150 lines
4.9 KiB
Plaintext
150 lines
4.9 KiB
Plaintext
Complements of NAC-Jack BBS from USENET
|
||
|
||
|
||
From the Journal of Irreproducible Results, Vol. 35, No. 1, without
|
||
permission. In exchange for that, you can subscribe by writing to them
|
||
at Blackwell Scientific Publications, Inc., Three Cambridge Center,
|
||
Cambridge, MA 02142, USA. Generally mildly humorous to outright silly.
|
||
(No affiliation, just a 20 year happy subscriber).
|
||
|
||
Note that since I cannot show overstricken or underlined characters, I
|
||
have taken to placing carats under the digits in question. Any errors are
|
||
almost certainly mine, and not the author's.
|
||
|
||
|
||
Simplified Mathematics
|
||
|
||
Ben Ruekberg
|
||
Cranston, Rhode Island
|
||
|
||
As John Allen Paulos points out in his book _Innumeracy, mathematical
|
||
illiteracy and its consequences_, "[some of the blame [for the inability
|
||
of students to do math, aside from `extreme intellectual lethargy' (p.89)]
|
||
...must ultimately lie with teachers who aren't sufficiently capable...
|
||
[themselves]."(p.75) He suggests that others, who are capable in math-
|
||
ematics, help instruct. Toward this noble end, I submit a convenient
|
||
and useful shortcut in dealing with fractions which are the bane of young
|
||
students.
|
||
|
||
While teachers often instruct students in simplifying fractions, they
|
||
usually restrict their instruction in this method to cancelling zeroes.
|
||
An example of this is:
|
||
|
||
10/100 = 10/100 = 1/10
|
||
^ ^
|
||
|
||
What they fail to point out is the much more useful expedient of
|
||
cancelling non-zero digits or groups of digits that lie adjacent to the
|
||
slash. Since this method may be novel to many readers and because of
|
||
possible confusion between the slash indicating "division" and the slash
|
||
indicating "crossing out," (and also because it is a great way to save
|
||
space) the cancelled digits will be underlined.
|
||
|
||
For the first example, consider the fraction 16/64. A "6" lies on
|
||
each side of the slash, and, thus, can be cancelled.
|
||
|
||
16/64 = 1/4
|
||
^ ^
|
||
|
||
Another example of one digit simplification of fractions is 19/95.
|
||
|
||
19/95 = 1/5
|
||
^ ^
|
||
|
||
The simplification can also be done with two digits which lie in the
|
||
same order on either side of the slash. In 133/3325, the 33's cancel.
|
||
|
||
133/3325 = 1/25
|
||
^^ ^^
|
||
|
||
Similarly, in 181/8145, the 81's on either side of the slash cancel.
|
||
|
||
181/8145 = 1/45
|
||
^^ ^^
|
||
|
||
The method works well with larger cancellations also, as illustrated
|
||
by a few random examples.
|
||
|
||
30405/40540 = 30/40
|
||
^^^ ^^^
|
||
(by conventional cancellation, 3/4)
|
||
|
||
1501/501334 = 1/334
|
||
^^^ ^^^
|
||
|
||
47641/641633 = 47/633
|
||
^^^ ^^^
|
||
|
||
467956/956955 = 467/955
|
||
^^^ ^^^
|
||
|
||
32727/272725 = 3/25
|
||
^^^^ ^^^^
|
||
|
||
129612/961289 = 12/89
|
||
^^^^ ^^^^
|
||
|
||
277777/7777756 = 2/56 = 1/28
|
||
^^^^^ ^^^^^
|
||
|
||
4848484/84848470 = 4/70 = 2/35
|
||
^^^^^^ ^^^^^^
|
||
|
||
7407407/407407385 = 7/385
|
||
^^^^^^ ^^^^^^
|
||
|
||
49999999999/99999999998 = 4/8 = 1/2
|
||
^^^^^^^^^^ ^^^^^^^^^^
|
||
|
||
This method works in simplifying improper fractions as well.
|
||
|
||
493991/99199 = 493/99
|
||
^^^ ^^^
|
||
|
||
4324/3243 = 4/3
|
||
^^^ ^^^
|
||
|
||
56504/5045 = 56/5
|
||
^^^ ^^^
|
||
|
||
67402/4024 = 67/4
|
||
^^^ ^^^
|
||
|
||
23828/8288 = 23/8
|
||
^^^ ^^^
|
||
|
||
At this point the ease of manipulation of this method for simplifying
|
||
otherwise daunting fractions must be obvious. But the skeptical reader
|
||
might believe that I have carefully picked cases that work, rather than
|
||
picking numbers at random. I will demonstrate with the cancellation of
|
||
two consecutive numbers and multiple examples using the second, all with
|
||
three-digit numbers cancelling; then with two fractions which simplify to
|
||
the same fraction, with four-digit numbers cancelling.
|
||
|
||
702 (followed by 7):
|
||
182702/7027 = 182/7
|
||
^^^ ^^^
|
||
|
||
703 (followed by 7):
|
||
133703/7037 = 133/7
|
||
^^^ ^^^
|
||
|
||
703 (followed by 57):
|
||
3703/70357 = 3/57
|
||
^^^ ^^^
|
||
|
||
Two fractions which simplify to the same value:
|
||
|
||
123762/37623648 = 12/3648 = 1/304
|
||
^^^^ ^^^^
|
||
|
||
103135/31353040 = 10/3040 = 1/304
|
||
^^^^ ^^^^
|
||
|
||
Only when our school systems adapt innovative approaches to handling
|
||
mathematics, such as this, can America hope to continue to compete in
|
||
today's world. With students in other countries showing superior
|
||
mathematical ability, we cannot afford to delay much longer.
|
||
|
||
|
||
|