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Chapter 2
Measurements and Calculations
Section 2.1 and 2.2
In these sections of chapter-2, you will learn about:
Types of Measurements (quantitative and qualitative)
Scientific Notation to express numbers
Measurements
• In Scientific Method, making observations is an
important part as you learned in Chapter 1.
• Observations can be Qualitative or Quantitative.
• Qualitative observation does not involve a number
(e.g., the book is heavy, the person is tall).
• Quantitative observation always involves a number
and a unit (e.g., the book is 1.50 kilograms heavy; the
person is 66.2 inches tall)
• Quantitative observation is called Measurement.
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Measurements (Quantitative Observations)
• Measurement has 2 Parts – the Number and
the Unit
Number tell us how much something measures and
gives us a Comparison (example, 5 cm vs. 10 cm)
Unit tells us the Scale that was used (example
centimeter, meter, etc for distance measurement)
Identify which is the number and which is the unit in the
following statements:
a) This box has 2 dozen marbles
b) This pencil measures 5 inches
c) This book weighs 1.6 kilograms 3
Scientific Notation
• The numbers associated with scientific
measurements can often be very large or very
small (e.g., 9452578; or 0.000023).
• Scientific Notation is a method for making very
large or very small numbers more compact and
easier to write.
• Scientific Notation expresses a number, as a
number between 1 and 10, multiplied by
appropriate power of 10.
• Based on Powers of 10
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Scientific Notation
• Example, 21523 can be written as 2.1523 x 104
• So, we have written the number 21523 as a number
between 1 and 10, without losing its original value!
• How? By multiplying the new decimal point containing
number, by appropriate power of 10.
• How do we know what this Power of 10 should be?
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Scientific Notation
• To determine the power number, You can put a decimal
point at the end of the last digit and then see how many
places you need to move it to the left. In this case it is
was by four places.
• 21523 = 2.1523 x 104
• So, when the decimal point is moved to the left, the
exponent is positive number.
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2. Scientific Notation
• Let us look at a few more examples.
• 724567845
• I first put at decimal point to the end, and then had to
move it by 8 times. Hence the exponent will be 8 and the
scientific notation will now be:
7.24567845 x 108
• Likewise, 242 would be 2.42 x 102
• Similarly, 4682 would be 4.682 x 103
• Notice that in all the cases where the decimal had to be
moved to the left, the exponent is a positive number.
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Scientific Notation
• What if a number already had decimal point and
you want to convert it to scientific notation as a
number between 1 and 10, but still moving to the
left?.
• Follow the same procedure, but this time start moving
the already existing decimal point. Example: 645.4
would be 6.454 x 102
• Exponent is still a positive number, because you have
moved the decimal to the left.
• Likewise, 61.06700 would be 6.106700x 101
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Scientific Notation for numbers >10
So, when you have a number greater than 10, first place
the decimal point at the very end of all the digits in that
number.
Then, keep moving the decimal point to the left, until you
make the old number to become a new number between
1 and 10.
Determine how many total times you moved the decimal
point.
Multiply the new number you have now obtained by 10n
(where n stands for the number of times you had to move
the decimal point)
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Scientific Notation for numbers >10
• Practice the following: What is the scientific notation
for the following numbers?
• 12
• 12.5
• 1258
• 125678
• 1256787865
• 12.0
• 12.06
• 1289
• 1.2 10
Scientific Notation of small numbers
• Now, let us look at how to convert very small
numbers that are less than 1, to scientific notation
by having to move the decimal point to the right.
• For example, the number 0.00235 can be written as
2.35 x 10 -3 by moving the decimal to the right.
• Notice that this time the exponent has a negative sign,
because we moved the decimal point to the right.
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Scientific Notation for numbers < 1
Therefore, when you have a number less than 1, this
number will already have a decimal point.
Move (or place) the decimal point to the right as many
times as you need to convert the original number to a
new number between 1 and 10
Determine how many total times you moved the decimal
point.
Multiply the new number you have now obtained by 10-n
(Note that the sign on the exponent is negative)
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3. Scientific Notation for standard numbers < 1
• For example, what is the scientific notation for
the following numbers?
