An Excerpt from Magnificent Mistakes in Mathematics

Alfred S. Posamentier and Ingmar Lehmann, who previously wrote the excellent book The Fabulous Fibonacci Numbers (1997), have teamed up once again to write Magnificent Mistakes in Mathematics (Prometheus Books, 2013).

It’s a book with a self-explanatory title, and you only need a high-school-level background in math to understand it.

The following are excerpts from the book, reprinted with permission of the publishers:

Stumbling Blocks: Numbering Mistakes

There are oftentimes counting errors that occur and remain unnoticed for a long time. On January 1, 2000, the New York Times corrected an error that had been made more than one hundred years earlier. On February 6, 1898, an employee of this newspaper noticed that the current day’s issue was number 14,499. And so he mistakenly gave the following day’s issue number 15,000, instead of 14,500. It was not until Saturday, January 1, 2000, that this mistake was corrected. On that day, issue number 51,254 was published, while the previous day’s issue was numbered 51,753. In case you’re wondering, issue number 1 of the New York Times was published on September 18, 1851.

Some printing errors are not as easy to correct as that of the New York Times’s numbering system. We have been shown through the Popeye comics that spinach brings a person extra strength. This is largely a result of some misunderstandings — or perhaps mistakes — about the iron content in spinach. Spinach has approximately 3.5 mg of iron per 100 g of spinach; while in its cooked version, it contains about 2 mg, which turns out to be a lot less than bread, meat, or fish. This misunderstanding of spinach’s iron value stems back to the 1930s, when there was a printing mistake. A decimal point was mistakenly moved one place to the right, which, of course, gives a value ten times that of the intended value. One might see this mistake as one where the value of iron in spinach was given ten times the correct amount, namely, 35 mg per 100 g of fresh spinach. With Popeye’s reinforcement, many children grew up imagining spinach as an instant source of power.

Rounding off Can Cause a Mistaken Answer

There are times when rounding off correctly leads to a wrong answer. Consider the following example: At an airport there are 963 stranded passengers. Busses are ordered to take these passengers on their way. Each bus can hold 59 passengers. The question is, how many buses are required to take these passengers on their way? Typically, a student would do the following calculation. 963 ÷ 59 ≈ 16.32203389. Since the number of buses must be a whole number, the student would round off the answer correctly to 16, since the 3 following the decimal point is less than 5. This answer clearly does not solve the problem. And although the calculation was correct, the problem was solved incorrectly. Obviously seventeen buses will be required, where the seventeenth bus will not be completely full. Here is an example of a mistaken answer despite the fact that correct calculations were made.

A Mistake Based on Prematurely Jumping to Conclusions

Suppose you take a circle, put some dots along the outside, and then connect them. If only two lines cross at any point, into how many regions will the circle be divided?

As we close this chapter on arithmetic mistakes, we should notice that sometimes what is seen as a mistake may, in fact, not be a mistake at all. Consider the following sequence and ask that for the next number: 1, 2, 4, 8, 16. Most people would assume that 32 will be the next number. Yes, that would be fine. However, when the next number is given as 31 (instead of the expected 32), cries of “wrong!” are usually heard.

Much to your amazement, this is can also be a correct answer, and 1, 2, 4, 8, 16, 31 can be a legitimate sequence, and not a mistake!

The task now is to be convinced of the legitimacy of this sequence. It would be nice if it could be done geometrically, as that would give evidence of a physical nature…

To be further convinced that this sequence is legitimate and not resulting from mistakenly replacing the “32” with a “31,” we shall consider the Pascal triangle. This triangle is formed by beginning on top with 1, then the second row has 1, 1, and the third row is obtained by placing 1s at the end and adding the two numbers in the second row (1 + 1 = 2) to get the 2; the fourth row is obtained the same way. After the end 1s are placed, the 3s are gotten from the sum of the two numbers above (to the right and left), that is, 1 + 2 = 3, and 2 + 1 = 3.

The horizontal sums of the rows of the Pascal triangle to the right of the bold line drawn: 1, 2, 4, 8, 16, 31, 57, 99, 163. This is again our newly developed sequence.

A geometric interpretation should further support the legitimacy of this sequence and support the beauty and consistency inherent in mathematics. To do this, we shall make a chart [see below] of the number of regions into which a circle can be partitioned by joining points on the circle, where no three lines meet at one point; otherwise a region would be lost.

Let’s focus on the case where n = 6. [See below]

We notice there is no thirty-second region.

