Download Scientific Notation

Scientific Notation
Definition
a x 10n ; 1 < a < 10 and n is an integer
Scientific notation is used as shorthand for extremely large numbers or extremely
small numbers. You will need to be able to change a number that is in scientific
notation into standard notation (the way it is normally written). You will also
need to be able to change a number that is in standard notation into a number
written in scientific notation. The steps for both of these processes are explained
below.
Scientific Notation Æ Standard Notation
1. Rewrite the decimal part.
2. find the absolute value of the exponent of the 10
3. Move the decimal that many places (left if exponent is negative and right if
exponent is positive)
4. Fill in empty loops with zeros.
5. Put commas and/or decimal wherever necessary.
1. rewrite decimal part
Examples
3.4 x 107 =
3.4
3.4
2. exponent of 10 is 7
3. move decimal 7 units to right
.
4. Fill in empty loops with zeros
3 4 0 0 0 0 0 0.
5. Put commas and/or decimal
3.4 x 107 = 34,000,000
5.06 x 10-8 =
1. rewrite decimal part
2. exponent of 10 is -8
3. move decimal 8 units to left
4. Fill in empty loops with zeros
5. Put commas and/or decimal
5.06
.
.
5.06
00000005 06
5.06 x 10-8 = 0.0000000506
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Standard Notation Æ Scientific Notation
1. Locate the decimal or insert one at the end of the number.
2. Move the decimal left or right towards the numbers that are not zeros using
loops.
3. Stop when you get to a number between 1 and 10.
4. Count the loops to determine exponent.
a. If you started with a number less than 1 (0.000…) or if you moved the
decimal to the right your exponent is negative.
b. If you started with a number greater than 10 or if you moved the decimal to
the left, your exponent is positive.
5. Rewrite the number with the number between 1 and 10 (ignoring all of the
0’s at the end) times 10 to the power that you determined.
Examples
437,000,000 =
1. Insert the decimal
2. Move the decimal left
3. Stop between 1 and 10
4. Count the loops
437,000,000.
4
. 3 7 0 0 0 0 0 0.
b. you moved decimal to the left, 8 places to the left
your exponent is positive.
b. Rewrite the number
437,000,000 = 4.37 x 108
0.00000063 =
0 .00000063
1. Locate the decimal
2. Move the decimal right
3. Stop between 1 and 10
4. Count the loops
0
a. you moved decimal to the right,
your exponent is negative.
5. Rewrite the number
.0
0 0 0 0 0 6.3
7 places to the right
0.00000063 = 6.3 x 10-7
Tip
Remember that the numbers that are greater than 1 (the large numbers) are
going to have a positive exponent. The numbers that are between 0 and 1 (the
decimals) will have the negative exponents. To figure out the exponent you
could also count how many digits are between where the decimal started off and
where it ended up.
35,000 = 35,000. = 3.5000. = 3.5 x 10
4 digits
between decimals
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4
Sample Problem Involving Scientific Notation
If the length of a rectangular plot of land is 6.2 x 107 ft and the width of
the plot is 2.0 x 105 ft, find the area of this plot of land.
Answer: The area of a rectangle is length times width. Using the commutative
property of multiplication and properties of exponents, the area can be calculated
as follows:
(6.2 x 107) x (2.0 x 105) = 6.2 x 2.0 x 107 x 105 = 12.4 x 1012
However 12.4 x 1012 is not in scientific notation since 12.4 is not less than 10.
So, rewrite 12.4 as 1.24 x 101. Now, rewrite the expression as
(1.24 x 101) x 1012 = 1.24 x 1013
So, the area of the rectangular plot is 1.24 x 1013 square feet.
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