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Numbers and Their Application - Lesson 8
More on Fractions
Lesson Overview
We have already assumed that multiplication occurs before
addition and exponentiation before that in our prior lesson on bases:
314 = 3×102 + 1×101
+ 4×100. We will summarize these rules here as follows.
- Operations within symbols of inclusion are done first.
- Exponentiation is done next right to left if stacked.
- Multiplication and Division are then performed left to right.
- Addition and Subtraction are next performed left to right.
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The most common symbols of inclusion are called parentheses (), but brackets
[], braces {}, vincula (plural of vinculum), and others (absolute value, radicals) are also
encountered. Some discussion regarding order of exponents is in order.
Although mathematicians for centuries have clearly intended
223 = 28 = 256 and not
43= 64, programming languages such as FORTRAN and C
and graphing calculators have not been as consistent.
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Be sure to use parentheses whenever encountering stacked exponents. |
The rules above are often remembered via the mnemonic (from the Greek meaning
a memory aid): PEMDAS- Please Excuse
My Dear Aunt Sally or
Please Eat Miss Daisy's AppleSauce.
Rule number 3 above deserves a little more ink since really only purists,
computer scientists, algebraic calculators, and perhaps high school teachers
seem to rigorously adhere to it. Consider expressions such as
3/2![[pi]](pi.gif)
or
3/2
where implicit multiplication might occur.
Some textbooks, especially those beyond the high school level,
might assume the 2 is first multiplied by the
in the first example, but not in the second.
It is for this reason that I highly recommend against the use of a solidus (/)
and for the use of a vinculum (-) especially when handwriting fractions.
"No authority decrees this, ...[but] this one rule [multiplication
indicated by juxtaposition is carried out before division]
is not universal agreement at the present time, but probably is
growing in acceptance." (Dr. Math)
When a student answer is an order of magnitude too large
I quickly check to see if a
in the denominator wandered above due to the lack of parentheses.
One can add to this the lack of agreement beyond the high school level
in evaluating -1n.
A common convention for organizing our numbers is to use a number line.
Some number line conventions will be noted as follows:
- A number line has larger numbers to the right and
smaller numbers to the left. At its center is zero.
- The integers are usually marked off with tick marks and labeled.
- Since numbers go on forever, but paper doesn't, arrows are put on each end.
Number lines can be used to show the solution set to certain problems,
especially those with infinite solution sets.
A sample number line is diagrammed below.
Mathematics deals not only with equality (=) but also with five
inequalities (<,
![[less than or equal to]](le.gif)
,
![[not equal]](ne.gif)
,
![[greater than or equal to]](ge.gif)
, and
>, known as less than, less than or equal to, not equal,
greater than or equal to, and greater than.
The big end or opening points toward the bigger quantity. (The alligator
is eating the big one, some of my students tell me.)
Two of these (<, >) are known as the strict inequalities,
because they do not include the end points.
All but
are called order inequalities. Number lines are useful to convey
such things as x > 2. To do this, another number line convention
should be noted.
- If a point is to be excluded at
the end of a group of numbers on the number line, an open circle is used.
Thus, a closed circle indicates inclusion of the endpoint.
Alternatively, a parenthesis is used to indicate exclusion and a bracket
to indicate inclusion. This convention is rooted in the practice of
specifying intervals as open, closed, or even half-open, such as 2 < x
5 as (2,5].
Inequalities are algebraically treated much like equalities
(what you do to one side, do also unto the other).
However, when an inequality is multiplied or divided by a negative number,
the direction the inequality points is reversed.
Inequalities involving absolute values are equivalent to
compound inequalities and are handled
similarily.
Example:
| 1 - x | > 2 |
| -x | > 1 | | subtract 1 from both sides |
| x | < -1 | | multiply by -1 both sides and reverse the inequality |
Division is usually the last of the four basic operations
(+, -, ×, ÷) to be mastered. Division is the inverse operation
of multiplication, but has an important exception as discussed below.
The division of one number by another can be represented as a fraction
with the dividend as the numerator and the divisor as the denominator.
One can simplify the fraction before doing the long division involved.
(Reminder: The divisor is out in front of the "box", the dividend is under
it and the quotient is on top of the "box").
|
____ ____ ____.____ ____ |
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| ____ ____ | ____ ____ ____ ____
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An example of a division problem is 441÷12.
After reducing, this is the same as 147÷4 or the fraction
147/4.
To find the quotient (or to find its mixed number), we divide thusly.
| 36 .75 |
(36 R3 or 36 ¾) |
| 4 | | 147 | .00 |
| 12 | |
| 27 | |
| 24 | |
| 3 | 0 |
| 2 | 8 |
| | 20 |
| | 20 |
| | 0 |
Fractions are often expressed with fairly arbitrary denominators:
1/2, 3/4, 2/3.
To compare them in magnitude, it is helpful to line them up on a number line:
1/2 < 2/3 < 3/4. To quantify the
difference between them, it is helpful to change the denominator to be
10 or a power of ten. Such fractions are called decimal fractions
or often just decimals.
