[G.Polya]
Setting up equations is like translation from one language
into another (NOTATION,1). This comparison, used
by Newton in his Arithmatica Univeresalis may help to
clarify the nature of certain difficulties often felt both
by students and by teachers.
1. To set up equations means to express in
mathematical symbols a condition that is stated in words; it is
translation from ordinary language into the language of
mathematical formulas. The difficulties which we may
have in setting up equations are difficulties of translation
In order to translate a sentence from English into
French two things are necessary. First, we must understand
thoroughly the English sentence. Second, we must
be familiar with the forms of expression peculiar to the
French language. The situation is very similar when we
attempt to express in mathematical symbols a condition
proposed in words. first we must understand thoroughly
the condition. Second, we must be familiar with the
forms of mathematical expression.
An English sentence is relatively easy to translate into
French if it can be translated word for word. But there
are English idioms which cannot be translated into
French word for word. If our sentence contains such
idiom, the translation becomes difficult; we have to pay
less attention to the separate words, and more attention
to the whole meaning; before translating the sentence,
we may have to rearrange it.
It is very much the same in setting up equations. In
easy cases, the verbal statement splits almost automatic
ally into successive parts, each of which can be immediately
written down in mathematical symbols. In more
difficult cases, the condition has parts which cannot be
immediately translated into mathematical symbols if
this is so , we must pay less attention to the verbal
statement, and concentrate more upon the meaning. Before
we start writing formulas, we may have to rearrange the
condition, and we should keep an eye on the resources of
mathematical notation while doing so.
In all cases, way or difficult, we have to understand the
condition, to Separate the various parts of the condition
and to ask Can you write them down, In easy cases
we succeed without hesitation in dividing the condition into
parts that can be written in mathematical symbols
in difficult cases, the appropriate division of the condition
is less obvious.
The foregoing explanation should be read again after
the study of the following examples
2. Find two quantities who's' sum is 78 and who's
product is 1296.
We divide the page by a vertical line. On one side, w
e write the verbal statement split into appropriate parts.
on the other side, we write algebraic signs, opposite to the
corresponding part of the verbal statement. The original
is on the left, the translation into symbols on the
right.
Stating the problem
in English in algebraic language
Find two quantities x y
who's sum is 78 and x + y = 78
whose product is 1296 xy=1296
In this case, the verbal statement splits almost
automatically into successive pars, each of which can be
immediately written down in mathematical
symbols
3. find the breadth and the height of a right prism
with a square base, being given the volume, 63 cu in and
the area of the surface 102 sq in
what are the unknown? the die of the base, say x
and the altitude of the prism, say y.
What are the data? the volume 63, and the area, 102.
What is the condition? the prism whose base is a
square with side x and whose altitude is y must have the
volume 63 and the are a02.
Separate the various parts of the condition. There are
two parts, one concerned with volume, the other with
the area.
We can scarcely hesitate in dividing the whole condition
just in these two parts; but we cannot write down
these parts "immediately" We must know how to calculate
the volume and the various parts of the area. yet if
we know that much geometry, we can easily restate both
parts of the condition so that the translation into equations
is feasible. We write on the left hand side of the page an
essentially rearranged and expanded statement of th problem,
ready for translation into algebraic language
Of a right prism with
a square base
find the side of the base x
and the altitude y
First the volume is given 63
the are of the base which
is a square with side x x^2
and the altitude y
determine the volume
which is their product x^2y=63
Second, the area of the
surface is given 102
the surface consists of two
squares with side x 2x^2
and of four rectangles each
with base x and altitude y
4 xy
and whose sum is the area 2x^2 + 4xy = 102
Coordinates of a point, find the point which is symmetrical to the given
point with respect to the given straight line.
this is a problem of plane analytic geometry.
what is the unknown? a point, which coordinates, say p,q.
What is given? the equation of a straight line, say
y= mx+n,
and a point with the coordinates, say a,b.
What is the condition? the points a(b, and p,q)
are symmetrical to each other with respect to the line
y=mx+n.
we now reach the essential difficulty which is to divide
the condition into parts each of which can be expressed in the
language of analytic geometry. The nature of this
difficulty must be well understood. A decomposition of the
condition into parts may be logically unobjectionable
able and nevertheless useless. what we need here is a
decomposition into parts which are fit for analytic
expression. in order to find such a decomposition, we must
go back to the definition of symmetry but keep an eye on the
resources of analytic geometry what is meant by symmetry
with respect tho what is meant by symmetry with respect
to a straight line ? What geometric
relations can we express simply in analytic geometry? we
concentrate upon the first question but we should not
forget the second. thus, eventually we may find the
decomposition which we are going to state.
The given point (a,b)
and the point required (p,q)
are so related that
first the line joining
them
is perpendicular (q-b)/(p-a) = -1/m
to the given line and
second the midpoint of
the line joining them lies
on the given line. p+q/2 = m a+p/2 +n