Friday, June 5, 2015

How to Add or Subtract Two Fractions by the Butterfly Method

The butterfly method is used in Japan to teach addition or subtraction of two fractions. Japanese schools teach this method and encourage students to "forget" the butterfly drawing as they get better using the method.

To add or subtract fractions the butterfly way,

1. Write the fractions side-by-side as usual and draw two wings along the diagonals made by the numerator of one fraction and the denominator of the other fraction and draw an antenna on each wing.

2. As suggested by the wings, that look like a multiplication sign, multiply the numbers in each wing and put the product in the antenna for the wing.

3. Think or say: “This poor butterfly needs a body.” To give it a body, connect the bottom parts of the wings with a body-like loop and multiply the two denominators it connects, putting the product inside the body.

4. Add or subtract the numbers in the antennae in keeping with what is being done to the fractions and put the result over the number in the body.

5. If necessary, reduce or simplify the result.

Thursday, June 4, 2015

Just How Many Stars are There?

People have been asking this question for a long time. Back before telescopes were invented by a Dutch eyeglass maker, Hans Lippershey in 1608, the number of stars was what we could see with our naked (or unaided) eyes. This number is estimated as no more than 10,000 stars. This number varies according to how you count them, as only 1/2 the stars are visible from a single place on Earth and then only 1/2 the time. You can't see stars through the Earth and during the daytime!

After the telescope was invented, the number of stars we could see increased as we could see more of the fainter stars we couldn't see before. This number is probably about 1,000,000,000 (1 billion), but no one could actually count them all. Instead, Jérôme Lalande published the Histoire Céleste Française in 1801, which contained an extensive star catalog. The observations made were made from the Paris Observatory and so it describes mostly Northern stars. This catalog contained the positions and magnitudes of 47,390 stars, out to magnitude 9 (very, very faint), and was the most complete catalog up to that time. This would give only about 100,000 stars of magnitude 9 or better if the Southern stars could be counted.
What we call the Milky Way was known long before telescopes, but it was not known that is was made up of stars. The name Milky Way is from a Greek fairy tale involving a goddess and her milk. After telescopes became commonplace, following the efforts of Galileo, it was soon realized that the Milky Way is an unusually dense collection of stars, all lined up.
When people first started looking through a telescope, they noticed fuzzy spiral objects which were grouped with other such "nebulae" and with star clusters. These were at first considered interesting curiosities in the sky, but not to be confused with really interesting things such as comets.
Edwin Powell Hubble (1889-1953) demonstrated that most "nebulae" are objects outside our galaxy, using the 100-inch telescope at Mt. Wilson (in 1924). With this powerful instrument, he was able to measure the distance to the Andromeda nebula itself, observing stars within that neighboring galaxy. Thus, the enormous distance to this neighbor became clear: Hubble gave it almost a million light years. As it turned out, this was still quite a bit short of the truth. The distance is greater than 1.5 million light years. With this greater distance, the information coming in from the Andromeda could be interpreted correctly, and it turned out to be a sister galaxy, quite similar to our own.

 Andromeda Galaxy

We now know that there are about 300,000,000,000 (300 billion) stars in the Milky Way Galaxy alone. The Milky Way is larger than average, an average galaxy has about 100 billion stars.

In 2012, Hubble scientists produced the Hubble Extreme Deep Field image. The area searched was an area of clear sky that was less than 1/100th the area of the full moon. In that little rectangle, scientists were able to count over 10,000 galaxies. If this area is typical, there are 100 billion galaxies in the universe. An average galaxy has 100 billion stars, so there are 10,000,000,000,000,000,000,000 stars in the universe (or 1022 in scientific notation).

Wednesday, June 3, 2015

Large Numbers With Names

The following table lists names of large numbers which are found in many English dictionaries and thus have a special claim to being "real words". The "Traditional British" values shown are not used in American English and are becoming very rare in British English, but their other-language variants are dominant in many non-English-speaking areas, including continental Europe and Spanish speaking countries in Latin America.

          English also has many words, such as "zillion", used informally to mean large but unspecified amounts.

Standard dictionary numbers

NameShort scale
(U.S., Canada and
modern British)
Long scale
(continental Europe,
older British)

















Sexdecillion (Sedecillion)10511096


Novemdecillion (Novendecillion)106010114



In the list, scientific notation is used. For instance, a trillion is listed as  1012. This is spoken as "ten to the twelth" and written as 1,000,000,000,000. Notice that when we write big numbers, we put a comma between every third digit.

The googol Family

The names googol and googolplex were introduced in Kasner and Newman's 1940 book, Mathematics and the Imagination, in the following passage:
The name "googol" was invented by a child (Dr. Kasner's nine-year-old nephew) who was asked to think up a name for a very big number, namely 1 with one hundred zeroes after it. He was very certain that this number was not infinite, and therefore equally certain that it had to have a name. At the same time that he suggested "googol" he gave a name for a still larger number: "Googolplex". A googolplex is much larger than a googol, but is still finite, as the inventor of the name was quick to point out. It was first suggested that a googolplex should be 1, followed by writing zeros until you got tired. This is a description of what would actually happen if one actually tried to write a googolplex, but different people get tired at different times and it would never do to have Carnera a better mathematician than Dr. Einstein, simply because he had more endurance. The googolplex is, then, a specific finite number, equal to 1 with a googol zeros after it.
                   10100GoogolKasner and Newman, dictionaries
10googol = \,\!10^{10^{100}}GoogolplexKasner and Newman, dictionaries

If you wish to see a googolplex written out, here's a link you can click on, . Of course it comes in multiple volumes, each containing 1 million digits (mostly zeros) of the written out number. A googolplex is so big that if you could read 1 million volumes of 1 million digits in only a 1/1000 of a second, it would still take you MUCH longer than the age of the universe to finish.