Arithmetic with big numbers (Part 1)

Ready for an elementary arithmetic problem? Here it is:

bigadd1

Nothing to it… just add the two numbers. Of course, we’d rather not add them by hand, so let’s use a calculator instead:

bigadd5

Uh oh… the calculator doesn’t give the complete answer. It does return the first nine significant digits, but it doesn’t return all 16 digits. Indeed, we can’t be sure that the final 7 in the answer is correct because of rounding.

So now what we do (other than buy a more expensive calculator)?

When I pose this question to students, the knee-jerk reaction is to just start adding one digit at a time. Though that’s not the worst possible response, it is possible to use modern technology to make ordinary grade-school addition move a lot quicker. One way to do this is to take three digits at a time while using a calculator:

 

bigadd2

Notice that the 1 in 1369 gets carried over to the next block of three digits in much the same way that a sum greater than 10 has the tens digit carried over to the next digit. Continuing:

bigadd3This is logically equivalent to using base 1000 to add these two numbers (as opposed to base 10) and is certainly a lot faster than using only one digit at a time. Of course, it’d go even faster if we use up to nine digits a time (which is equivalent to using base one billion).

bigadd4

Scientific research and false positives

I just read the following interesting article regarding the importance of replicating experiments to be sure that a result is scientifically valid: http://www.slate.com/articles/health_and_science/science/2014/07/replication_controversy_in_psychology_bullying_file_drawer_effect_blog_posts.single.html. This strikes me as an engaging way to introduce the importance of P-values to a statistics class. Among the many salient quotes:

Psychologists are up in arms over, of all things, the editorial process that led to the recent publication of a special issue of the journal Social Psychology. This may seem like a classic case of ivory tower navel gazing, but its impact extends far beyond academia. The issue attempts to replicate 27 “important findings in social psychology.” Replication—repeating an experiment as closely as possible to see whether you get the same results—is a cornerstone of the scientific method. Replication of experiments is vital not only because it can detect the rare cases of outright fraud, but also because it guards against uncritical acceptance of findings that were actually inadvertent false positives, helps researchers refine experimental techniques, and affirms the existence of new facts that scientific theories must be able to explain…

Unfortunately, published replications have been distressingly rare in psychology. A 2012 survey of the top 100 psychology journals found that barely 1 percent of papers published since 1900 were purely attempts to reproduce previous findings…

Since journal publications are valuable academic currency, researchers—especially those early in their careers—have strong incentives to conduct original work rather than to replicate the findings of others. Replication efforts that do happen but fail to find the expected effect are usually filed away rather than published. That makes the scientific record look more robust and complete than it is—a phenomenon known as the “file drawer problem.”

The emphasis on positive findings may also partly explain the fact that when original studies are subjected to replication, so many turn out to be false positives. The near-universal preference for counterintuitive, positive findings gives researchers an incentive to manipulate their methods or poke around in their data until a positive finding crops up, a common practice known as “p-hacking” because it can result in p-values, or measures of statistical significance, that make the results look stronger, and therefore more believable, than they really are.

I encourage teachers of statistics to read the entire article.

My favorite SSA problem

Last month, I had a series of posts on solving a triangle when two sides and a non-included angle are given. Here is my all-time favorite word problem along these lines:

Assume that Venus and Earth both have circular orbits around the sun with radii 68 million miles and 93 million miles, respectively. Just after sunrise, an astronomer sees Venus on the horizon and measures the angle between Venus and the sun to be 20 degrees. Find the possible distances from Venus to Earth at that moment.

venus

I won’t go through the solution of the problem… it’s a fairly straightforward application of SSA techniques. But I’ve always had a soft spot for this problem… probably because I have a soft spot for astronomy and the picture of the planets in their orbits makes perfectly clear why the information can narrow down the answer to two possible solutions, but more information is needed in order to figure out which one is actually correct.

How high can you count on two hands?

How high can you count on two hands? The answer is 1023, if you use binary. I made the following video to demonstrate this to my students.

True story: This is a trick that I came up with when I was 10 years old. As is common in classrooms, my teacher had had enough disruptions in class, and told us students to put our heads on our desks in silence for 15 or 20 minutes as punishment. Naturally, I was trying to think of something to do to pass the time, and I somehow came up with the idea that I could keep myself entertained by counting in binary using my fingers.

When I was a kid, I could count to 1023 in about 5 minutes. But I was a lot more dextrous then, and so it takes me about 6 or 7 minutes today.

Engaging students: Introducing the number e

Story of George Dantzig

Every math teacher should be familiar with this famous story concerning George B. Dantzig (1914-2005). Dantzig is universally hailed as the Father of Linear Programming for his development of the simplex method, which was named one of the top 10 algorithms of the 20th century. The following story happened while he was a graduate student at the University of California.

 

If you search the Web for “urban legend George Dantzig” you will probably find the first hit to be “Snopes.com, The Unsolvable Math Problem.” That site recounts the story of how George, coming in late for class, mistakenly thought two problems written on the board by Neyman were homework problems. After a few days of struggling, George turned his answers in. About six weeks later, at 8 a.m. on a Sunday morning, he and Anne were awakened by someone banging on their front door. It was Neyman who said, “I have just written an introduction to one of your papers. Read it so I can send it out right away for publication.”

