SPH3U1
Introduction
SIGNIFICANT DIGITS AND SCIENTIFIC NOTATION
LEARNING GOALS
Students will:



Identify the number of significant digits in a value.
Be able to determine the number of significant digits produced in a calculation.
Learn how to express values in scientific notation.
SIGNIFICANT FIGURES
1.
2.
3.
4.
5.
Non-zero digits are always significant, eg. 4.53 is 3 SD
Zeros at the beginning are never significant, eg. 0.00234 is 3 SD
Zeros between non-zeros are always significant, eg. 23.04 is 4 SD
Zeros at the end of an integer are usually not significant, eg. 4300 is 2 SD
Zeros following a decimal and behind a non-zero are significant, eg. 2.3500 is 5 SD
Remember that a measurement is an approximation of an exact number.
A measurement is accurate if it is close to the known value.
Precision is measured by the last significant digit.
COUNTING SIG FIGS
When you’re given a number, how do you know how many sig figs it has?
Leading zeroes are
never significant
(just placeholders)
Captured zeroes, between
2 non-zeroes, are always
significant
0.00200600
Trailing zeroes are
significant only if the
number has a decimal
EXAMPLES
7050000  3 SD
0.00200600  6 SD
705000.0  7 SD
If a number contains a decimal point, any number (including zero) after the first non-zero is
significant!
Q1. How many students are in this room? How many significant digits is that?
Q2. I measure a piece of fabric for curtains to be 12 meters long. How many significant
digits is this value?
Q3. What is the purpose of significant digits? (i.e. why is the concept necessary?)
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SPH3U1
Introduction
ROUNDING
1. If the first digit to be dropped is less than 5, leave the previous digit
eg. 4.274 RO to 3 SD becomes 4.27
2. If the first digit to be dropped is 5 or more, increase the previous digit by 1
eg. 4.278 RO to 3 SD becomes 4.28
Q4. Percent error can be calculated by the equation,
|experimental value − theoretical value|
× 100%
theoretical value
Use this equation to determine the percent deviation if the experimental value is
12.2 m/s and the accepted value is 12.4 m/s. Round your answer to the correct number
of significant digits.
SCIENTIFIC NOTATION
A number written as the product of a decimal number, equal to or greater than one and less
than ten, and a power of 10, is said to be expressed in scientific notation.
EXAMPLE
32400 becomes 3.24 x 104
or
0.00324 becomes 3.24 x 10-3
The first factor (eg. 3.24) indicates the accuracy of the number and the second factor (eg. 10 4)
indicates the magnitude of the number. All digits in the first factor are significant.
SIG FIGS IN CALCULATIONS (PROPOGATION OF SIG FIGS)
When you do a calculation using numbers with some uncertainty, your answer must reflect that uncertainty.
MULTIPLICATION/DIVISION
The result should have the same number of sig figs as the term with the least number of sig figs.
(38.7)(0.005525) = 0.2138175 = 0.214
308.7⁄2.5 = 123.48 = 1.2348 × 102 = 1.2 × 102
(2.700 × 108 )(3300) = 8.91 × 1011 = 8.9 × 1011
(Answer has 3 SD limited by the 3 SD of 38.7)
(Answer has 2 SD limited by the 2 SD of 2.5)
(Answer has 2 SD limited by the 2 SD of 3300)
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SPH3U1
Introduction
ADDITION/SUBTRACTION
The result has the same precision (places after the decimal) as the least precise term.
12.11 + 1200.8868 = 1212.9968 = 1213.00
(2 dec. places, limited by 12.11)
0.0425 1.2 = 1.1575 = 1.2
(1 dec. place, limited by 1.2)
150 + 75.5 =
(an advanced example….give it a try! Hint: convert the numbers to scientific notation)
Thought Key: For multi./div., count sig figs; for add./sub., count decimal places
MIXED CALCULATIONS – MULT/DIV AND ADD/SUB IN THE SAME PROBLEM
*** Note the order of operations – after you do the first step, note the correct number of sig
figs/decimal places, then apply the correct sig figs/decimal places in the next step ***
During a lab period, one student determined the density of an unknown liquid to be 1.21 g/mL
and another student found 1.42 g/mL. What is the percent error of each student’s measurement
if the actual density of the liquid is 1.267 g/mL?
Percent error is an important concept in science!
