a measurable distance between the curve and the perpendicular line.

That distance is the small but ever declining distance known to all carpenters as the mythical “cunt hair”. That’s a phrase I and my lead hand, a carpenter by trade, have used for the 15 years he has worked for me and we’re not giving it up no matter how much the feminazis screech.

1. GT3Ted says:

In the Mechanics trade , the measurement is an RCH — somewhat smaller and finer.

1. ralphgizzip says:

And Machinists refer to the smallest possible measurement as a VFRCH.

5. Mark D says:

“In this case, allow me to be the devil’s advocate: as geometric shapes get more and more sides (triangle, square, pentagon, hexagon, octagon, decagon, etc etc) – they begin to resemble a circle as the number of sides increases. In theory (so I’ve heard), all a circle is, is a polygon with an infinite number of sides. If that is true, then your diagram has two right angles.”

Actually, no. We COULD consider a circle to be an equilateral polygon with an infinite number of sides. HOWEVER, the only equilateral polygon where the sides form right angles is the square. Think if it this way: a circle is all points at a given distance (radius) from one central point. The point where the arc and line intersect. The very next point on the circle which is on the circle but NOT on the line is the same distance from the center, and is therefore NOT on a (non-pictured) line at right angles to the line, therefore there’s no right angle.

As for the jackwagon in the article who said he could write a dissertation on the topic, my old High School Geometry teacher Miss Phimister (yes, really) would’ve given you an F had you turned it in as an essay, let alone a term paper.

6. Leander says:

Regarding the hyphen, the Word spellchecker flags lots of phrases that it believes are correct if a hyphen is added. I wonder if that is involved. Also, how much the Word spellchecker is changing what is perceived to be grammatically correct.

1. Kim du Toit says:

DO NOT get me started…

2. Windy Wilson says:

I’m ready for the abomination known as Spell-checker correctness.

Eye halve a spelling chequer
It came with my pea sea
It plainly marques four my revue Miss steaks eye kin knot sea.
Eye strike a quay and type a word And weight four it two say
Weather eye am wrong oar write It shows me strait a weigh.
As soon as a mist ache is maid It nose bee fore two long
And eye can put the error rite Its really ever wrong.
Eye have run this poem threw it I am shore your pleased two no
Its letter perfect in it’s weigh My chequer tolled me sew.
(Sauce unknown)

7. GT3Ted says:

OK…… Because it’s the correct answer. This is Math class, not philosophy class. ( or Maths class if you live on the wrong side of the pond. )

…. and a Polygon with infinite sides is still a polygon and not a circle. Circles don’t have sides.

1. JQ says:

circles have two sides, inside and outside

8. Wallace says:

This is what happens after decades of –
There are no absolutes.
There is no right or wrong.
There is no black and white.
EVERYTHING is grey.
etc.
Math, among other things, becoming more and more confusing and less and less useful for ANYTHING. I’m sorry, two plus two still equals four and I don’t care how many degrees you have
in Underwater Basket Weaving and Determining the Gender of Ping Pong Balls that claim otherwise !!
Mixing philosophy with EVERYTHING.
Higher education my sweet ……………….backside.

“Higher Education” as it delves further and further into the realm of “Philosophy” becomes more and more delusional as it cannot any longer see the forest for the trees.

2. Roy says:

For about the last twenty years or so, I have been looking in college catalogs all over for that course. I think “Underwater Basket-weaving” would be an interesting class to take – especially if there were a lot of college age girls enrolled. So far, I have been unsuccessful.

1. Windy Wilson says:

Ph.D’s in Bayberry Candlemaking? (ten points if you identify the reference).

My university merged with a women’s college about two years before I got there, and people have said they could tell the difference. Before the merger, the men’s University had a philosophy class EVERY GODDAMN SEMESTER. After? An Upper Division and a Lower Division class, unless you wanted your degree in philosophy.
William Peter Blatty dedicated the novel that became the film, “The Exorcist”, “To the Jesuits who taught me to think.”

9. Riddle me this: What is the angle between the line and the tangent of the circle at the point that it meets the line?

1. Jack Brown says:

That would be a right angle. It is also not depicted in the image, so, no right angle in the image.

10. RandyGC says:

I actually looked at that picture much longer than I should, as the answer was so obvious I assumed it was trick question and I was missing something.

It turns out what I missed was “learning” math after it’s teaching had been “modernized”.

That and spending years performing tasks such as aerial navigation and ballistics where how you feel about the final answer doesn’t matter when your miss the target or you run out of fuel on final approach.

