Topics in Algebra
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Following are some topics in algebra.

Contents of this Page
Algebra in Nature
A History of Algebra Symbols
Boolean Algebra
The Hailstone (3n + 1 ) Problem
Functions
Half-Life
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 


Algebra in Nature

 

Over the centuries, as mathematical concepts have developed, mathematicians have discovered links from their work to nature. Here are a few topics with their link to the natural world.

Fibonacci Numbers

When we look a Fibonacci Numbers, we can quickly see the pattern.

1, (1+0) 1, (1+1) 2, (1+2) 3, (2+3) 5, (3+5) 8, ....

Many flower species have been found that produce petals that follow this sequence.
For example:
 Enchanter's Nightshade flowers = 2 petals
 Lilies = 3 petals
 Wild Geranium = 5 petals
  Delphinium = 8 petals
 Corn Merigold = 13 petals

Also, pineapple scales and pine cones spiral in two different directions. The number of spirals are Fibonacci numbers.
 Pineapple = 5 & 8, 8 & 13
 Pineapple = 5 & 8, 8 & 13

The Golden Ratio

The regular pentagon bears a very close relationship to the Golden Ratio. When you draw two diagonal lines from each verticee, you form Golden Triangles. This forms a pentagram.

In nature we find a variety of examples that hold true to this concept. Five petaled flower blossoms have the shape of a pentagram. There are more flowers that have five petals than any other number of petals. If you measure the distance from the tip of one petal to the tip of a nonadjacent petal and then divide that distance by the distance between two adjacent petal tips, you will get an approximation of the Golden Ratio.

This is also true of the star fish, the cross section of an apple seedbed, and the sand dollar.

Finite Space

The packing industry has surely spent much time and effort trying to find the best was to pack products into boxes for shipment. The goal usually is to allow for the least amount of wasted space and hold maximum capacity. All along all they had to do was turn to bee keeping.

Bees have chosen what appears to be the most efficient and economically shaped packing container, a regular hexagonal prism. When calculating the densities of this tessellation and comparing it with those of a square prism or an equilateral triangular prism, you will find the bee made the correct choice by sticking with the regular hexagonal prism.

Contributed by Jennifer Garretson

References

Dalton, LeRoy C. Algebra in the Real World, Palo Alto, California, 1983.


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A History of Algebra Symbols

Algebra, often described as the language of mathematics, is a branch of mathematics in which letters are used to represent basic arithmetic relations. Classical algebra is concerned with solving equations, using symbols instead of specific numbers and using arithemetic operations to establish ways of handling symbols.

Though the beginnings of some algebraic symbols are not completely clear, history shows us that ancient civilizations wrote out algebraic expressions using only occasional abbreviations. The early Egyptians had symbols for addition and equality. The Greeks, Hindus, and Arabs had symbols for equality and the unknown quantity. However, during that time there was such limited use of math symbols that working with mathematics was very cumbersome. Mathematical processes were either written out in full or indicated with word abbreviations.

The later Greeks, Hindus and Nemorarius Jordanus, a German born mathematician, expressed addition by juxtaposition. The Italian algebraists eventually began to use the initial letter P or p , sometimes with a line drawn through it for plus. The practice, however, was not uniform. Some mathematicians denoted plus by , and sometimes by e, and the mathematician Niccolo Tartaglia commonly used . In 1489, the German mathematician Johann Widman first introduced the symbol +, referring to it as "signum additorum" and used it only to indicate excess. By 1630 the + sign was a part of the recognized notation of algebra and was used as a symbol of operation.

Subtraction was indicated by the Greek mathematician by an inverted and truncated or . The Hindus used a dot while the Italian algebraists denoted minus by M or m, sometimes with a line drawn through the letter or sometimes by . The German and English algebraists introduced the present symbol - and described it as "signum subtractorum." The symbol was in general use as a symbol of operation about 1630.

