## All posts in category **Elementary**

# Dividing fractions

*Posted by John Quintanilla on January 5, 2018*

https://meangreenmath.com/2018/01/05/dividing-fractions/

# Repunit prime

In the United States, today is abbreviated 10/31. Define the th repunit number as

,

a base-10 number consisting of consecutive 1s. For example,

,

and so on.

It turns out that is the largest known prime repunit number.

*Posted by John Quintanilla on October 31, 2017*

https://meangreenmath.com/2017/10/31/repunit-prime/

# Pizza Hut Pi Day Challenge: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series on the 2016 Pizza Hut Pi Day Challenge.

Part 1: Statement of the problem.

Part 2: Using the divisibility rules for 1, 5, 9, 10 to reduce the number of possibilities from 3,628,800 to 40,320.

Part 3: Using the divisibility rule for 2 to reduce the number of possibilities to 576.

Part 4: Using the divisibility rule for 3 to reduce the number of possibilities to 192.

Part 5: Using the divisibility rule for 4 to reduce the number of possibilities to 96.

Part 6: Using the divisibility rule for 8 to reduce the number of possibilities to 24.

Part 7: Reusing the divisibility rule for 3 to reduce the number of possibilities to 10.

Part 8: Dividing by 7 to find the answer.

*Posted by John Quintanilla on August 22, 2017*

https://meangreenmath.com/2017/08/22/pizza-hut-pi-day-challenge-index/

# Euler and 1,000,009

Here’s a tale of one the great mathematicians of all time that I heard for the first time this year: the great mathematician published a mistake… which, when it occurs today, is highly professionally embarrassing to modern mathematicians. From *Mathematics in Ancient Greece*:

In a paper published in the year 1774, [Leonhard] Euler listed [1,000,009] as prime. In a subsequent paper Euler corrected his error and gave the prime factors of the integer, adding that one time he had been under the impression that the integer in question admitted of the unique partition

but that he had since discovered a second partition, namely

,

which revealed the composite character of the number.

See Wikipedia and/or Mathworld for the details of how this allowed Euler to factor .

*Posted by John Quintanilla on August 3, 2017*

https://meangreenmath.com/2017/08/03/euler-and-1000009/

# Factoring Mersenne “primes”

I love hearing and telling tales of legendary mathematicians. Today’s tale comes from Frank Nelson Cole and definitely comes from the era before calculators or computers. From Wikipedia:

On October 31, 1903, Cole famously made a presentation to a meeting of the American Mathematical Society where he identified the factors of the Mersenne number 2

^{67}− 1, or M_{67}. Édouard Lucas had demonstrated in 1876 that M_{67}must have factors (i.e., is not prime), but he was unable to determine what those factors were. During Cole’s so-called “lecture”, he approached the chalkboard and in complete silence proceeded to calculate the value of M_{67}, with the result being 147,573,952,589,676,412,927. Cole then moved to the other side of the board and wrote 193,707,721 × 761,838,257,287, and worked through the tedious calculations by hand. Upon completing the multiplication and demonstrating that the result equaled M_{67}, Cole returned to his seat, not having uttered a word during the hour-long presentation. His audience greeted the presentation with a standing ovation. Cole later admitted that finding the factors had taken “three years of Sundays.”

*Posted by John Quintanilla on August 2, 2017*

https://meangreenmath.com/2017/08/02/factoring-mersenne-primes/

# My Favorite One-Liners: Part 92

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

This is one of my favorite quote from Alice in Wonderland that I’ll use whenever discussing the difference between the ring axioms (integers are closed under addition, subtraction, and multiplication, but not division) and the field axioms (closed under division except for division by zero):

‘I only took the regular course [in school,’ said the Mock Turtle.]

‘What was that?’ inquired Alice.

‘Reeling and Writhing, of course, to begin with,’ the Mock Turtle replied; ‘and then the different branches of Arithmetic — Ambition, Distraction, Uglification, and Derision.’

*Posted by John Quintanilla on May 3, 2017*

https://meangreenmath.com/2017/05/03/my-favorite-one-liners-part-92/

# My Favorite One-Liners: Part 18

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them. This is a quip that I’ll use when a theoretical calculation can be easily confirmed with a calculator.

