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Note: the symbol 
means "for all"
and the symbol 
means "there exists".
Groups are an important mathematical structure which form the basis of the study of abstract algebra, known to mathematicians just as algebra. The axioms above depend of the concept of a set G with elements a, b, c, etc. and one operation (• above) such as addition, multiplication, reflection, etc.
Note how the familiar set of natural numbers are closed under both addition and multiplication. Both multiplication and addition are associative, and each has an identity element. The additive identity element is zero (0), whereas the multiplicative identity element is one (1).
Group axiom 4 requires inverses. We have seen our number system grow from natural numbers to integers when the operation of subtraction or additive inverses was introduced. When the operation of multiplication is used and the concept of multiplicative inverses is required, the concept of division is the result and the number system must expand to include fractions.
An important restriction, the fact that 0 has no multiplicative inverse, will be developed later in lesson 8. We thus see that the integers form a group under addition, but not under the operation of multiplication!
| A rational number is a number which can be expressed as the ratio of two integers. |
The set of rational numbers is denoted by Q, as in quotient.
| A vinculum is an overhead line as is used for fractions, radicals, and for repeating decimal fractions. |
| The numerator is the portion of a fraction above the vinculum. |
| The denominator is the part of a fraction below the vinculum. |
| Percentage is the numerator of a fraction with a denominator of 100. |
| Millage or permille is the numerator of a fraction with a denominator of 1000. |
Percentages are written with a percent sign (%) and permille are written with a permille sign (‰). Similar higher order fractions are parts per million (ppm), parts per billion (ppb), and parts per trillion (ppt).
| A unit fraction is a fraction with a numerator of 1. |
Historically, unit fractions were the first to be developed. Ancient Egyptians would add long series of unit fractions to generate other values. It was a historic event when 2/3's came into usage! An application of unit (Egyptian) fractions will be examined in the homework. Today, fractions come in many forms: mixed numbers, improper fraction, decimal fractions, etc..
| An improper fraction has a numerator larger than the denominator, a proper fraction does not. |
An interpretation of improper fractions is that the denominator says how each whole piece is divided, and the numerator says how many total pieces we have. Improper fractions are quite acceptable in high school and beyond and are, in fact, often the preferred form of answer. Too bad elementary/middle school teachers always consider them wrong! However, in their defense, for those less numerically inclined, converting to a mixed number may give a better sense of the number's magnitude. (Converting to a decimal approximation doesn't necessarily do that so clearly!)
| A mixed number has an integer part and proper fraction part. |
A mixed number is generated by dividing the denominator into the numerator to determine how many whole parts there are. The remainder is the numerator of the fractional part.
| A complex fraction has fractions in the numerator or the denominator. |
| Partial fractions describes a technique for splitting a fraction into pieces. |
This technique will be more formally introduced in Algebra II and is
often used in Calculus to simplify a complex expression for ease in integration.
| 5 | = | -49 + 54 | = | -7 | + | 6 | and | 5x - 1 | = | 2 | + | 3 |
| 63 | 63 | 9 | 7 | x2 - x - 2 | x + 1 | x - 2 |
The following is an example of a continued fraction. |
Continued fractions
can arise due to recursive definitions.
Consider the example above as the solution to the equation:
x=2+1/x. Early methods of expressing and extracting square roots
depended on this method so it was well developed. It can also be useful for
finding integer solutions.
|
OR |
|
Yet this, perhaps in a slightly complicated situation, is a very common mistake.
Our Algebra II book calls it "freshman cancellation"!
Consider what disaster happens when this was done to the examples below.
If in doubt, try letting x=2 and compare the before and after results.
|
OR |
|
| 2 | + | 4 | |
| 3 | 5 |
| 2 ·5 | + | 4 ·3 |
| 3 ·5 | 5 ·3 |
| 10 + 12 | = | 22 | = | 1 7/15 |
| 15 | 15 |
| 10 | • | 22 | = | 220 | = 4 |
| 11 | 5 | 55 |
Of course, after you are done multiplying (or adding, etc.),
you should always simplify!!!
Another way to do it is to reduce as you go:
| • | = | 4 | = 4 | ||
| 1 |
| The reciprocal of a number is it's multiplicative inverse. |
For fractions, this can be obtained by exchanging the numerator with the denominator. The -1 key on the calculator does this as well. Whole numbers are nonnegative fractions with a denominator of 1. (Thus unit fractions are the reciprocals of whole numbers.) Division is equivalent to multiplying by the reciprocal. On many very early computers, this was the only form of division implemented! For example: 2/3 divided by 1/6.
The reason can be seen by simplifying the complex fraction.
Proportions are two or more ratios set equal: 2/6 = 1/3 = 12/36. If a proportion has a missing term, we can simply cross-multiply and solve for the missing term. For example:
| x | = | 1 | becomes | 4x = 16 which gives x = 4 |
| 16 | 4 |
| BACK | HOMEWORK | ACTIVITY | CONTINUE |
|---|
G,
then a•b