The English mathematician William Oughtred, in 1631, first used the symbol x for multiplication, while Thomas Harriot used a dot. The German Mathematician Gottfried Wilhelm Leibnitz used a period to indicate the operation and French mathematician Rene Descartes used juxtaposition in 1637. In 1686 Leibnitz used the sign to denote multiplication and to denote division. The Arabic notation for division had been in the form of a fraction with a line such as a - b, a/b, or . Leibnitz used the colon (:) in the form of a ratio a:b. The current symbol for division , which is a combination of the - symbol and the : symbol, was used by Johann Heinrich Rahn at Zurich in 1659 and by John Pell in London in 1668.

In 1557, the current symbol of equality = was introduced by the English mathematician Robert Recorde in his Whetstone of Witte, "because no two things can be more equal than parallel lines." Wilhelm Xylander used two vertical lines in 1575. But for the most part, the word was written at length until the year 1600.

The symbols for greater than and less than, > and < , originated with Thomas Harriot in 1631, eventually replacing the symbols and introduced by Oughtred for the same purpose.

Rafaello Bombelli is credited with the earliest symbolic notation for exponential notation in 1572. He represented the unknown quantity by , its square by , its cube by , and so on. In 1586, simon Stevinus used , , , etc. in a similar manner. In mathematical writings published in 1634, P.Herigonus wrote a, a2, a3, ...for a , a2 , a3 ,... . Rene Descartes presentd the idea for the modern day notation of using exponents to mark the power to which a quantity was raised.

English mathematician John Wallis explained the meaning of negative and fractional exponents in 1659 and first employed the symbol for infinity in 1655.

Contributed by Karolee Weller

References:

  1. Ball, W.W. Rouse. (1960). A Short Account of the History of Mathematics.New York, NY: Dover Publications Inc.
  2. Bowen, J. (1995, April) . Early English Algebra [On-line] . Available: http://www.comlab.ox.ac.uk/oucl/users/jonathan.bowen/algebra/section3_2.html
  3. Bowen, J. (1995, April) . The Origins of Algebra [On-line] . Available: http://www.comlab.ox.ac.uk/oucl/users/jonathan.bowen/algebra/section3_2.html
  4. Boyer, C. (1994) . Mathematical Symbols. Microsoft (R) Encarta Encyclopedia 94. Funk & Wagnalls Corporation.
  5. Dauben, J., & Berggren, J. (1994) . Algebra. Microsoft (R) Encarta Encyclopedia 94. Funk & Wagnalls Corporation.

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Boolean Algebra

The predecessor to symbolic logic was classical logic more commonly referred to as Aristotelian logic.

Aristotle's (384-322 B.C.) logic held three principles.

  • The principle of identity: A thing is itself: A is A
  • The principle of the excluded middle: Is either true or false: either A or not A.
  • The principle of contradiction: no proposition can be both true and false. A cannot be both A & not A.
Aristotelian logic dominated scientific reasoning in the Western world for 200 years.

In 1847 the English mathematician and logician George Boole (1815-1864) demonstrated in his Mathematical Analysis of Logic that a system of algebra can be used to express logical relations. Originally devised (Gullberg, 1997) as a system for logical reasoning, Boole developed symbolic logic as a means of clarifying difficult Aristotelian logic.

His system has been used as a tool to help sound reasoning. Boole’s basic idea was that if the simple propositions of logic could be represented by precise symbols, the relations between two propositions could be read as precisely as an algebraic equation (Life, 1963).We refer to it today as Boolean Algebra or symbolic logic.

Boolean algebra (Gullberg, 1997) is concerned with ideas or objects that have only two possible stable states - e.g. on/off, closed/open, yes/no, true/false. Although it was devised by Boole in 1847.

In the late 17th century the foundation for symbolic logic was made by the German mathematician and philosopher G. W. von Leibniz, (Gullberg, 1997) who tried to find a lingua universali, a language where errors in thinking would be equivalent to arithmetical errors.

