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Blown Out of Proportion at Zero?

Date: 09/22/2016 at 09:02:43
From: Liz
Subject: The definition of proportion.

The definition of "proportional" states that if a is (directly)
proportional to b, then a/b is a constant. The relationship is written "a
proportional b," which implies a = cb, for some constant c, known as the
constant of proportionality.

Now, when you graph a proportional relationship, the line goes through the
origin, (0, 0). At that point, the "a" and "b" are both equal to 0.

But 0/0 is not a constant. It's not even defined!

The mathematical definition seems to conflict with the real world
application of the concept. Pick any proportional relationship from
everyday life, such as between the number of hours you work and the pay
you earn from it. If you work 0 hours, you get $0 pay. But the definition
of "proportional" would mean that zero is proportional to zero, forcing us
to reconcile with indeterminacy.

So how to do you rectify the discrepancy between the definition -- which
says that a/b equals some constant -- and the reality that 0/0 is

Date: 09/22/2016 at 10:15:34
From: Doctor Peterson
Subject: Re: The definition of proportion.

Hi, Liz.

This is a great question.

If y is proportional to x, that means that y = cx for some constant c.
This will be true for all pairs of values of the related variables. Given
this relationship between the variables, we can conclude that, in
particular, when x = 0, y = 0.

But this is true regardless of the value of the constant c; so from the
pair (0, 0), you can't determine that constant. This matches up with the
fact that 0/0 is indeterminate, and therefore undefined.

Moreover, we do not precisely say that "0 is proportional to 0." We don't
even say "6 is proportional to 2": the proportionality is down to the
relationship between the *variables* -- that is, between ALL pairs of
numbers in the relation.

Now, if we already know that y is proportional to x, then we can use the
pair (2, 6) to determine the constant of proportionality; and as I said
above, it happens that we can't use the pair (0, 0) in the same way. Zero
is just special.

The other point I have to make is that there is a problem in some
statements of what proportionality means. You defined it as "if a is
(directly) proportional to b, then a/b is a constant." Above, I opened by
presenting it as "if y is proportional to x, that means that y = cx for
some constant c." This is a cleaner definition for two reason: (1) it
focuses on the variables x and y, not (as one might assume) on specific
values a and b; and (2) it does not require that the specific values be

The definition of proportionality coming from your teacher or your
textbook author effectively swapped the proper definition for an
implication. The version "If a is proportional to b, then a/b is a
constant" is an idea that I might introduce after giving my definition. I
would do this to connect with what students have learned previously about
"proportions," which relate two specific PAIRS of numbers, a/b = c/d.
This, too, does not apply when any of the numbers is zero (unless you
interpret it not as fractions but as ratios, in which case it is, in some
views at least, proper to say 3:0 = 4:0, or perhaps even 0:0 = 0:0).

Avoiding the use of division in the definition allows us to include zero.
Then the only issue remaining is to recognize that zero is a special case.
It is the intersection of all the straight-line graphs of proportions, so
that in this one case we can't infer a statement about division.

And we do want to account for zero, because, as you say, if your pay is
proportional to the hours you work, that does include the case of not
working at all, and as a result getting paid nothing.

But the main point here is that proportionality, in the sense we are
discussing, is a relationship between two variables -- not between
individual numbers.

Does that help?

- Doctor Peterson, The Math Forum at NCTM 

Date: 09/22/2016 at 12:02:49
From: Liz
Subject: Thank you (The definition of proportion.)

Thank you!
Associated Topics:
High School Definitions
High School Number Theory

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