Irrational Numbers with Patterns

img_20160919_163743715I'm not really posting anymore but I had to make an exception. The bad quality picture above is from a student's math class and the worksheet says: "Decimals that never end and have no pattern, think MUMBO JUMBO NUMBERS" in describing irrational numbers. Some context: the student had prealgebra last year where they learned that rational numbers were numbers that can be written as an integer divided by a nonzero integer and that irrational numbers were real numbers that weren't rational. More importantly, the student had learned the decimal characterization of rational and irrational numbers: rational numbers are terminating or nonrepeating and irrational numbers are nonterminating and nonrepeating.

So when the student, now in algebra, went through the beginning of the school year review of the real number system with this worksheet you have to cringe. First, the algebra class has learned the important definitions for rational and irrational numbers, so why not review those terms? That SHOULD be the goal. Second, the characterization of irrational numbers as "have no pattern" is wrong and she uses pi as an the only example to back up her claim. The teacher has given the class a worksheet with wrong information and doesn't know it. Welcome to a quality public school.

So let's give some quick examples to of irrational numbers which have decimals that never end BUT have a pattern.


Yes, ... is ambiguous, so let's describe the pattern. Just count the positive integers and put them together after the decimal. Understanding the pattern you know the next digits are ....161718192021..... and you can calculate whatever place after the decimal you want if you need to because there is a pattern. Here's another example:


The pattern is to have separate one "1" from two "1"s from three "1"s from...using a 0. Under that pattern, you know the next digits are 11111110111111110..... and so on. Both of these decimal numbers are nonterminating and nonrepeating. You can't shouldn't make up your own math, especially when it's wrong. It causes unnecessary confusion.

Here are some stories that caught my eye recently:

  • From the Daily Mail, "Meet the boy geniuses who developed a math theorem that calculates problems faster than a COMPUTER - despite still being in high school"
  • Russian chess announces the passing of chess trainer Mark Dvoretsky. As someone fortunate enough to have attended some of his training sessions, I appreciate just what a tragic loss this is for chess.
  • Poor Chicago teachers! Well, not poor exactly. ZeroHedge reports they are ready to strike yet again. "Chicago teachers have a 13% raise (over four years) offer on the table, but that is not enough. They set a strike date of October 11 because the city wants the union to contribute more than 2% for their underfunded pensions, among the worst funded pensions in the nation. The Chicago public school system is bankrupt. Its bonds are deep in junk affiliate of the American Federation of Teachers, is gearing up for a strike. It would be the union’s second in four years, despite the fact that the median salary for a teacher in Chicago is $78,169. When you add another $27,564 for various benefits, the total compensation for a teacher – good, bad or middling – becomes almost $106K per annum. (Please keep in mind teachers work 180 days a year, while employees in other professions typically work for 240 to 250 days.) In retirement, the average Chicago teacher receives a hefty $50,000 a year.
  • Truthdig with a sobering look at the status of adjunct teachers. With the hurdles to teach at public high school too onerous they are the working poor.
  • with a well covered story on a Michigan student who, "...noticed nasty discolored water coming from the sink in her school, so she took a picture and shared it with social media. Hazel Juco was hoping that she could raise awareness about the potentially dangerous issue, and perhaps get the problem fixed. Unfortunately, the school was not impressed with the student's whistleblowing and actually suspended her for violating the school's policy of taking photos in the bathroom."
  • Propublica on "Education Department Terminates Agency That Allowed Predatory For-profit Colleges to Thrive"
  • Daily Mail with a viral image. You can't trust your senses.
  • Remember the poor Yale professor who was screamed at by hysterical brats? The Federalist has more video of what led up to the incident. "Altogether, the footage is nearly 25 minutes long — during which Nicholas manages to keep his cool in an attempt to rationally discuss his wife’s email with a cluster of impassioned students. The scene is chaotic — students verbally attack Nicholas, demanding he apologize for his wife’s “racist” comments. Ultimately, he does not decry the content of Erika’s email, but he emphatically and repeatedly apologizes for any pain her words caused....Based on this footage, it’s clear the students at Yale were incapable of having a rational discussion with another individual who dared to push back against their hasty accusations of racism. Despite what others have tried to claim, it’s plain as day many of those students who loudly disrupted the campus with their protest really were crybabies that threw tizzies over an email about Halloween costumes."
  • Lew Rockwell with a nice video on the value of a college degree.
  • American Thinker has some thoughts on reforming higher education: "We need to increase by 40% over the next ten years the number of people with college degrees, a state official once lectured us. Sitting in the audience, I thought, we know how to give people degrees. We no longer know how to give them a college education...."
  • Quartz with a piece on "Seven Rhode Island universities, including Brown and Rhode Island College, will move to open-license textbooks in a bid to save students $5 million over the next five years, the governor announced Tuesday"
  • The Federalist spotlights just how worthless a PhD in education is in reporting on  a PhD candidate who claims "Science Is Sexist Because It’s Not Subjective". "Throughout her dissertation, Parson assumes and asserts that women and minorities are uniquely challenged by the idea that science can provide objective information about the natural world. This is an unfair assumption, she says, because the concept of objectivity is too hard for women and minorities to understand."


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