Scientific Notation
Significant Digits
January 2011
Scientific Notation
Significant Digits
Objective: The student will learn to use
scientific notation and significant digits.
Scientific Notation
Scientific notation is used to make very large
numbers more understandable.
In one year light travels approximately,
31,000,000,000,000,000 feet
We can use scientific notation to represent
this number as,
3.11016 ft
Scientific Notation
Scientific notation also makes very small
numbers more understandable.
The diameter of an atom’s nucleus is about,
0.000000000000010 m
We can use scientific notation to represent
this number as,
1.01014 m
Scientific Notation
In Scientific Notation, a number is expressed
in the form,
m  10n
Where
1  m < 10
n is an integer
Scientific Notation
A number written in scientific notation must
have exactly one non-zero digit to the left of
the decimal point.
It is written as a number between 1 and 10
times 10 raised to an integral power.
4006 = 4.006103
0.00203 = 2.03103
Scientific Notation
How do we know want the exponent is?
Consider
4006.
We move the decimal point three places to
the left:
4 006
This decreases the value (size) of the number
by three factors of ten. We must compensate
by multiplying by three powers of 10:
4.006103
Scientific Notation
How do we know want the exponent is?
Consider
0.00203
We move the decimal point three places to
the right:
0.002 03
This increases the value (size) of the number
by three factors of ten. We must compensate
by dividing by three powers of 10:
2.03103
Let’s take a 5 minute break.
Yes, 5 minutes.
Precision
No measurement is perfect.
Whenever we take a measurement, there are
always errors, no matter how careful we are.
We call these errors the uncertainty of the
measurement.
We indicate the precision of a measurement
by using significant digits.
Precision
Measurement errors arise from two sources:
1. The intrinsic difficulty of performing the
measurement.
How would you measure the circumference of
the earth?
2. The limits of our measuring instrument.
If we use a stick divided into centimeter
segments, we cannot expect to get an accurate
measurement to the nearest millimeter.
Calculating Using Measurements
What is the width of this rectangle?
7.55 cm
What is the height of this rectangle?
2.70 cm
What is the area? 7.55 cm × 2.70 cm = 20.385 cm2
7.54 cm × 2.69 cm = 20.2826 cm2
7.56 cm × 2.71 cm = 20.4876 cm2
Multiplying with Significant Digits
When you multiply (or divide) two numbers that have
limited precision, the resulting number has no more
significant digits than the number in the product (or
quotient) that has the fewest significant digits.
In the previous slide, each number has three significant
digits. Therefore the answer has only three significant
digits.
7.55 cm × 2.70 cm = 20.385 cm2 → 20.4 cm2
As the previous slide shows, the third digit has some
uncertainty.
Adding with Significant Digits
When adding (or subtracting) two numbers with
limited precision,
1. Line up the decimal points.
2. Perform the operation.
3. Identify the number that has the largest, least
significant digit.
4. Round the answer at that digit.
Example
Subtract, 269.479 − 81.5
Line up the decimal points:
Perform the subtraction:
269.479
− 81.5
187.979
Identify the largest least
significant digit:
Round at that decimal place:
188.0
Notice the answer has four significant digits.
Adding with Scientific Notation
When adding (or subtracting) two numbers in
scientific notation, You must perform two
additional steps.
1.
2.
3.
4.
Make the powers of 10 equal.
Line up the decimal points.
Perform the operation.
Identify the number that has the largest, least
significant digit.
5. Round the answer at that digit.
6. Convert the result to scientific notation, if
necessary.
Example
Add 1.35  103 + 5.26  104
1.35  103 + 52.6103
1.35103
+ 52.6103
53.95103
= 54.0103
= 5.40104
=
Let’s do some data mining.
Class work:
Homework:
p 223: Oral Exercises 1-12
p 223: 2-30 even
p 234 prob: 1-5, 13-15