Sometimes I teach my students how people converted decimal expansions into fractions before there was a button on a calculator to do this for them. For example, to convert into a fraction, the first step (from the Bag of Tricks) is to multiply by 1000: How do we change this into a decimal? Let’s call this number .

Notice that the decimal parts of both and are the same. Subtracting, the decimal parts cancel, leaving

or

In my experience, most students — even senior math majors who have taken a few theorem-proof classes and hence are no dummies — are a little stunned when they see this procedure for the first time.

To make this more real and believable to them, I then tell them my one-liner: “I can see that no one believes me. OK, let’s try something that you *will* believe. Pop out your calculators. Then punch in 16 divided by 37.”

Indeed, my experience many students really do need this technological confirmation to be psychologically sure that it really did work. Then I’ll tease them that, by pulling out their calculators, I’m trying to speak my students’ language.

See also my fuller post on this topic as well as the index for the entire series.

*Posted by John Quintanilla on February 18, 2017*

https://meangreenmath.com/2017/02/18/my-favorite-one-liners-part-18/

# My Favorite One-Liners: Part 14

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them. This quip is similar to the “bag of tricks” one-liner, and I’ll use this one if the “bag of tricks” line is starting to get a little dry.

Sometimes in math, there’s a step in a derivation that, to the novice, appears to make absolutely no sense. For example, to find the antiderivative of , the first step is far from obvious:

While that’s certainly correct, it’s from from obvious to a student that this such a “simplification” is actually helpful.

To give a simpler example, to convert

into a decimal, the first step is to multiply by :

Students often give skeptical, quizzical, and/or frustrated looks about this non-intuitive next step… they’re thinking, “How did you know to do that?” To lighten the mood, I’ll explain with a big smile that I’m clairvoyant… when I got my Ph.D., I walked across the stage, got my diploma, someone waved a magic wand at me, and *poof*! I became clairvoyant.

Clairvoyance is wonderful; I highly recommend it.

The joke, of course, is that the only reason that I multiplied by 1000 is that someone figured out that multiplying by 1000 at this juncture would actually be helpful. Subtracting from , the decimal parts cancel, leaving

or

.

In my experience, most students — even senior math majors who have taken a few theorem-proof classes and hence are no dummies — are a little stunned when they see this procedure for the first time. I learned this procedure when I was very young; however, in modern times, this procedure appears to be a dying art. I’m guessing that this algorithm is a dying art because of the ease and convenience of modern calculators. As always, I hold my students blameless for the things that they were simply not taught at a younger age, and part of my job is repairing these odd holes in their mathematical backgrounds so that they’ll have their best chance at becoming excellent high school math teachers.

For further reading, here’s my series on rational numbers and decimal expansions.

*Posted by John Quintanilla on February 14, 2017*

https://meangreenmath.com/2017/02/14/my-favorite-one-liners-part-14/

# 15-square puzzle

From the category “This Is Completely Useless”: here’s what a 15-square puzzle looks like when you arrange the tiles in order of how many factors they have.

*Posted by John Quintanilla on December 12, 2016*

https://meangreenmath.com/2016/12/12/15-square-puzzle/

# My Mathematical Magic Show: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The links below show the mathematical magic show that I’ll perform from time to time.

Part 1: Introduction.

Part 2a, Part 2b, and Part 2c: The 1089 trick.

Part 3a, Part 3b, and Part 3c: A geometric magic trick.

Part 4a: Part 4b, Part 4c, and Part 4d: A trick using binary numbers.

Part 5a, Part 5b, Part 5c, and Part 5d: A trick using the rule for checking if a number is a multiple of 9.

Part 7: The Fitch-Cheney card trick, which is perhaps the slickest mathematical card trick ever devised.

Part 8a, Part 8b, and Part 8c: A trick using Pascal’s triangle.

Part 9: Mentally computing given if .

Part 6: The Grand Finale.

And, for the sake of completeness, here’s a recent picture of me just before I performed an abbreviated version of this show for UNT’s Preview Day for high school students thinking about enrolling at my university.

*Posted by John Quintanilla on December 10, 2016*

https://meangreenmath.com/2016/12/10/my-mathematical-magic-show-index-3/