Transcript Addition and subtraction

Significant Figures
What is so
significant about
numbers??
Just which numbers are
significant?

All nonzero numbers (all numbers 1-9) –
289
 573
 119

Just which numbers are
significant?
All nonzero numbers (all numbers 1-9) - 289
 All zeros b/w numbers – 909

501
 3002
 4000001

Just which numbers are
significant?
All nonzero numbers (all numbers 1-9) - 289
 All zeros b/w numbers – 909
 Zeros to the right of a number AND
to the left of a written decimal
point –

250.
 700.
 10.

Just which numbers are
significant?




All nonzero numbers (all numbers 1-9) - 289
All zeros b/w numbers – 909
Zeros to the right of a number AND to the left of a
written decimal point – 250. 700.
10.
Zeros to the right of a number AND to
the right or a written decimal point –
 5.0
 236.70
 3.0 x 108
These are NEVER significant
 Zeros
to the left of the
decimal in numbers less
than one
 0.8
 0.222
*** these are placeholders
only
These are NEVER significant
 Zeros
to the left of the decimal in
numbers less than one
 Zeros to the right of a decimal,
but to the left of the first
number
 0.008
 0.02
Exceptions to the rule
 Conversion
factors –
unlimited sig. fig.
1
km/1000 m
 60 min/1 hr
 100 cm/1 m
Exceptions to the rule
 Conversion
factors – unlimited sig.
fig.
 Counting numbers
 30
days
 12 in one dozen
Calculations with sig. figs.
The
following rules must be
followed so that results
reflect sig. figs. from your
measurements taken
originally…
Addition & subtraction with
sig. figs.
 Your
answer can’t be MORE
precise than your LEAST precise
measurement
(what in the world does
that mean??)
Addition & subtraction with
sig. figs.
Your answer can’t be MORE precise
than your LEAST precise measurement
 Your answer must be rounded to the
same number of decimal places from
the original problem…

34.2
+ 2.002 = ?
Addition & subtraction
 34.2
+ 2.002 =
 Actual answer would be 36.202
Addition & subtraction

Actual answer would be 36.202
 The
least precise number is 34.2,
therefore your answer MUST be to
one decimal place
Addition & subtraction
Actual answer would be 36.202
 The least precise number is 34.2,
therefore your answer MUST be to one
decimal place

 Correct
answer = 36.2
Addition & subtraction
Example
3.999
#2
– 1.77 = ?
Addition & subtraction
 3.999
– 1.77 =
 Actual answer would be 2.229
Addition & subtraction
Actual answer would be 2.229
 The least precise number is 1.77,
therefore your answer MUST be to two
decimal places

Addition & subtraction
Actual answer would be 2.229
 The least precise number is 1.77,
therefore your answer MUST be to two
decimal places

 The
correct answer would be
2.23
Multiplication & division
Answer must be in the fewest number
of sig. figs. from the original problem.
 Example:


45.6 x 1.009 = ?
Multiplication & division
Answer must be in the fewest number
of sig. figs. from the original problem.
 Example:

45.6 x 1.009 = ?
 Actual is 46.0104

Multiplication & division
Answer must be in the fewest number
of sig. figs. from the original problem.
 Example:

45.6 x 1.009 = ?
 Actual is 46.0104
 Smallest # of sig figs is 3… so answer must
be in 3 sig figs.

Multiplication & division
Answer must be in the fewest number
of sig. figs. from the original problem.
 Example:

45.6 x 1.009 = ?
 Actual is 46.0104
 Smallest # of sig figs is 3… so answer must
be in 3 sig figs.
 Correct answer is 46.0 (not just 46)

Multiplication & division

Example #2
 505
/7=?
Multiplication & division

Example #2

505 / 7 = ?
 Actual
answer is 72.14285714
Multiplication & division

Example #2
505 / 7 = ?
 Actual answer is 72.14285714

 Least
number of sig figs is 1, so
answer can only have one sig fig.
Multiplication & division

Example #2
505 / 7 = ?
 Actual answer is 72.14285714
 Least number of sig figs is 1, so answer
can only have on sig fig.

 Correct
answer is 70 or it could
be 7 x 101
Rounding…
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If the number right past the one you
want to keep is:

Greater than 5  go up by one
295.46 to 4 sig figs would be 295.5
 999.97 to four sig figs would be 1000.

Rounding…

If the number right past the one you
want to keep is:
Greater than 5  go up by one
 Less than 5  no change

999.94 to four sig figs would be 999.9
 564.44 to three sig figs would be 564

Rounding…

If the number right past the one you
want to keep is:
Greater than 5  go up by one
 Less than 5  no change
 5 followed by a number  go up by
one

2.352 to two sig figs would be 2.4
 4.156 to two sig figs would be 4.2
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Rounding…

If the number right past the one you want to
keep is:

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
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Greater than 5  go up by one
Less than 5  no change
5 followed by a number  go up by one
5 followed by nothing… look at the
number before it… if it is ODD  go
up by one
3.375 to three sig figs would be 3.38
 0.035 to one sig fig would be 0.04

Rounding…

If the number right past the one you want to
keep is:





Greater than 5  go up by one
Less than 5  no change
5 followed by a number  go up by one
5 followed by nothing… look at the number before it… if it is
ODD  go up by one
5 followed by nothing… look at the
number before it… if it is EVEN  no
change
4.8785 to four sig figs would be 4.878
 399.345 to five sig figs would be 399.34

