Transcript Number # Significant Digits

Significant Digits
1
Measurement needs to be
precise and accurate
 Precision:
 How
closely multiple
measurements of the same
quantity come to each other.
 This will depend on the measuring
device. For example, a
thermometer that shows degrees
in tenths is more precise than one
that only shows single degrees.
2
Accuracy
 Refers
to how close a measurement
comes to the true or accepted value.
 This depends on both the measuring
device and the skill of the person using
the measuring device.
 This can be determined by comparing
the measured value to the known or
accepted value.
3
Precision and Accuracy
Good precision
Good Precision
Poor Precision
Good Accuracy
Poor Accuracy
Poor Accuracy
4
Measurement and numbers

There are two kinds of numbers
 Counted or defined - exact, not subject to
estimate. Ex: number of eggs in a carton.
 Measured - always carries some amount of
uncertainty because measurement involves
estimation. The size of uncertainty depends
on the precision of the measuring device
AND the skill of the person using the device
5
6
Measurement and numbers
 Measurements
consist of two
parts
 The number itself (the quantity)
 The units (the nature of the
quantity measured)
7
Estimation in measurement
1 cm
2 cm
A




3 cm
4 cm
B
5 cm
C
When we measure, the quantity rarely falls exactly
on the calibration marks of the scale we are using.
Because of this we are estimating the last digit.
For instance, we could measure “A” above as about
2.3 cm. We are certain of the digit “2”, but the “.3”
part is a guess - an estimate.
What is your estimate for B and C?
8
Higher Precision
1 cm
2 cm
A




B
3 cm
C
D
A measuring device with more marks on the scale is more
precise. I.e., we are estimating less, and get a more
accurate reading. Here we are estimating the hundredths
place instead of the tenths.
Here, we can measure A as 1.25 cm. Only the last digit is
uncertain.
Usually we assume the last digit is accurate ± 1.
How would you read B, C, and D?
9
The Problem
Area of a rectangle = length x width
We measure:
Length = 14.26 cm Width = 11.70 cm
Punch this into a calculator and we find the area as:
14.26 cm x 11.70 cm = 166.842 cm2




But there is a problem here!
This answer makes it seem like our measurements were more
accurate than they really were.
By expressing the answer this way we imply that we estimated the
thousandths position, when in fact we were less accurate than that!
A much better answer would be that the area is 166.84 cm2
because that keeps the same accuracy as our original
measurements.
10
Significant Figures (aka Significant Digits)
To help keep track of (and communicate
to others) the precision and accuracy of
our measurements, we use Significant
Figures
 These are the digits in any measurement
that are known with certainty plus one
digit that is uncertain (but usually
assumed to be accurate ± 1)

11
Rules for Significant Figures
1.
2.
3.
4.
Digits from 1-9 are always significant.
Zeros between two other significant
digits are always significant
One or more additional zeros to the
right of both the decimal place and
another significant digit are significant.
Zeros used solely for spacing the
decimal point (placeholders) are not
significant.
12
Examples
EXAMPLES
# OF SIG.
DIG.
COMMENT
453
3
All non-zero digits are always
significant.
5057
4.06
4
3
Zeros between two significant digits
are significant.
5.00
106.00
114.050
3
5
6
Additional zeros to the right of
decimal and a significant digit are
significant.
0.007
1
Placeholders are not significant
2
Trailing zeros in numbers with no
decimal point are not significant (=
placeholder)
12000
13
Practice
How many significant digits in the following?
Number
1.4682
110256.002
0.000000003
114.00000006
110
120600
# Significant Digits
5
9
1
11
2
4
14
Multiplication and Division with Significant Digits
The rule for multiplying or dividing significant digits is that the answer must
have only as many significant digits as the original measurement with the least
number of significant digits.
Our measurements, 14.26 and 11.70 each have four significant digits. Our calculator told us
the answer was 166.842, but we need to round it off. Do we round up or do we round it down?
166.842
166.84
If our original measurements had been 14.26 and 11.7, what happens?
166.842
167
How many significant digits would the answer to each of these have?
Problem
#Sig. Digits in Result?
114.6 x 2.0004
4
0.0006 x 14.63
1
12.901 / 6.23
3
15
Addition and Subtraction with Significant Digits
The rule for adding or subtracting with significant digits is that the answer must
have only as many digits past the decimal point as the measurement with the
least number of digits past the decimal.
How many significant digits would the answer to each of these have?
Problem
#Digits Past the Decimal?
114.6 + 2.0004
1
0.0006 + 14.63
2
12.901 - 6.23
2
One complication: doing math on numbers without decimal points does
not follow the above rules, it just works like it normally does.
So: 1200 x 56 = 67200 not 67!
16