0.1 = 1 x 10-1 0.01 = 1 x 10-2
0.12 = 1.2 x 10-1 0.002 = 2 x 10 -3
0.123 = 1.23 x 10-1 0.00042 = 4.2 x 10-4
0.8256 = 8.256 x 10-1 0.000000089= 8.9 x10-8
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To Determine the sign on the exponent n
• If the decimal point was moved to the left, n is +
• If the decimal point was moved right, n is –
• If the decimal point was not moved at all
(when the original number was already
between 1 and 10) then n = 0. No need to
multiply be a power of 10
• Practice the many examples given in the ebook
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Summary
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Converting scientific notation to Standard Form
1. Determine the sign of n in 10n
If n is + the decimal point will move to the right
If n is – the decimal point will move to the left
2. Determine the value of the exponent of 10
Tells the number of places to move the decimal
point
3. Move the decimal point and rewrite the number
4. Practice the examples shown in your text-book
and workout the problems at the end of the
chapter and in the following website
5. http://www.aaamath.com/B/g6_71gx1.htm
Converting scientific notation to Standard Form
• So far, you converted standard numbers to
scientific notation.
• Can you do the reverse? That is, if you are
given a scientific notation, can you figure out
what the standard number is?
• For example, what does 1.24 x 102 represent?
= 124
• What does 1.5 x 10-6 represent?= 0.00000156
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Scientific Notation in Addition and Subtraction
For addition and subtraction:
1. When you have numbers that are in scientific
notations but with different exponents, first
they all should be converted to having the
same value for the exponents.
2. Only then the addition or subtraction operation
can be carried out.
Let us work on a few examples.
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Scientific Notation in Addition and Subtraction
Example-1: (1.36 x 102) + (4.73 x 103)
Notice that the first number has exponent of 2 and the
second number has exponent of 3.
You can choose to convert second one also to exponent of
2, or the first one to exponent of 3.
(1.36×102)+(47.3×102)=(1.36+47.3)×102=48.66×102
or
(0.136×103)+(4.73×103)=(0.136+4.73)×103)=4.866×103
In terms of the value, they both are same answers.
Because 48.66x 102 = 4.866 x 103 same as second answer.
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Scientific Notation in Addition and Subtraction
Example-2 (subtraction): (((6.923×10−3) − (8.756×10−4)
Notice that the first number has exponent of -3 and the second
number has exponent of -4.
You can choose to convert second one also to exponent of -3, or
the first one to exponent of -4. And then subtract.
• (6.923×10−3)−(0.8756×10−3)=(6.923−0.8756)×10−3= 6.0474 x
10 -3
• or
• (69.23×10−4)−(8.756×10−4)=(69.23−8.756)×10−4=60.474×10−4
In terms of the value, they both are same answers.
Because 60.474 x 10-4 = 6.0474 x 10-3 same as first answer
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Scientific Notation in Multiplication and Division
For Multiplication and Division
1. There is no need to convert the numbers to
have the same e
2. For multiplication, just multiply the two
numbers, but the exponents get added.
3. For Division, just divide the two numbers
but the exponents get subtracted.
Let us work on a few examples.
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Scientific Notation in Multiplication and Division
For Multiplication:
Example-1: (6.022×1023) (6.42×102)
(6.022 x 6.42) x 10 [ (23)+2)] = 38.66×10 25 = 3.866 x 1026
Example-2: Even if one of the exponents had negative sign, you
should still do the addition. For example (6.022×1023) (6.42×10-2)
(6.022 x 6.42) x 10 [ (23)+ (-2)] = 38.66×10 21 = 3 .866 x 1022
Example-3:
(7.63×10−34)(5.0×10) = (7.63 x 5.0) x 10[(-34) + 1)] =38.15 x 10-33
or 3.815 x 10 -32
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Scientific Notation in Multiplication and Division
For Division: Remember, you need to subtract the
exponents in this case.
Example-1: 1.67×10−24 = 1.67 × 10[(−24) - (-28) = 0.183 x 10 +4
9.12×10−28 9.12
Because -(-28) = + 28 . Therefore, -24 + 28 = +4
Example-2: 0.58×1020 = 0.58 × 10[(20) - (18) = 0.213 x 102
2.72×1018 2.72
Example-3: 6.022×1023 = 6.022 x 10[23)-(-2)] = 0.938 x 1025
6.42×10−2 6.42
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Scientific Notation in more than one operation
When you have multiple operations in the same
problem. Example: both multiplication and
division both involved.
Involve the addition of exponents and then the subtraction of the
denomintaor exponent)
• Example: (6.63×10−34)(6.0×10) = (6.63 x 6.0) x 10[ (-34)+1 (-(-2) ]
(8.52×10−2) (8.52)
= 4.67 x 10-31
Because -34 + 1 – (-2) = -34 + 3 = -31
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Scientific Notation in more than one operation
The multiple operations can be any
combination.
Example: Addition with multiplication (or division)
or subtraction with division (or multiplication), or
multiplication and division together.
Just follow the rules of each accordingly and combine
them.
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