Category Archives: Number Theory

Transcendental Numbers

In a previous post, “Magnificent Pi”, I mentioned that pi is an irrational number, but it is also a transcendental number. A transcendental number is a number that is not algebraic; it is not a root of a non-zero polynomial equation with rational coefficients.  What does that mean?

Numberphile has a video that explains what it means for a number to be algebraic or transcendental.

Special Numbers: 6174

Try this:

  1. Take any four-digit number, using at least two different digits. Repdigits, such as 1111, will not work, because you will just end up with 0 after step 3.
  2. Arrange the digits in ascending and then in descending order, adding leading zeros if necessary. Add leading zeros if necessary – for example, 4560 in ascending order is 0456 and 6540.
  3. Subtract the smaller number from the bigger number.
  4. Go back to step 2 and repeat the process.

This process, known as the Kaprekar routine, will always reach the number 6174, within 7 iterations. Once 6174 is reached, the process will continue yielding 6174 because 7641 – 1467 = 6174.

For example, choose 6532:

6532 – 2356 = 4176

7641 – 1467 = 6174

Another example, choose 4905:

9640 – 0469 = 9171

9711 – 1179 = 8532

8532 – 2358 = 6174

7641 – 1467 = 6174

6174 is known as Kaprekar’s constant, named after Indian mathematician D. R. Kaprekar.

Cicadas: Mathematicians of the Insect World

800px-Magicicada_species

Source: Wikimedia Commons (Author: Bruce Marlin)

Cicadas are coming to the East Coast this year. This year marks the end of the 17-year life-cycle for this particular population of cicadas. Periodical cicadas (Magicicada septendecim) spend most of their lives underground sucking fluids from roots, but emerge after 13 or 17 years (depending on the population) to mate and then die.

What does this have to do with math?

13 and 17 are both prime numbers, and therefore cannot be divided evenly by any other numbers (besides 1 and themselves). Researchers believe that prime number life-cycles help the periodical cicadas avoid predators and parasites. For example, a cicada with a 17-year cycle and a parasite with a two-year cycle would meet only twice in a century (year 34 and year 68). A 17-year cicada would only meet a 5-year parasite once in a century, year 85 (5 x 17). Also, the prime number cycles keep different populations from meeting and interbreeding. Since the life-cycle is determined by genes, interbreeding between 13-year and 17-year cicada populations would throw off their clockwork cycle.

How do the cicadas know when to emerge? They count, of course. Cicadas feed on the roots of trees that flower every year. Scientists at University of California-Davis were able to make some cicadas hatch a year early by transplanting them onto potted trees and forcing the trees to flower twice in one year.

Read more at:

Science News for Kids: Prime Time for Cicadas

The Baltimore Sun: Mathematicians explore cicada’s mysterious link with primes

 

Very Big and Very Small Numbers

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What’s the biggest number you can think of? Depending on a kid’s age, they may answer 100 or a million or infinity (infinity is a concept, not a number). What’s the smallest number you can think of? Sometimes kids answer this with 1 or zero or a negative number. With elementary aged kids, I often have to remind them that there are many numbers possible between 1 and 0 (the fractions or decimals).

The problems with very big and very small numbers are (1) How do you express them? and (2) How do you conceptualize or understand them?

1. To deal with the first problem, we have scientific notation. Scientific notation is a way to write numbers that are too big or two small to write in the conventional decimal notation. (If you are doing a calculation using very big or very small numbers, it can be inconvenient to write all those zeros.) As an example, 3,000,000 written in scientific notation is 3 x 106, because this is the same as 3 x 10 x 10 x 10 x 10 x 10 x 10. Similarly, a small number such as 0.000003 would be written as 3 x 10-6, because it is the same as 3 / (10 x 10 x 10 x 10 x 10 x 10).

The table below shows some numbers in decimal notation, their American names (Did you know that a billion does not mean the same thing in all countries?), and the scientific notation. Do you have a large number that you don’t know how to name?  Try this interactive site.

ScientificNotationTable

2. To address the second problem with very large and very small numbers – how to conceptualize them – I like to have kids play with this fun interactive website: Scale of the Universe 2. This site shows the relative sizes of things from theoretical strings and quantum foam (on the order of 10-35 meters) to the observable universe (on the order of 1027 meters). Seeing the differences in scale is pretty mind-blowing, and students enjoy finding out that the site was developed by a 14-year-old boy with help from his twin brother.

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Screenshot from Scale of the Universe 2

Pi Video Round-up

I’ve been looking for some good videos to show the kids in our after-school math club. (We are lucky that we meet on Pi Day this year.) So, I thought I would post what I had found.

This short video explains that pi = circumference/diameter:

Another brief video. This one tells about some of the places, outside of circles, that we find pi:

This one is a bit longer. The first 10 minutes give a good history of pi, but the last 5 minutes devolves into a series of bad song and TV show clips:

This video (with a very old-school feel) has some nice animations illustrating the value of pi:

All of Vi Hart’s math videos are great.  Here are 2 of them involving pi:

Here is the video by Numberphile, explaining that you only need 39 digits of pi:

And, of course, the pi song:

39 Digits of Pi

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The more digits of pi you use, the more accurate your calculation. For example, using 3.14 is more accurate than using a value of 3. The value of pi has been calculated to trillions of digits (the record at the time of this post is a little over 10 trillion digits). However, the push for more and more digits of pi has more to do with a desire to break records and test computer algorithms than any need to be more accurate. It would never be necessary to use more than a few digits of pi. In fact, you only need 39 digits of pi to calculate the circumference of the known universe to an accuracy that is within the width of a single hydrogen atom.

Pi rounded to 39 digits: 3.14159265358979323846264338327950288420

Numberphile explains the calculation in his video:

Pop Culture Pi

  • In the Star Trek episode, “Wolf in the Fold,” the ships computer is taken over by an evil entity named Redjac. Spock thwarts Redjac, by commanding the computer to “compute to last digit the value of pi”.
  • In Carl Sagan’s novel Contact, scientists compute pi to a record number of decimal places and find a pattern in its digits believed to be a message from the creators of the human race.
  • During the O.J. Simpson trial, FBI agent, Robert Martz, misstated the value of pi, undermining his value as a witness.
  • In Darren Aronofsky’s 1998 psychological thriller Pi, the main character is driven mad by his attempt to find a number that will unlock the patterns in nature.
  • In an episode of The Simpsons, Professor Frink gets the attention of a group of scientists by shouting Pi is exactly 3!”.