Gottlob Frege (1848-1925) from Germany, founder of modern symbolic logic, in 1879 constructed an elaborate logico-mathematical system, (Gullberg, 1997) now known as predicate logic. He was not interested in higher mathematics but, was rather interested in philosophy, logic, and law which were the bases of his career as a diplomat. His goal was to reduce all truths of reason to a simple system of arithmetic lead into infinitesimal calculus.

Bertrand Russell (1872-1970) in the early 1900’s carried on the work of Frege. Together with Alfred North Whitehead tried to derive mathematics from self-evident logical principles. They did not reach their goal.

Kurt Godel (1906-1976) a Czech-Austrian-American mathematician and logician published his famous incompleteness theorem that states any consistent formal system adequate to describe arithmetic must contain statements which can either be proved nor disproved within this system. He showed that there is no systematic way to list the true statements in arithmetic.

Boolean algebra (symbolic logic) remained dormant until the middle of the 20th century. In the 1950’s (Life, 1963) it was used for telephone switching units and the new up and coming electronic computers.

Symbolic logic is used not only in genuinely logical or mathematical domains but also in the natural sciences, and in disciplines such as linguistics, law, and computer technology.

Today Boolean algebra is used everyday to help people when doing searches on the Internet. It is more commonly referred to as a Boolean search. The three Boolean operators used today are as follows: AND, OR, NOT

When you search the internet for specific words you will get many, often thousands of hits for the words your typed in and still your frustrated because it isn't what you want. The reason for that is the default setting on many search engines, such as AltaVista, Excite, Infoseek, and MetaCrawler us the Boolean operator OR.

What is actually happening is when you type in { lesson plans} your search engine will give you sites with the word lesson and others with the word plans. Since OR is the default it is assuming you want either lesson or plans not (lesson plans).

To modify your search to be more specific use AND between words if you are interested in dyslexia in teens type: dyslexia AND teens. This will give you sites that have both words in their titles.

You will want to use the Boolean operator NOT when you want to search for say… radiation yet you don't want nuclear radiation. You will type: radiation NOT nuclear and it will give you just the radiation sites.

To find out more: [Boolean Searching on the Internet]

Note: Happy Searching

Contributed by A. Motter

References:

  1. Bergamini, David. (1963) Mathematics. New York, NY: Time Life Books
  2. Gullberg, Jan. (1997) Mathematics: from the birth of numbers. New York, NY: Norton Co.

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Hailstone Numbers

Choose a positive integer. If your number is even, divide it by two. If your number is odd, multiply it by three and add 1. Take your new number as the starting number, and repeat until you can't go any farther. The process sounds easy enough, but how to predict what will happen may be less obvious. Welcome to hailstones.

Hailstone numbers are generated by the simple mathematical process described above. The process is called the 3n + 1 problem, and is also known as the Collatz problem or the Syracuse problem. A small integer value of n can demonstrate the process. Take n = 5, for example, and the sequence is 5, 16, 8, 4, 2, 1, 4, 2, 1… The pattern repeats at 4, 2, 1. To compare hailstone sequences from different starting numbers, it is useful to note both the largest value attained, and the number of computations needed to reach 1. For n = 5, the largest value is 16, and the number of computations is 5. To compare, change n by a small amount and note what happens to the sequence. The pattern for n = 6 is 6, 3, 10, 5, 16, 8, 4, 2, 1. Here the largest value is again 16, but the number of steps is 8. For n = 7, the pattern of numbers produced is 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1. For n = 7, the largest value is 52, and the number of steps is 16. How high will hailstone numbers go? And just how many steps are needed?

Some larger examples may show greater variation attained in hailstone number sequences. Let n = 25. The pattern is 25, 76, 38, 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1. Here the maximum value is 88, and the length is 23. It may be expected that n = 26 will produce an even higher value, and a longer sequence. This is where the surprises of hailstone numbers become more obvious. Let n = 26. The sequence is 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1. The maximum value is 40, and the number of computations only 10. To be surprised again, try to compute the sequence for n = 27. The sequence has a maximum value of 9,232, and takes 111 computations to reach 1!