1/2 = 5/10 = 0.5
2/3 = 0.66666...
3/4 = 7.5/10
= 75/100 = 0.75
So 2/3 is closer to 3/4
than to 1/2. Of course, if we obtained a common
denominator of 12, that would have been clear as well:
6/12
< 8/12 < 9/12.
The choice of base 10 is very common, although
basimal fractions related to powers of two are commonly
encountered with computers. In fact a marvelous
algorithm for calculating
was recently discovered, but involves hexadecimal fractions only.
The number of digits in the repeating unit of a nonterminating but
repeating decimal fraction is an area of
interesting study. The biggest
unit fraction (i.e. smallest denominator) with much interest is
1/7 = .142857142857....
As can be seen in the table below, all multiple of 1/7
have the same digits in the same order, just a different starting point.
In an earlier homework, you already did the equivalent of finding the decimal
fraction for 1/7 (7 into 1,000,000; NL1).
Note how there can be seven different remainders (0-6) when dividing something
by 7. However, if the remainder of 0 is obtained, the fraction terminates
(i.e. 7/7 = 1.0). This is
part of the reason
the cycle length is six for the fraction 1/7.
In today's activity you will derive the exact decimal fractions for
1/17 and 1/19
which exceed the calculator's accuracy. Of course you could also attempt this
by long division like your teacher did since calculators were not common until
he was in high school.
Terminating decimals are decimals that have an ending. These numbers do not go
on forever or repeat. They are clearly rational numbers since you can express
them as the ratio of two integers: the decimal values over the power of ten
(what the last digit of the decimal represents).
Don't forget to reduce, because this result is not unique.
For example, you could multiply the numerator and denominator by 2.
It should be clear that fractions with denominators containing only
powers of 2 and 5 (the prime factors of our base 10) terminate, whereas
those containing other prime factors do not.
| 45.336 = |
45 | 336 |
= 45 |
42 |
| 1000 | 125 |
|
Knowing all repeating decimals are rational numbers,
or the ratio of two integers, leaves us with the task of
finding these integers when presented with an arbitrary example.
Suppose you are asked to find two integers whose ratio is 0.586586586586....
One way is to use the FRAC key on your calculator, but another involves
just a little algebra.
| Let | 1000x | = 586.586... |
| Where the 1000 was chosen since it is 103 |
and there are three digits in our repeat unit. |
| |
|
| Subtracting off | 1x |
= 0.586 |
| We are left with | 999x |
= 586.000... |
| or | x | = 586/999. |
For fun, you might try this method on 0.143434343!
We stated in the previous lesson
that zero does not have a multiplicative inverse.
This is equivalent to the concept that zero multiplied by anything is always zero.
If we examine this further, we discover that sometimes things are not
quite exactly zero and if multiplied by something big enough, unity will result.
Examine the sequence of 0.1•10 = 1; 0.01•100 = 1; 0.001•1000 = 1; ....
Next examine the same thing but as a division problem:
1÷0.1 = 10; 1÷0.01 = 100; 1÷0.001 = 1000; ....
The denomonator approaches zero and the quotient approaches
![[infinity]"](infty.gif)
.
However, if we approach zero from the other side:
1÷-0.1 = -10; 1÷-0.01 = -100; 1÷-0.001 = -1000; ....
the result is at the other "end" of our number line.
For this reason, it is usual to call division by zero undefined (ill-defined).
For some applications, it is useful to join our number line at
the two infinities, thus closing our unbounded interval!
Thus the complete number line (interval between plus and minus
infinity) is termed both open and closed.
Another important consideration is how many rational numbers are there?
The answer may surprise you.
Start by listing the natural numbers with one as a denominator.
For every successive row, increase the denominator. Then you will have
completed a chart containing all the positive rational numbers.
| 1/1 | 2/1 | 3/1 | 4/1 | 5/1 | ... |
| 1/2 | 2/2 | 3/2 | 4/2 | 5/2 | ... |
| 1/3 | 2/3 | 3/3 | 4/3 | 5/3 | ... |
| 1/4 | 2/4 | 3/4 | 4/4 | 5/4 | ... |
| 1/5 | 2/5 | 3/5 | 4/5 | 5/5 | ... |
| 1/6 | 2/6 | 3/6 | 4/6 | 5/6 | ... |
| ... | ... | ... | ... | ... | ... |
Some of them appear more than once
(1/2 = 2/4 = 3/6....).
We then count the fractions in this order:
1/1, 2/1, 1/2, 1/3, 2/2, 3/1....
Since we have put the natural numbers into a one-to-one correspondance
with the positive (unsigned) rational numbers, they are countable or
there are "just as many" as natural numbers.
This is commonly recognized as the
lowest order of infinity,
termed (
0) or aleph null,
after the first letter of the Hebrew alphabet.
There are other arrangements possible, such as sorted by "height"
(numerator plus denominator) then by numerator, for example.
However, fractions cannot be put in a strictly increasing
order, because in between each pair is always another!
The rational numbers are thus termed dense.
However, we will see later
there are still gaps between them!