Metric System and Fractions
Date: November 2008
I am a math educator and just learned from a science educator that
students are not allowed to represent the metric system numbers as fractions.
They are only allowed to represent them as decimals. Is it wrong to allow them
to represent 3.2 as possibly 3 1/5? In math we teach them number sense and
being able to represent the different forms of numbers as percents, fractions,
Having been a teacher of both Math and Science I appreciate your concern.
As a mathematician, all forms of a fraction are viable, and the ability to
convert from one form to another is a very useful mathematical skill.
It is scientific convention, however, that ALL fractions are represented as
decimals, and in the case of large factors, then Exponential notation is used -
e.g. Avogadro's Number = 6.02 x 10E23
Is it wrong - No. It allows a free exchange of scientific discoveries by the
use of a uniform "language" of science. By convention even American Scientists
use the metric system of units. No one else in the USA would use millimetres,
but its use by scientists is not wrong. (In Scientific circles by the way the
system of measurements is called SI - System International )
Tennant Creek High School
Being also an educator and living in a country( Brasil) where the metric system
is the legal and used one, maybe I can be of some help.
The metric(decimal system) is based on the decimal number system, base 10
of course and all multiples and sub multiples are base 10 determined, at
all type of measurements (length, surface, volume). Then it is more practical
and I would say logic to use the decimal fractions. To remember also it is
easier since the prefixes are all similar and taken from the Latin language:
meter, decimeter, centimeter, millimeter, kilometer; and so on: liter,
deciliter, centiliter, milliliter, kiloliter...
Maybe it is not "wrong" to use other kind of fractions but it breaks the logic
behind the decimal system.
Thanks for asking NEWTON!
(Dr. Mabel Rodrigues)
You sound like a very thoughtful teacher.
I think the answer to your dilemma lies in your goals and objectives for the
lesson. It sounds as if you are using metric system numbers in a lesson to
teach number sense. The numbers without the units are just numbers. If you
want to be a purist about the metric system, drop the units to teach the number
sense and relationship between decimals, fractions, etc. The concept of units
can be added later. OR keep doing what you are doing, and let the students
know that when they do science, they use decimals only. The science teacher
can reinforce this (perhaps you can take the opportunity to develop an
interdisciplinary unit so students are doing similar work in math and science---
perhaps a measurement lab or similar measurement-intensive work with the same
sorts of math operations required in both classes).
Thank you for your efforts!
One of the reasons might be that in science we quantify things based
on observations and measurements. A fraction is a very precise thing -
an object can be split mathematically into exactly three one-third
pieces. However, when we measure something physically, we can never
achieve the precision that a fraction represents. The degree of
precision depends on the instrument used for the measurement. For
example, if I give you a ruler and ask you to measure the cover of a
textbook, you could probably measure it within a sixteenth of an inch
or so - a small percentage of the size of the cover. If I then give
you a dime and ask you to measure the thickness with the same ruler,
you will have difficulty achieving the same degree of accuracy, as
the marks on the ruler are much too coarse compared to the thickness
of the dime. So we need a better ruler. Think now about the ruler
required to measure precisely one-third of something. How can we ever
resolve 0.3333333... ? Hope this helps.
I am not sure who claims that 3.2 does not equal to 3 1/5 or 16/5 for that
matter. However, I can think of reasons why decimals are more convenient, but
First, in addition and subtraction there is no need to find a "common
Second, in multiplication things get even more messy. Example:
(3 1/5)^2 = (3 + 1/5)x(3 + 1/5)=
3x3 + 2x(3x 1/5) + (1/5)^2 = 9 + 6/5 + 1/25 = 9 + 30/25 +1/25 = 9 + 31/25 =
9 + 1 + 6/25 = 10 6/25.
You have to admit that is messy and prone to numerical mistakes.
In general life, it is good to know both fractions and decimals. However,
decimals are the vast, vast preference for science. There are many reasons
why fractions are inappropriate in science, especially when used with the
Perhaps the easiest reason has to do with comparing and ordering numbers.
With fractions, there are many different options for what denominator to use
-- and it becomes much more difficult to compare two values if they are not
using the same denominator (such as if you reduced them to their simplest
form). It is quick and easy to know that 10.4 is smaller than 10.42, but it
takes a split second to think about if 10 2/5 is smaller than 10 3/7.
Second is the prevalence of computational equipment (such as calculators).
It is much easier to perform calculations using decimals as opposed to
fractions, and most computers and calculators default to decimal output.