George’s answers to the homework problems were proofs of then two unproven theorems in statistics. The Web site gives all the details about how George’s experiences ended up as a sermon for a Lutheran minister and the basis for the film, “Good Will Hunting.” The solution to the second homework problem became part of a joint paper with Abraham Wald who proved it in 1950, unaware that George had solved it until it was called to his attention by a journal referee. Neyman had George submit his answers to the “homework” problems as his doctoral dissertation.

Source: http://www.orms-today.org/orms-8-05/dantzig.html

True story: my own paths actually overlapped with Dantzig’s once. When I was a sophomore in college and he was a professor emeritus, we both attended the same seminar, and he was stick as sharp as a tack. However, I couldn’t build up enough courage to introduce myself to the great man.

 

Sphere

Source: http://www.xkcd.com/1248/

Collaborative Mathematics: Challenge 11

I’m a couple months late with this, but my colleague Jason Ermer at Collaborative Mathematics has published Challenge 11 on his website: http://www.collaborativemathematics.org/

Engaging students: Computing trigonometric functions

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Nataly Arias. Her topic, from Precalculus: computing trigonometric functions.

green line

How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

 

Trigonometry does not only relate to mathematics, trigonometry is also used in real life. Many people don’t know that trigonometry is involved in video games. In game development, there are many situations where you will need to use trig functions. Video games are full of triangles. For example in order to calculate the direction the player is heading you will form a triangle and use sine, cosine, or tangent to solve. The trig function used depends on the values given. For example if the opposite and adjacent values are given (the xSpeed and ySpeed), the function you will need to calculate the direction of the player is tangent. This is represented by the equation Tan( Dir ) = xSpeed /ySpeed. Again, by applying the inverted function of tan to both sides of the equal sign, we get an equation that will return the player’s direction. In a spaceship game you will need to use trigonometric functions to have one ship shoot a laser in the direction of the other ship, play a warning sound effect if an enemy ship is getting too close, or have one ship start moving in the direction of another ship to chase. Trig is used in several situations in video games some more examples include calculating a new trajectory after a collision between two objects such as billiard balls, rotating a spaceship or other vehicle, properly handling the trajectory of projectiles shot from a rotated weapon, and determining if a collision between two objects is happening.

 

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How has this topic appeared in high culture (art, classical music, theatre, etc.)?

 

The “unit circle” is a circle with a radius of 1 that is centered at the origin in the Cartesian coordinate system in the Euclidean plane. Because the radius is 1 we can directly measure sine, cosine, and tangent. The unit circle has made parts of mathematics easier and neater. The concepts of the unit circle go far back into the past. Not only do we use and see circles in mathematics we also can see circles in art form. We can also use trigonometric functions to determine the best position to view a painting hanging on an art gallery wall. For example you can determine the angle between a person’s eye and the top and base of the painting when a person is standing 1m away, 2 m away, 3 m away and so on. By comparing your data you can estimate the best position for a person to stand in front of the painting. Also using trig functions and your handy calculator you can develop a formula that describes the relationship between the distance away from the painting and the angle that exists between the person’s eye and the top and bottom of the painting.

 

 

 

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How have different cultures throughout time used this topic in their society?

 

Today the unit circle is used as a helpful tool to help calculate trig functions. Trig functions are taught in trigonometry, pre-calculus and are frequently used in advanced math classes. Many people don’t realize that not only are trig functions learned and used in school but throughout time several cultures have used trig functions in their society. The main application of trigonometry in past cultures was in astronomy. In 1900 BC the Babylonians kept details of stars, the motion of planets, and solar eclipses by using angular distance measured on the celestial sphere. In 1680-1620 BC the Egyptians used ancient forms of trigonometry for building pyramids. The idea of dividing a circle into 360 equal pieces goes back to the sexagesimal counting system of the ancient Sumerians. Early astronomical calculations wedded the sexagesimal system to circles and the rest is history. Today in trigonometry the unit circle has a radius of 1 unlike the Greek, Indian, Arabic, and early Europeans who used a circle of some other convenient radius. In today’s society trigonometry is everywhere. The mathematics used behind trigonometry is the same mathematics that allows us to store sound waves digitally onto a CD. We use it without even knowing it. When we plug something into the wall there is trigonometry involved. The sine and cosine wave are the waves that are running through the electrical circuit known as alternating current.


References

 

http://www.math.ucdenver.edu/~jloats/Student%20pdfs/40_Trigonometry_Trenkamp.pdf

 

http://www.math.dartmouth.edu/~matc/math5.geometry/unit9/unit9.html

 

http://en.wikipedia.org/wiki/Trigonometric_functions

 

http://aleph0.clarku.edu/~djoyce/ma105/trighist.html

 

http://www.slideshare.net/mgeis784/building-the-unit-circle

 

http://www.softlion.nl/download/article/Trigonometry.pdf

 

http://www.raywenderlich.com/35866/trigonometry-for-game-programming-part-1

 

http://stackoverflow.com/questions/3946892/trigonometry-and-game-development

 

100 years ago…

Sadly, a current conundrum in secondary mathematics education was very much on the minds of mathematicians in 1914.

Source: https://www.facebook.com/AmerMathMonthly/photos/a.250425975006394.53155.241224542593204/737393606309626/?type=1&theater

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