Now that you see that this unusual sequence (1, 2, 4, 8, 16, 31, 57, 99, 163…) appears in various other contexts, you should be convinced that even though there appeared to be a mistake at the original introduction of the “31,” there was, in fact, no mistake. Thus, even mistakes can be deceptive — or mistakenly identified as mistakes!

Magnificent Mistakes in Mathematics is available for pre-order right now and it’ll be in bookstores in mid-August.

If you’d like to win your own copy of the book, just let us know what math topic (e.g. Fractions, Trigonometry) you enjoyed the most! Use the hashtag #Matheist at the end of your comment and you’ll be automatically entered in the contest. I’ll contact one random winner next week.

About Hemant Mehta

Hemant Mehta is the editor of Friendly Atheist, appears on the Atheist Voice channel on YouTube, and co-hosts the uniquely-named Friendly Atheist Podcast. You can read much more about him here.

  • Buckley

    As a historian and history teacher….znooze…wait where am I?

    • Buckley

      Oh, for cripe sake, it was a joke…lighten up people.

      • allein

        As an English major, I gave you an up. :)

  • ChanaM

    Looks wonderful! I will probably buy this for students.

    • Guest

      Calculus & Probability ftw. #matheist

  • primenumbers

    Looks to be a great popular math book – thanks for the info. I enjoyed number theory the most. #Matheist

  • Michael Harrison

    So we read it for a chance to get a copy? That seems backwards from the usual order of things.

  • http://coolingtwilight.com/ Dan Wilkinson

    Calculus was by far my favorite area of math! #Matheist

  • The Other Weirdo

    I tend to freak out when I encounter math beyond retail store arithmetic, but I’d love to read this book. #Matheist

  • Mairianna

    You lost me at the number sequence…….

    • MariaO

      Dont worry. ANY number series can be correctly continued with ANY numer. Just use a polynomial equation of high enough degree. Number series are stupid – because they can only be correctly solved by the mathematically challenged.

      • Epinephrine

        Exactly – every finite sequence has infinitely many correct solutions , with pretty much any number being the next in sequence. It’s nearly as bad as those “which of the following words is not like the others,” questions.

  • Joe Geiger

    Have always loved math books for lay folks. Like primenumbers, my favorite math discipline has to be number theory. In college, I took classes in Combinatorics and Abstract Algebra — stuggled in both but found them fascinating! #Matheist

  • GubbaBumpkin

    This is largely a result of some misunderstandings — or perhaps mistakes
    — about the iron content in spinach. Spinach has approximately 3.5 mg
    of iron per 100 g of spinach…

    What’s more. spinach has a lot of chelators, which prevent your GI tract from absorbing much of that iron.

    #Matheist

  • GubbaBumpkin

    Much to your amazement, this is can also be a correct answer,…

    Much to my amazement, this is cannot be correct grammar.
    #Matheist

    • flyb

      Their follow-up book:

      “Egregious Errors in English”

    • rhodent

      What’s wrong with it?

      • GubbaBumpkin

        Correct: “this is also a correct answer.”
        Correct: “this can also be a correct answer.”
        Incorrect: “this is can also be a correct answer.”

        • rhodent

          Ah. I completely missed the “is”.

          • Chakolate

            Me, too.

            Or should I say, “I, also.”

            • Jude

              [voice="Te'alc"]
              As did I.
              [/voice]

              • sane37

                as did I

  • GubbaBumpkin

    My favourite field of maths: Fourier analysis. My career is built on it.
    #Matheist

  • JET

    As someone who felt that high school math was akin to being tortured on the rack, I’ll leave this book for others. But we English majors really do appreciate that there are people in the world who find this stuff interesting!

    • allein

      I’m an English major but I still find this stuff interesting. My reading habits are all over the place but I’ve been on a science-y kick for a while now. (That and the Holocaust, lately. Currently in the middle of a Neil DeGrasse Tyson book and also Milkweed by Jerry Spinelli, and just finished The Boy in the Striped Pajamas.)

      • JET

        I like reading the science-y stuff, too. But pure mathematics makes my eyes glaze over. :)

    • Lagerbaer

      High school math to university math is what memorizing the irregular verb forms is to reading and analyzing a Shakespear play.