Student 1:
% error =
=
1.21 − 1.267
× 100%
1.267
−0.057
× 100%
1.267
2 decimal places from the
subtraction operation
leaves only 1 SD in the
numerator for the division.
= −4%
Student 2:
% error =
=
1.42 − 1.267
× 100%
1.267
0.153
× 100%
1.267
2 decimal places from the
subtraction operation
leaves 2 SD in the
numerator for the division.
= 12%
Example 2:
24.6500 − (92.30)(0.260) = 24.6500 − 23.998 = 0.652 = 0.7
3 SD in multiplication operation leaves only 1 decimal place as significant after subtraction.
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SPH3U1
Introduction
MEASUREMENTS
A qualitative measurement is just an observation with no number.  e.g. “It’s hot outside”
A quantitative measurement includes a number and a unit.  e.g. “It’s 105 F outside”
A measurement is accurate if it is close to the true value.
A set of measurements is precise if the measurements are closely spaced.
READING A MEASURING DEVICE
When you are reading a measuring device, how do you know how many numbers to write down?
The general rule for reading a device to the correct number of sig figs is to record the number of
certain digits plus one uncertain digit.
EXAMPLE: GRADUATED CYLINDER
The MAJOR division side is the difference between the numbers on the device:
8 mL – 7mL = 1 mL
We count 10 spaces in the major division, so the actual division size is:
1 mL/10 = 0.1 mL
In this example, the meniscus (i.e. liquid level) lies between 7.3 mL and
7.4 mL. We are certain of the 7.3, and the rule of sig figs allows us to
estimate one more uncertain digit. A possible answer is thus 7.32 mL. You
could also say 7.31 mL or 7.33 mL. The last digit is uncertain. In this example, the device is
“precise to 2 decimal places.”
Here’s another way to figure out how many decimal places you should go to:
First, round the division size to 1 non-zero digit, then divide by 10. This result is the error of
device. HOWEVER MANY DECIMAL PLACES THE ERROR HAS, THAT’S HOW MANY DECIMAL
PLACES YOUR READING SHOULD HAVE. (Note: this method only works if the non-zero digit in
the error is 4 or less; if it’s 5 or greater, reduce the decimal places by 1)
For our example we have 0.1 mL/10 = 0.01 mL. So the error is 0.01 mL, which is 2 decimal
places, so every reading of the device should be to 2 decimal places.
PRACTICE
What is the division size?
What is the division size?
What is the error?
What is the error?
What is the correct reading?
What is the correct reading?
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SPH3U1
Introduction
PRACTICE QUESTIONS (COMPLETE ON A SEPARATE PAGE)
1. How many significant figures are there in each of the following?
2.
3.
4.
5.
6.
a)
146.32 cm
b)
1.8 cm
c)
28.0 cm
d)
28 cm
e)
700 m
f)
700. m
g)
1.00346 kg
h)
0.00346 kg
i)
3.46 x 10-3 kg
j)
3.00 x 108 m/s
k)
2 640 m/s
l)
2.640 m/s
Round off the following to two significant digits:
a)
5.934
b)
4.863
c)
76.9
d)
0.345 1
e)
0.018 1
f)
0.034 6
Express the following numbers in scientific notation with the proper number of significant
digits:
a)
954
b)
321.0
c)
6 874
d)
6 400
e)
6 400.
f)
0.000 64
g)
0.800
h)
0.000 15
i)
300 000 000
Express the following numbers in decimal notation:
a)
3.2 x 105
b)
3.2 x 10-5
c)
3.2 x 100
d)
8.791 x 102
e)
4.96 x 10-4
f)
3 x 108
Calculate the following expressing your answers with the proper number of significant
figures:
a)
21.36 + 2.8 + 35.348 + 2.9164
c)
54.612

3.0
d)
b)
147.845 - 121.1
the area of a rectangle whose length is 35.66 cm and
whose width is 9.1 cm.
Calculate the values of the following expressing your answers in scientific notation with the
proper number of significant digits:
 (2.0 x 103)
a)
4.0 x 105 x 2.0 x 103
b)
4.0 x 105
c)
(2.02)3 x 102
d)
(5.0 x 4.0)2 x 104
e)
2.7 x 103
f)
8.5 x 103 - 4.5 x 102
 (3.0 x 10-4)
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