There are certain constants that become deadly when ignored.

11. valine76 says:

During my four years (1957-1961) at an all boys math/science public high school, located in a 50 year old building in a rough neighborhood of a large city, the school was ranked #2 nationally behind Brooklyn Technical High School. It was what is now called a magnet school, and even with 2,800 boys, there were no discipline problems for the staff of about 100 teachers and administration.
This year, the same school, now coed and located on multi-million dollar a park-like campus, is ranked only #12 in the metropolitan area, ranked #995 nationally, with a bloated staff of 128 for only 1,500 students.
Their reading proficiency is only 70% and mathematics a dismal 42% proficiency.
A double header of innumeracy and illiteracy. And the bloated administration brags about how great the school is. They really don’t have a clue what excellence is.

Why the change?
While I was there,
1. Admission was by test scores.
2. The school was all boys, with raging hormones, with no girls to distract them.
3. The teachers were strict, 90% of them men who were subject matter experts with degrees in their subject, not education degrees, and there were no social promotions. If you failed, you went to summer school or repeated a semester.
4. The principal was a tough old man with a PhD in engineering who demanded and got excellence in academics and behavior.
5. The student body had perhaps 60 blacks, about 2% of the total enrolled, who were there because they met all the academic requirements. There was no such thing as social promotion or affirmative action.

Now,
1. Admission test scores are calculated with double weighted junior high grades favored over standardized tests. With social promotion the current rule du jour, grades from a dumbed down school are worthless.
2. The school is coed because feminazis demanded the experience and prestige the school had. Their presence changed what they were seeking. Welcome to hormone distractions and PC sensitivity.
3. The teachers are 55% female. I don’t want to sound sexist, but I found women teachers on average were not as demanding as men.
4. The school no longer has a principal, but a female Director with a degree in education. I guess job title inflation goes along with grade inflation and administration inflation. There are now 36 people in administration; in 1961 there were only 13 for nearly twice as many students.
5. And last but certainly not least, the school is now 83% “minority.” Being “minority” at 83% must be New Math. ‘Nuf said.

PC and *spit* diversity*spit* drag all down to the lowest common denominator.

Wymyn count as a minority.

12. stablecross says:

If the figure is on a curved surface (i.e. non-Euclidean geometry) then the two top interior angles could be right angles.

13. Sorry, Kim et al, but no. Thale’s theorem.

Let the line segment (from the left side of the hemisphere to the right side of the hemisphere) be AC. For any angle ABC, where B is any other point on the circle (only half of the circle has been drawn here), ABC is a right angle.

If B=C, it’s still a right angle. But B=C means it’s not a triangle! Yes, it is. It just has one side whose length is zero.

This is assuming that we’re dealing with the geometric concepts of lines, points, and circles here. If we’re talking about pictures drawn on paper or on web pages, all bets are off.

(I first ran into this thing in Frances Graham’s classroom, in 1973. Canal Zone Schools. Mrs. Graham was a fourteen-carat bitch, and I hated her with passion at the time. Today, she is one of the top five of my very favorite instructors from the time I was crapping my shorts in Kindergarten until I walked the stage at SMU to take my Masters [thank you, Texas Instruments]; perhaps #2 or even #1. Mrs. Graham was a WWII veteran; a Drill Sgt. for the WAC, and was further hardened by the experience of losing her husband during the 1964 Panama riots. She used to bring her Dobermans to school with her in her VW camper van, and NObody fucked with that vehicle.)

1. Kim du Toit says:

Sorry, but Thale’s Theorem is an abstraction. If in any angled line ABC the value of B is zero, then ABC = AC and therefore ABC is a straight line.

And we are dealing with a picture.

1. Well, consider it with the calculus rather than Euclidean geometry. As B approaches C on the circle, the tangent at B approaches 90° with AB (extended to intersect that tangent). So, in the limit, there is indeed a right angle there.

As you say, the triangle ABC collapses into a line segment when B=C. But the mathematics still works out. Wikipedia (yeah, I know) has a pretty good article on Thales’s Theorem.

14. Kristophr Anon says:

Heh. Only true if this is a flat surface, and not a spherical surface.

If that picture is a view of half the Earth, looking down, with the curved line representing the Equator, the ends of the curve would measure 90 degrees each.

1. Windy Wilson says:

What seventh grader will know that?

15. geekWithA.45 says:

It occurs to me that the conflict on display is NOT about geometry and the underlying conceptual structures.

It is about whether such structures have any consistent validity at all, and who gets to set the terms by which they can be set aside.

Because racist. Or whatever.