      • JET

        Worked my butt off in high school to get all the math requirements out of the way so I could spend college studying all those wonderful books! :)

  • bassplr19

    Life is all probabilistic… #Matheist

  • katiehippie

    I loved analytic geometry in high school. Not that I can remember much of it. #Matheist

  • Erin Pendleton

    Love this kind of stuff, especially trig! Also, while I can sympathize with commentators who aren’t interested in the book because they don’t care for math, it’s not quite fair to assume that all English majors dislike math. I happen to love math and I hold a master’s in Renaissance literature. #Matheist

  • CultOfReason

    I was a math major and enjoyed all the topics. #Matheist

  • randomfactor

    It was trig for me. Because I took it for fun, as a liberal arts major. Loved my slide rule with the trig functions engraved on the back. My shining moment was coming up with the right answer instantly when my instructor drew a right triangle and asked for the hypotenuse…she’d accidentally drawn a 3:4:5 ratio.
    #Matheist

  • Garrett Williams

    Calculus & Probability ftw! #matheist

  • rhodent

    I took a number theory course my senior year of high school, and for some reason thought that working in bases other than 10 and learning about the uses of this were pretty cool.

    #Matheist

    • Mario Strada

      You know, in high school I was terrible at math. Terrible. But at the same time I was fascinated by number theory and especially, like you, bases other than 10. There is a great video about base 12 math on youtube in one of the Math channels. You should check it out.

      #Matheist

  • Composer 99

    Excerpt from the book:

    With Popeye’s reinforcement, many children grew up imagining spinach as an instant source of power.

    Really? Hadn’t heard that. Certainly when I was a child there was no such talk of things (except perhaps when discussing Popeye).

    • Composer 99

      Also, the niftiest topic in the OP was the one discussing number sequences and circle geometry. #matheist

      • allein

        OK, the pyramid thing I can get the gist of even though I’m at work and don’t have time to really try to digest it, but the circle thing is beyond me right now…

  • allein

    I never did well in math until I got to high school and wound up with the right teacher. Algebra and geometry finally made sense! (Not that I remember much.) I still majored in English and took the bare minimum of math to graduate college, though. Oddly, I ended up working for a publisher of academic math journals for a while and proofreading was part of my job. That was kinda fun once I got used to identifying when a whole equation was a noun and when the = or > or whatever was the only verb in the whole sentence. My current job (and the one before it) is pretty number-heavy, too. How that happened I still can’t fathom. ;) #matheist

  • http://v1car.wordpress.com/ The Vicar

    Sounds like an interesting book. I liked basic calculus best of all the classes I took, because I had a terrific teacher (who you might actually have met, Hemant, although it would have been a while ago since I’m sure he’s retired by now, and I’m not naming names for the sake of my own online anonymity!). But I really enjoyed reading explanations of transfinite group theory à la Cantor on my own. (Too bad my set/group theory classes appeared to be designed to utterly annul any enjoyment anyone might possibly have in the topic.) (And yes, I chose the word “annul” over “destroy” for the faint-echo-of-a-pun.) #Matheist

  • rg57

    1. “On that day, issue number 51,254 was published, while the previous day’s issue was numbered 51,753.”

    If you’re paying attention, you’ll realize that the attempted solution is in fact just another mistake. It treats a series of nominal numbers as if it were a series of cardinal numbers.

    If the 14,500-14,999 gap for some reason absolutely must be filled, then those are the actual numbers that should have been used, followed by 51,754 at which point the numbers would finally be correct again for both nominal and cardinal interpretations.

    I’m guessing that the authors were not paying attention.

    2. The ceiling function of x (where x is a positive non-integer) in the example is equivalent to rounding off x + 1/2. Rounding, like addition, works when you use it properly.

    3. “Suppose you take a circle, put some dots along the outside, and then
    connect them. If only two lines cross at any point, into how many
    regions will the circle be divided?” Read it a few times. The answer is not possible to know.

    Now, if this was about line segments, then sure you could go ahead as the authors did. But any high school student knows that lines extend infinitely in both directions. If “only two lines cross”, then there can be only two lines period. If they intersect “at any point”, then such intersection could happen outside the circle. None of the lines are tangent to the circle because they are drawn by connecting dots on the circle. Therefore each line is defined by two dots on the circle. Ignoring identical lines, there are two cases: a shared dot, and no shared dots. In the shared-dot case, the area inside the circle is divided into three areas. In other case, if the intersection is outside the circle, the answer is also three. If the intersection in the area inside the circle you get four.

  • http://squeakysoapbox.com/ Rich Wilson

    Number theory. Hurt my brain, but mind blowing.

    #Matheist

  • DCF

    I had a bad Algebra II teacher, and then fell apart emotionally the year I took trig. I hated & feared math until I had an epiphany and now I’m a statistician. LOVE this stuff, and love showing middle & high school students how “dry” math can be fun & useful. #statistics!