So far, it appears that the maximum values and numbers of computations needed are in some way related to the size of the starting number. Perhaps more notable is the contrast between even and odd starting numbers. There is, as yet, no complete understanding of the apparently chaotic nature of the sequences produced by this simple procedure. Also, another puzzling question remains.

Will any starting number used for a hailstone sequence eventually reach 1? From the examples above, it would appear so. Certainly the hailstones these numbers are named for have somewhat predictable behavior. Hailstones go up and down inside of clouds, gaining size and mass with each pass through. Then, when heavy enough, hailstones fall out of the clouds and tumble to earth. It seems that our hailstone numbers have their own earth. They increase and decrease, and eventually fall to the value of 1. Don't they?

Simple as it may seem, this 3n + 1 question remains unsolved. Studies have been carried out to try larger and larger starting numbers, and so far each number tried has led to the repeating cycle of 4, 2, 1. No one has been able to prove, however, that every number used as the starting number for a hailstone sequence will eventually reach 1. While the sequences can be used to study many patterns and comparisons, the final question may remain unanswered.

Contributed by Laurie Kiss

References:

  1. Banks, Robert. (1999). Slicing pizzas, racing turtles, and further adventures in applied mathematics. Princeton, NJ: Princeton University Press.
  2. Hailstone numbers - the 3N+1 problem. Available http://www.frii.com/~dboll/hailston.htm [2000 June 11]
  3. Magic pi - the magic of numbers. Available http://www.cs.man.uk/aig/staff/toby/writing/PCadvisor/numbers.htm [2000 June 11]
  4. Mathematical mysteries - hailstone sequences. Available http://pass.maths.org.uk/issue1/xfile/index.html [2000 June 11]

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Functions

One of the most fascinating topics in middle school, junior high, or senior high mathematics today is that of a function. Although the definition and uses of the term differ somewhat today, there is still a need for the concept. The term function was first used by Rene’ Descartes, a French philosopher, scientist, and mathematician, in 1637 to designate a power x^n of a variable x. Gottfried Leibniz applied the term to various aspects of a curve in 1694. Peter Dirichlet, in 1829, conceived of a function as a variable y having its value fixed or determined in some definite manner by the values assigned to a variable x or to several values of x. At the time, the values for the variables were real or complex numbers.

There was a need to develop ideas in the days of Descartes, Leibniz, and Dirichlet to better understand their surroundings. Dirichlet studied progressions, evaluating integrals, gravitational attraction, the solar system, harmonic motion, and trigonometric series.

The concept of function today compared to the times of Descartes, Leibniz, and Dirichlet has been altered. By definition, a function is a correspondence, or rule, that pairs each element of a set (the domain) with exactly one element of another set (the range). An alternative definition is that a function is a set of ordered pairs in which no two ordered pairs have the same first component. It is interesting to note that the elements of the sets need not be numbers and that the elements of the domain do not have to be of the same type as those of the range.

Although the uses for the concept of functions at the high school level or below is not as involved as Dirichlet applied them, there is a great need to study them. A tremendous vocabulary of mathematical terms evolves. Learning the meaning and uses of words such as domain, range, graph, vertical line test, increasing/decreasing, maximum/minimum, even/odd, intercepts, symmetry, translations, reflections, and others contributes greatly to the understanding of algebra and other math courses. Understanding the graphs of functions relates to the ability to use other graphs in other subjects concerning relationships between knowns or unknowns. Basic operations on functions, compositions of functions, or piecing functions together produce other more complex and interesting functions.

There are so many kinds of functions today that can be studied at the various levels. Linear, quadratic, polynomial, constant, trigonometric, logarithmic, exponential, step, inverse, and piecewise are some to choose from and apply to their level. Applications of the variations of direct, indirect or inverse, joint, and combined can be used. There are applications in business such so cost and profit functions. The sciences use Charle’s Law, the Ideal Gas Law, Kepler’s Law, Hooke’s Law, to name a few that can be written as functions.

