Scientific Notation
The Savior of Science
Notation
There are many ways to write numbers.
• Standard Notation
10
20 4,000
4.0
4,562,234.54
Standard Notation
Using standard notation, numbers are written in decimal form and rounded to a position.
This method works fine for general book keeping and everyday life in the general world.
But in science, we have a problem……
Standard Notation
The numbers used in science for measurements and calculations can be very long with over 20 places.
0.00000000073 m is a typical measurement when working with light.
5980000000000000000000000 kg is the mass of the Earth
While this format can be impressive, it is not fun when math needs to be done.
Scientific Notation
Numbers can be written in small sizes which lead to less writing and simpler arithmetic.
Scientific notation reduces a value to powers of ten.
10,000,000 = 1.0x107 0.00000000073 m = 7.3 x 10­10 m
5980000000000000000000000 kg = 5.98 x10 24 kg
How it is done.
• To convert, all numbers written in scientific notation are to have only one digit to the left of the decimal point.
4567 = 4.567 x 10?
• The power of ten is based on how far the decimal point has been moved
4567 = 4.567 x 103
The new number means 4.567 is
multiplied by 10 three times
Conversions and Powers
• When converting to scientific notation,
If the decimal point must move left, the power of ten will be positive.
6,234 m = 6.234 x103 m If the decimal point must move to the right, the power of ten will be negative.
0.00836 g = 8.36 x 10­3 g
Math
When adding or subtracting numbers in scientific notation, the powers of ten must be the same.
While adding or subtracting numbers, the number in front of the power of ten can have the decimal wherever it needs to be to have the equal powers of ten.
0.345 x 10
4
+ 2.3 x 104 = 2.645 x104
Multiplication When multiplying, the method is simpler.
The front digits multiply as regular numbers.
The powers of ten add together to give the new result.
Ex:
(4.34 x 103)(2.03 x 104) = (4.34)(2.03) x 10(3 +4) = 8.81 x 107
Significant Figures apply.
n
o
i
s
i
Div
• When dividing, the method is similar.
The front digits divide as regular numbers.
The powers of ten are subtracted from each other to give the new result.
Ex:
(4.34 x 103 )/(2.03 x 104 ) = (4.34)/(2.03) x 10(3 ­ 4) =2.14 x 10­1
Significant Figures apply.
Positive and negative powers
The sign of the power of ten simply tells which way the decimal needs to move to return the number to regular notation and the number of places it must move.
4.567 x 103 = 4567
5.23 x 10­4 = 0.000523