  • Noelle

    And all this time I thought the Popeye thing was a ploy to get kids to eat their vegetables.

  • Nancy

    Combinatorics, graph theory, coding theory. I will perhaps buy the book.

    • http://squeakysoapbox.com/ Rich Wilson

      Don’t forget the #Matheist

  • Lori Ford

    I loved calculus! #Matheist

  • cr0sh

    In high school, and for some time afterward, my favorite was trigonometry (and to a certain extent geometry) – but then again, I was also playing around with coding 3D engines and understanding them. I also delved into chaos theory, fractals, and cellular automata…

    Lately, though, I have rediscovered and fell in love with linear algebra, particularly how it applies to machine learning and artificial intelligence; that said, I have also found that I need to seriously study up on probability and statistics (found out all of this after taking Stanford’s original AI and ML MOOCs in 2011, then later the Udacity CS373 course).

    #Matheist

  • Matt H.

    I majored in math in community college, and by far my favorite course was Linear Algebra…something to do with the way it reached into lots of other areas of math is really cool to me. #Mathiest

  • Matt H.

    Also, I’m just now getting that Mathiest is athiest with an m, not just “most mathy”…

    • Kodie

      It’s spelled atheist. A-the-ist. Break it down. “Mathiest” would be most mathy, but that’s not how anyone else spelled it.

  • Lagerbaer

    I absolutely loved Linear Algebra in university. I actually liked how it was abstract and then all those concepts started popping up everywhere. In my physics courses as well as the computer science courses. Actually being good at linear algebra once allowed to very briefly read up on a topic (Distributed Computing) and get two nice conference papers out in no time.

    But complex analysis (theory of functions) is also pretty darn cool. Some of the theorems just sound so unbelievably strong.

    What I personally don’t like are differential equations. Somehow I never got the hang of “guessing” the right ansatz.

    #Matheist

  • Chakolate

    Calculus was my favorite tortur… I mean math subject. There’s something so sweet about conquering something so tricky. :-) #Matheist

  • ImRike

    Thanks for the heads-up! It will also have a Kindle edition! My favorite Math topic was Trig. #Matheist

  • Ouigui

    I loved linear algebra and group theory: so many useful connections to physics! #Matheist

  • http://exconvert.blogspot.com/ Kacy

    I loved Geometry in high school because I enjoyed working with manipulatives and viewed proofs as fun logic puzzles. #Matheist

  • Hanna

    It’s seems like most people don’t really like algebra, but that has always been my favorite type of math! #matheist

  • sane37

    I actually quite enjoyed calculus. Once I got the hang of it.

    #Matheist

  • Rhiar

    I loved all my math classes save one but especially favored geometry. I’m a very visual learner and I loved manipulating all that stuff in my head. That led me to topology which I had to learn on my own since it wasn’t covered in my school. I ordered books directly from publishers since none were to be found otherwise. In college I took scientific glassblowing and made Klein bottles. Currently I am getting back into calculus on my own – my year of calculus was taught by a very ineffective teacher which was a shame since calculus is quite lovely and elegant. That was the only math class I didn’t like – not because of the subject but because of the teacher. #Matheist

  • duke_of_omnium

    I think the area of math that I enjoyed the most was logarithms. I just found logarithms and their manipulation to be so elegant. And you never know: some day, you might need to log a few rhythms …

    #matheist

  • Epinephrine

    Loved game theory, but also enjoyed anything remotely connected to axiomatic systems (e.g., non-Euclidean geometry) #Matheist

  • Gringa123

    How to pick just one? I enjoyed identities #Matheist

  • cary_w

    Ahhh… Hemant! I just love it when you start talking math!

    I’ve allways considered the bus question as a classic example of why division with a remainder is not just a simplified form of division used to teach children who have not yet learned decimals, but rather a useful form of division that should be used when the objects being divided must remain whole objects. For example, it makes no sense to have .322 of a bus, buses only exist as wholes. So if you have 963 people, and each bus hold 59 people, we can use division with a remainder to get the answer: 963/59 = 16 r19 which mean we will fill 16 buses and have 19 people left over, so you will need 17 busses. This answer is more useful than just automatically rounding up to get 17 buses because, in the real world, you may have access to some smaller buses and now you know that if you want to use one smaller bus, it must hold at least 19 people.

    My favorite branch of mathematics is the conceptual side of Calculus, I just love how things like the area under a curve, the slope at a point, and limits fit so beautifully into mechanical physics, statistics, and other sciences. To me, this is where math truly becomes elegant.

    #Matheist

  • hotshoe

    Geometry!

    #Matheist


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