Associated Topics || Dr. Math Home || Search Dr. Math

### Decimal Places and Significant Figures

```
Date: 02/05/99 at 04:04:13
From: zakia ali
Subject: Number skills

If a length of 5.738km is rounded to 5.7km, then the accuracy is
either "to .....d.p." or "to .....s.f."

I am stuck on this problem. Could you help me?
```

```
Date: 02/05/99 at 12:46:29
From: Doctor Peterson
Subject: Re: Number skills

Welcome to the Doctor's office, Zakia.

I'm happy to help, and I'll give you a little more than you asked for
- an explanation of why "d.p." and "s.f." matter.

"Accurate to N decimal places" means the number of digits to the right
of the decimal point that you can trust is N. For example, if I
measure a length with a ruler that shows millimeters, the measurement
will be accurate to one millimeter, or 3 decimal places if I write it
in meters (0.001 m). If I claimed to have measured it as 1.1293 m,
you'd know I was guessing about the 3, and would round it off to the
nearest thousandth: 1.129. The ruler will always produce the same
number of decimal places.

So your number, 5.7 km, is presumed to be accurate to one decimal
place.

"Accurate to N significant figures" means the total number of
meaningful digits that you can trust is N. For example, in my 1.129 m,
there are four digits I consider dependable, based on how I measured.
If I had measured 0.024 m with the same ruler, there would be only two
significant figures. (The zeroes are there only to show the place value
of the other digits, and aren't "significant.") The ruler doesn't
always produce the same number of significant figures, because it is
better at measuring larger things. If I tried to measure something
smaller than a millimeter, it would be useless - it wouldn't give me
any significant figures at all!

In your number there are two significant figures, 5 and 7.

The number of decimal places matters when you are adding the numbers.
For example, if I add 1.2 and 3.45, with different numbers of decimal
places, I don't know what the hundredths place of 1.2 is, or what the
thousandths place of 3.45 is. I can put an X for the unknown digits
and see what happens:

actual    with X's

1.2       1.2XX
+ 3.45    + 3.45X
------    -------
4.65      4.6XX

You see, since I don't know all the hundredths I'm adding, I have no
idea what the hundredths place of the result is, so I have to drop the
5, and call the answer 4.6, accurate to only one decimal place. (Even
the tenths might be wrong because of a carry, but it won't be too far
off.) When I add numbers, the result is only accurate to the smallest
number of decimal places I'm adding. In this case, since 1.2 has only
2 decimal places, that's all I can keep in my answer.

On the other hand, if I multiply numbers, what counts is the number of
significant figures. Suppose I run for 1.45 hours at 6.1 miles per
hour. Then I've gone 1.45 * 6.1 miles. How accurate is that? Again I'll
put an X for the unknown places and see what happens:

actual       with X's

1.4 5        1.4 5 X
x     6.1    x     6.1 X
---------    -----------
1 4 5        X X X X
8 7 0        1 4 5 X
---------    8 7 0 X
8.8 4 5    -----------
8.8 X X X X

You can see that the number of significant figures in the result (2)
is the smaller of the significant figures for the two multiplicands (3
and 2), so I have to write my answer as 8.8, rounding it to two
significant figures and dropping two digits that I worked hard for and
would otherwise have thought were good. Since 6.1 has only two
significant figures, I can't keep more than that in my answer.

So that's why decimal places and significant figures are both useful.
I hope that gives you a better feel for this concept. It's especially
important in the age of calculators, which give us a false sense of
accuracy in cases like this!

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
Elementary Place Value

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search