With how important computational equipment is in science, decimals are the
Third, the metric system is designed to work with base-10 numbering system
to not only be easily scaled (from very large to very small). It is much
easier to start with 42cm and move the decimal place to 0.42m instead of
converting from 42cm to 3/7m (or 21/50m).
Last, and perhaps the most important, there is the concept of precision. In
science, we always try to measure values precisely. However, we should never
represent a measurement as having more precision than it actually has. With
fractions, there is no way to represent precision. You would say 10 2/5 for
either 10.4cm or 10.400000 -- there is no way to distinguish the two. Let me
illustrate this with an example: if I use a ruler to measure the size of a,
say, turtle, it would not be appropriate to say it was 10.43593 cm long
because the tool (a ruler) is not appropriate for that kind of precision. It
is more likely that I measured only 10.4cm, or even just 10cm if it was
squirming around. Suppose you used a laser to measure a distance very, very
precisely, and suppose it ended up exactly 10.40000cm. You really measured
all those zeroes, and it is important to represent them. With fractions,
you would just end up using 40000/100000, at which point you may as well use
English/standard system measurements are not in base-10 units (e.g.
pints-quarts-gallons, inches-feet-yards-miles). For this system, in
every-day use (e.g. not science), fractions are very important to know.
People are used to thinking of things in terms of quarter-inches and
half-pints, so a student should be able to speak in those terms as well.
Hope this helps,
As one who instructs both math and science in the same semester, I appreciate
In science, we are concerned with the accuracy of real world measurements.
For example 3.2 is a far less accurate measurement than 3.2000. A science book
that addresses "significant digits" will give you more information than I can
fit in here.
In math, a fraction such as 1/5 is a precise, rational number. It would be
inappropriate to list it as a fraction in the science lab because it would
not give us any idea of the accuracy of the measurement.
In math, 1/2 + 1/2 = 1 is an exact problem, and one assumes infinite accuracy
if the topic ever comes up. But in science, we ask, "How sure are you of the
measurements of those halves?". Are they measured to the nearest tenth (0.5),
the nearest thousandth (0.500) or nearest hundred thousandths (0.50000) of a
My advice? Just keep on what you are doing in your math classes and let science
teachers explain why they want only decimals. (It all becomes pretty apparent
to the students when they hit high school science.)
Hope this helps.
Oklahoma State University Institute of Technology
Strictly speaking, there is nothing stopping you from saying 3 1/5 concerning
metric measurements, but as the system is based on 10 as opposed to doubling
(2, 4, 8, 16, etc) as the Imperial pounds and ounces are, then decimals are
far better in the metric system. e.g 2 miles equal 3.2 kilometres.
As a retired math and science teacher, I, like you, wrestled with this issue
many times. I often questioned teaching the algorithms of adding,
subtracting, dividing and multiplying fractions is a waste of time within a
math curricula. I also insisted in fractions not being used in science class.
I have been reminded by my other jobs (teachers never made much
money) that USA construction projects is still stuck with manipulating fractions.
Working with tile installations, I was amazed at the amount of fraction
calculations that were necessary to plan the tile placement and cutting
determination. It was done with fractions. I brought up that there was an
easier method, but contruction workers find the SI system too difficult. It is
a sad case, but points out that in our society we still use fractions because we
have failed to completely adopt the SI system.
But I think the key to understanding fractions and their manipulations comes
more focused with dealing with algebra and calculus functions. Here in the abstract
students are placed in the position of manipulating fractions; so the skill can
be of benefit with higher mathematics.
Now with this said, one of my colleagues insists that one of the prime reasons students
do not like working math and have trouble with algebra and calculus is the confusion
they develop with the time they frustrated themselves in grade school with fraction
Her opinion may or may not be justified, but it shows that the US student is
confronted with a series of algorithums that may contradict or confuse mathematical
A great deal of this is caused by our duel system of measurement.
Science does not represent data as fractions, but science uses fraction all the same.
Students are often asked to cut quantities in half or quarters in experiments, and
oral explainings often use fractions instead of 0.125. Fraction have their use.
Ok, let's portion our pie into 0.25.
In my class, I am delighted if my students exhibit number sense. If they describe the
numbers using fractions, decimals, percentiles, or appropriate descriptors like "greater
than" or "less than," it's fine with me. I will say "half a meter" WITH no guilt
Department of Physics and Astronomy
University of Wyoming
The biggest reason that comes to mind about decimals vs. fractions in
science is the idea of significant figures. Using fractions you cannot
easily express the significant figures of a number, which can be
critical when performing math related to measurement and testing.
I am not sure about "allowed", but to a scientist the
number 3 1/5 has a very different meaning, at least in our
context, than the number 3.2. The number 3 1/5 is an exact
number, whereas in science most things are not exact.
