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### Changing Units ... and Significant Figures?

```Date: 08/22/2011 at 09:29:32
From: Amie
Subject: Number of significant figures changes with unit conversion.

Suppose I measured the length of a table and got 2 feet, which is
60.96cm.

My teacher says I need to round my result according to the number of
significant figures in my measured data. Well, the first measurement has
one significant digit. The second has four significant digits! So, how
many significant digits are there in the above measurement?

I think that when I change from feet to centimeters, the number of
significant figures changes, and I have to re-write my result according to
that unit. But I have searched Wikipedia for clarity on significant digits
and unit conversion, to no avail.

```

```
Date: 08/22/2011 at 10:39:04
From: Doctor Peterson
Subject: Re: Number of significant figures changes with unit conversion.

Hi, Amie.

It looks like you've missed the central concept of significant digits
(which is not uncommon, if they are not taught in the right context).

What you've done is just to WRITE the quantities "2 feet" and "60.96 cm"
with different numbers of significant digits. They don't actually *have*
that many. Essentially, you are "lying" about the measurement. Significant
digits are supposed to represent how accurately something was MEASURED.

When you write "2 feet," with only one significant digit, you are implying
that you don't know how many inches; the length might really be anything
that rounds to 2, from 1.5 feet to 2.5 feet. I very much doubt that that
is what you did. More likely, you measured with a ruler or yardstick, and
found that it was 2 feet TO THE NEAREST 1/8TH INCH, say -- that is, if the
ruler is marked with eighths of an inch, you couldn't be sure it wasn't 2
feet 0 inches and 1/16th of an inch, but you know it's not more than that.
So in order to express this, you should say you measured it as 2 feet, 0.0
inches.

(Fractional measurements like this are really hard to express in terms of
significant digits, and in reality people making such measurements would
just say, "2 feet, plus or minus 1/16 inch" to express this precision.)

Now, when you convert (or do any other calculations), you are supposed to
fact that your calculator or whatever shows 4 digits doesn't mean they are
all valid.

If I were forced to use significant digits, I would do the conversion
something like this:

2 feet 0 inches to the nearest 1/8 inch
= 2*12 + 0 = 24 inches +/- 1/16, that is, between 23.9375 and 24.0625.

Call it 24.0, since the hundredths are uncertain but the tenths are
more or less sure. This gives three significant digits.

24.0 in * 2.54 cm/in = 60.96 cm
Round this to three significant digits, giving 61.0 cm.

Now, what I'd really do is to measure in centimeters, and avoid all the
trouble with fractions. Most likely, the ruler would be marked in
millimeters, so it would be accurate to the nearest mm (tenth of a cm),
and I would get exactly the same result: 61.0 cm.

Does that help?

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/

```

```
Date: 08/23/2011 at 08:36:20
From: Amie
Subject: Thank you (Number of significant figures changes with unit conversion.)

Thank you, Dr. Peterson! It's really helped! Thank you, indeed!
```
Associated Topics:
Middle School Measurement

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