Transcript 0.000 000 000 000 000 000 000 000 000 000 911 kg

SCIENTIFIC
NOTATION
What is it?
And
How it works?
Much of the data collected and used in
Physics is either very large or very small.
When we talk about data, we are talking
about the measurements or numbers used
to represent what we are looking for.
Let’s look for example at the distance from
the Sun to Mars.
The mean distance from the Sun to Mars is:
227 billion, 800 million meters
OR
227 800 000 000 m
How about the mass of an electron:
0.000 000 000 000 000 000 000 000 000 000 911 kg
Because of this problem, Scientist have
developed a type of short hand to work with
these numbers.
“SCIENTIFIC NOTATION”
Do you remember the the distance from the
Sun to Mars ?
227 800 000 000 m
Written in scientific notation, it would look like this.
2.278 x 1011 m
How about the mass of an electron ?
0.000 000 000 000 000 000 000 000 000 000 911 kg
Written in scientific notation, it would look like this.
9.11 x 10-31 kg
Scientific Notation Rules:
•When moving the decimal to the left, the
exponent will increase.
•When moving the decimal to the right, the
exponent will decrease.
•Only one digit should be to the left of the
decimal.
Convert from Scientific Notation to Real
Number:
5.14 x 105 = 514000
•Scientific notation consists of a coefficient (here
5.14) multiplied by 10 raised to an exponent (here 5).
•To convert to a real number, start with the
coefficient and multiply by 5 tens like this:
5.14 x 10 x 10 x 10 x 10 x 10 = 514000 . Multiplying
by tens is easy: one simply moves the decimal point
in the base (5.14) 5 places to the right, adding extra
zeroes as needed.
•Convert from Real Number to Scientific
Notation:
0.000 345 = 3.45 x 10-4
•Here we wish to write the number 0.000345 as a
coefficient times 10 raised to an exponent.
•To convert to scientific notation, start by moving
the decimal place in the number until you have a
number equal to or greater 1 and less than 10; here
it is 3.45.
•We move the decimal 4 places to the right, so the
exponent decreases to -4.
•Examples:
Express in Scientific Notation
1.
5800
5.8 x 103
2.
450 000
4.5 x 105
3.
86 000 000 000
8.6 x 1010
4.
0.000 508
5.08 x 10-4
5.
6.
0.000 360
0.004
3.60 x 10-4
4 x 10-3
•Examples:
Express in Real Numbers
1.
6.3 x 103
6300
2.
9.723 x 109
9 723 000 000
3.
5.8 x 101
58
4.
4.75 x 10-4
0.000 475
5.
6.
3.56 x 10-7
6.3 x 10-1
0.000 000 356
0.63
Calculating with Scientific Notation
Not only does scientific notation give us a
way of writing very large and very small
numbers, it allows us to easily do calculations
as well. Calculators are very helpful tools, but
unless you can do these calculations without
them, you can never check to see if your
answers make sense. Any calculation should
be checked using your logic, so don't just
assume an answer is correct.
Rule for Multiplication
When you multiply numbers with scientific notation,
multiply the coefficients together and add the
exponents. The base will remain 10.
Example:
Multiply (3 x 107) x (6 x 105)
First rewrite the problem as:
(3 x 6) x (107 x 105)
Then multiply the coefficients and add the
exponents: 18 x 1012
Then change to correct scientific notation:
1.8 x 1013
Remember that correct scientific notation has a
coefficient that is less than 10, but greater than or
equal to one to the left of the decimal.
Rule for Division
When dividing with scientific notation, divide the
coefficients and subtract the exponents. The base
will remain 10.
Example:
Divide 3.0 x 108 by 6.0 x 104
3.0 x 108
6.0 x 104
Divide the coefficients and subtract the exponents
to get:
0.5 x 104
Rewrite the problem as:
Then change to correct scientific notation:
5 x 103
Remember that correct scientific notation has a
coefficient that is less than 10, but greater than or
equal to one to the left of the decimal.
Rule for Addition and Subtraction
When adding or subtracting in scientific notation,
you must express the numbers as the same power
of 10. The exponents must match before you do any
math. This will often involve changing the decimal
place of the coefficient.
Example:
Add 3.5 x 104 and 5.5 x 104
Rewrite the problem as: (3.5 + 5.5) x 104
Add the coefficients and leave the base and
exponent the same: 3.5 + 5.5 = 9 x 104
Example:
Add 2.75 x 104 and 5.5 x 103
First we must pick one of the factors and move the
decimal to make the exponents match. Let’s change
5.5 x 103 to 0.55 x 104
Rewrite the problems as: (2.75 + 0.55) x 104
Add the coefficients and leave the base and
exponent the same: 2.75 + 0.55 = 3.3 x 104
Example:
Subtract (4.8 x 105) - (9.7 x 104)
Pick one of the factors and move the decimal to
make the exponents match. Let’s change 9.7 x 104
to 0.97 x 105
Rewrite the problems as: (4.8 – 0.97) x 105
Subtract the coefficients and leave the base and
exponent the same: 4.8 - 0.97 = 3.83 x 105
•Examples:
1.
(3.2 x 103) + (4.8 x 103) =
8 x 103
2.
(3.2 x 103) - (4.8 x 10-3) =
3.1999952 x 103
3.
(5 x 103) X (12 x 104) =
6 x 108
4.
(6.6 x 103) ÷ (2 x 104) =
3.3 x 10-1