Functions have application in all walks of life such as business, science, mathematics, homemaking, sports, and construction, just to name a few. So often there are those correspondences between sets of objects and functions deal with these. Much of the language of algebra can be learned from the study of functions. Many of the manipu- lations we do with numbers can be learned and reinforced while learning of functions. There are so many real-life situations that can be defined, solved, and understood if they are related to functions. Functions can be graphed, thereby visualized, and other concepts and discoveries can be observed and made.

Contributed by Rodney S. Karjala

References:

  1. Aufman, Barker, and Nation. College Algebra, 2nd Edition., Houghton-Miflin, Boston, 1993
  2. Dolciani, Brown, Cole. Algebra, Houghton-Mifflin, Boston, 1988
  3. Funk and Wagnalls New Encyclopedia, Book 11, USA, MCMLXXI
  4. http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Dirichlet.html

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Half-Life

What is half-life? Scientists use it to define radioactive elements. The time it takes an element to lose 50% of its radioactivity is its half-life. The most commonly known use of half-life is radiocarbon dating. The radiocarbon method of dating was discovered by Professor Willard F. Libby and his team of scientists. Professor Libby won a Nobel Prize in Chemistry in 1960 because of his work with radiocarbons.

All living organisms are made mostly of carbon and continue to maintain their carbon levels while they are still living. Once they die they stop using carbon and the carbon begins to decay. One of the types of carbon is carbon-14. C-14 is called radiocarbon because it is radioactive, which means we can measure its half-life. Professor Libby found that it takes 5568 years for half of the carbon-14 to disappear. Therefore its half-life is 5568 years. So every 5568 years half of the remaining carbon in a sample would disappear. After about 50,000 years all of the radiocarbon will have disappeared. So carbon-14 dating cannot date anything older than 60 or 70 thousand years. Often because of the difficulty in dating materials older than 50,000 years scientists will just classify something as older than 45,000 years or older than 50,000 years.

Anything that was once living can be carbon dated, here are some examples:

  • Charcoal, wood, twigs, and seeds
  • Lake mud and sediments
  • Peat
  • Bone
  • Leather
  • Soil
  • Hair
  • Pottery
  • Metal casting ores
  • Wall paintings
  • Pollen
  • Ice cores
  • Meteorites
  • Coral
  • Bird eggshell
  • Blood residues
  • Fabrics and textiles
  • Paper and parchment
  • Resins and glues
  • Insect remains
  • Antler and horn
  • Water

To test his method Professor Libby carbon dated objects whose dates were known and compared his results. The test proved that Professor Libby had discovered an accurate way to date articles of unknown origin. Because of more discoveries in the 1970's the sample size needed to carbon date an object have grown smaller. You can now use radiocarbons to date something as small as a grain of rice or a strand of hair.

Several famous artifacts have been dated using carbon-14 dating. The Dead Sea Scrolls were a famous archaeological find that was carbon dated. The Iceman, a frozen body found in Northern Italy in 1991 was radiocarbon dated and he was found to be almost 5500 years old, but perfectly preserved in a block of ice.

Student Questions?

  1. So let's pretend that you have found an artifact that has only 6.125% of its carbon-14 remaining, how old is the sample?
  2. What if ¼ of the carbon was remaining, how old is the sample?
  3. What if you knew that a piece of wood was 11,100 years old, approximately how much of the radiocarbon would you expect to find remaining in the sample when you applied carbon dating? (Express your answer as a percentage)
  4. What if one-thirtysecond of the carbon was remaining, how old is the sample you have?
  5. What if the wood was known to be 16,700 years old, how much carbon-14 would you expect to find remaining? (Express your answer as a fraction)

Contributed by Nora Rayl

References:

  1. "Dating Methods," Microsoft® Encarta® Online Encyclopedia 2000
  2. http://encarta.msn.com © 1997-2000 Microsoft Corporation. All rights reserved.
  3. http://c14.sci.waikato.ac.nz/

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