Virtually all measured quantities
have uncertainty in them. Some have very little uncertainty and
some have a lot of uncertainty, and so we write numbers down in a way
which explicitly shows this fact of life. In that sense the number 3.2
(which experimentally might be 3.1 or 3.3 depending on who does the
measuring) is not the same as the number 3.200 (which might be 3.202 or
3.198 if someone else measures it). However, the number 3 1/5 is an exact
number representing the fraction 16/5, or 3.20000000000000...
However, there are exact numbers in a few experimental contexts and
it would undesirable, sometimes even incorrect, for a
scientist to use decimals to represent them.
As a simple example: if I have 20 eggs, how
many dozens of eggs do I have? Answer: 20/12 = 1 2/3 dozen, exactly.
Not 1.7, or 1.67, or 1.667. I suppose 1.666666... would be okay, but
it would never be used in a journal article by a scientist,
since 1 2/3 gives the exact answer in a more compact form.
All of this, by the way, has nothing to do with the metric system. The
issue would still be there whether we measured length in meters, yards or
ells. It simply has to do with scientists' preference for numeric
representations which reflect the confidence that other scientists
can have in getting the same result under the same conditions.
In the end, students need to be able to represent numbers as
fractions, percentages, decimals...and science students also need to
learn scientific notation, which we use a lot (2344.55 = 2.34455 x 10^3)
Hope this is helpful.
I imagine that the confusion comes from a valid point, though it may not
appear at first to be one. When I first saw your question my reaction
(probably typical) was to think "of course you can write them as fractions."
However, I think there is a reason you are being asked to have students write
out in decimals even if it has not been made clear.
The reason to write decimals instead of fractions has to do with measurement.
So long as you are dealing with pure mathematics, it does not matter if you
write 3.2 or 3 1/5. However, once you are talking about a measurement, the
number can tell you more than just the value. One of the most common ways to
convey both the magnitude and precision of a measured quantity is to use
Suppose we are talking about length. If I say an object is 3.2m (meters),
that implies the precision in my measurement of the object is no better than
0.1m (which is pretty poor!). If on the other hand I say the object is 3.200
meters, then you know I measured it down to 0.001m precision. The reason for
not using fractions is now clear, if I say the object is 3 1/5 meters long,
then you will not know how precise that value is. It could be 3.2, 3.20000,
or anything else.
What has perhaps not been made clear by the science educator is that this is
not a question of using the metric system, but its applicable to any system
of units/measurements. So if this person maintains that it is OK to say 3 1/5
inches, then they are wrong.
Another way to say it is that you could in fact use fractions, but specify both
the value and the precision. So the length of the object could be written as
"3 1/5 meters, to the nearest 1/1000th meter." That answer is then entirely
equivalent to the decimal answer.
I had almost decided to neglect this as it is one of my personal beafs, but on
second thought I will go ahead and stick my neck out. While the basic idea of
"significant figures" is useful (more so than not having any measure of
precision), it is not the best thing in practice. The motivation for using it
is good and in the right place, but in practice it is better (I think) to just
go with the "full" answer.
My problem with significant figures is that if I write 3.20 meters it states a
precision to 0.01 meters while 3.200 implies 0.001 meters, but there is nothing
in between (which is nonsense). As a practicing scientist (and I would suspect
in any endeavor where precision and accuracy are important), there are many cases
where the precision is in-between and it is important to know just how precise the
value is. And thankfully there is a much better way to express it.
I can write the number as measured and then using either Â± or parentheses writing
So, 3.20 Â± 0.02 meters, or equivalently (should the Â± plus/minus symbol not work)
Even when it comes to doing arithmetic with measured quantities, I will still say
it is better (and not much more complicated) to teach students the correct way than
the "significant figures" and "significant arithmetic" way.
Michael S. Pierce
"Fractions" are a very inefficient way of expressing numbers. This would
not be so confusing if the "denominator" was always "100" in which case the
fraction and decimal notation would be the same. Confusion results because
even simple computations, for example:3/7 + 1/19 = ???
The underlying mathematical issue is that while rational numbers can always
be expressed as a repeating decimal, the length of the repeating string may
be very long.
The important concept is that every rational number can be expressed as a
repeating decimal, and every repeating decimal number is rational. Physical
measurements -- lengths, widths, areas, volumes, etc. -- are almost always
expressed in "base 10".
It is a sad commentary that not only has the U.S. (and a few other
countries) clung to the "English" units of "inches, feet, yards, etc., but
that conversions even within of these units of measure are a different
conversion factor. What sense does that make???
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Update: June 2012