Scientific notation

Very large or very small numbers are
expressed in scientific notation
Scientific notation
Very large or very small numbers are
expressed in scientific notation
n
 M x 10

Scientific notation
Very large or very small numbers are
expressed in scientific notation
n
 M x 10
 “M” must be 1 or greater, but less than
10


2.2 x 105
Scientific notation
Very large or very small numbers are
expressed in scientific notation
n
 M x 10
 “M” must be 1 or greater, but less than 10
 All numbers that represent “M” are significant
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
3.10 x 109
9.98 x 103
Scientific notation

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Very large or very small numbers are
expressed in scientific notation
n
M x 10
“M” must be 1 or greater, but less than 10
All numbers that represent “M” are significant
Numbers that represent “n” are whole
numbers, positive or negative


6.022 x 1023
2.1 x 10-5
Scientific notation

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
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
Very large or very small numbers are expressed in scientific
notation
n
M x 10
“M” must be 1 or greater, but less than 10
All numbers that represent “M” are significant
Numbers that represent “n” are whole numbers, positive or
negative
Examples:



5
270,000 would be 2.7 x 10
The understood decimal is behind the last
zero…move the decimal until it makes the number
represent “M”
0.000000505 would be 5.05 x 10-7
Calculations w/ scientific
notation
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Addition and subtraction:
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All numbers must have the same exponent before
“M” can be added or subtracted
Answer will have the same exponent value as
originals
 Don’t
forget rules w/ sig figs… they
still apply!!
 2.4
x 102 + 5.7 x 102 = 8.1 x 102
 9.05 x 105 – 5.5 x 105 = actual 3.55 x
105

Correct is 3.6 x 105
Calculations w/ scientific
notation

Addition and subtraction:
All numbers must have the same exponent
before “M” can be added or subtracted
 The problem comes when exponents are
DIFFERENT… 

You must MAKE the exponents the same…
 LL / RR = left larger or right reduce 
when you move the decimal it changes the
exponent -- ALWAYS

Calculations w/ scientific
notation

Addition and subtraction:


2.5 x 104 + 5.2 x 103 =
Choose which number you want to deal with…
either will work out
Calculations w/ scientific
notation

Addition and subtraction:



2.5 x 104 + 5.2 x 103 =
Choose which number you want to deal with…
either will work out
I like to change the smaller exponent to the larger

5.2 x 103
Calculations w/ scientific
notation

Addition and subtraction:




2.5 x 104 + 5.2 x 103 =
Choose which number you want to deal with…
either will work out
I like to change the smaller exponent to the larger

5.2 x 103
I want to make the exp. larger, so I move the
decimal one place to the left
Calculations w/ scientific
notation

Addition and subtraction:





2.5 x 104 + 5.2 x 103 =
Choose which number you want to deal with…
either will work out
I like to change the smaller exponent to the larger

5.2 x 103
I want to make the exp. larger, so I move the
decimal one place to the left
This makes it .52 x 104
Calculations w/ scientific
notation

Addition and subtraction:






2.5 x 104 + 5.2 x 103 =
Choose which number you want to deal with…
either will work out
I like to change the smaller exponent to the larger

5.2 x 103
I want to make the exp. larger, so I move the
decimal one place to the left
This makes it .52 x 104
Now “M” can be added because exp are the same


Actual answer = 3.02 x 104
Correct answer = 3.0 x 104
Calculations w/ scientific
notation
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Multiplication – this one is much easier 

3.0 x 103 x 4.0 x 104
Calculations w/ scientific
notation

Multiplication – this one is much easier 
3.0 x 103 x 4.0 x 104
 You will multiply your “M”s together
(12)

Calculations w/ scientific
notation

Multiplication – this one is much easier 
3.0 x 103 x 4.0 x 104
 You will multiply your “M”s together (12)
 Then add your “n”s together (7)

Calculations w/ scientific
notation

Multiplication – this one is much easier 
3.0 x 103 x 4.0 x 104
 You will multiply your “M”s together (12)
 Then add your “n”s together (7)
 CAREFUL… you will often need to
move the decimal to keep correct
scientific notation

Calculations w/ scientific
notation

Multiplication – this one is much easier 
3.0 x 103 x 4.0 x 104
 You will multiply your “M”s together (12)
 Then add your “n”s together (7)
 CAREFUL… you will often need to move the
decimal to keep correct scientific notation
 Actual answer would be 12 x 107

Calculations w/ scientific
notation

Multiplication – this one is much easier 



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
3.0 x 103 x 4.0 x 104
You will multiply your “M”s together (12)
Then add your “n”s together (7)
CAREFUL… you will often need to move the
decimal to keep correct scientific notation
Actual answer would be 12 x 107
Move the decimal one place to the LEFT to
make 1.2
Calculations w/ scientific
notation

Multiplication – this one is much easier 







3.0 x 103 x 4.0 x 104
You will multiply your “M”s together (12)
Then add your “n”s together (7)
CAREFUL… you will often need to move the
decimal to keep correct scientific notation
Actual answer would be 12 x 107
Move the decimal one place to the LEFT to make
1.2
This makes the exponent go up (larger) to 8
Calculations w/ scientific
notation

Multiplication – this one is much easier 

3.0 x 103 x 4.0 x 104
You will multiply your “M”s together (12)
Then add your “n”s together (7)
CAREFUL… you will often need to move the
decimal to keep correct scientific notation
Actual answer would be 12 x 107
Move the decimal one place to the LEFT to make
1.2
This makes the exponent go up (larger) to 8

Now you answer is 1.2 x 108 (2 sig figs)

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