Scientific Notation

Scientific notation is all about decimals. Especially very big and very small decimals. Scientists deal with some very large numbers (like the distance of the Earth from the sun, 93,000,000 miles) or very small numbers (the diameter of an atom: about 0.00000003 centimeter). It is awkward to deal with numbers in this form (all those zeros!), so mathematicians (and scientists) developed a way to express these numbers without all those zeros. It involves exponents, specifically powers of 10.

Every time you move the decimal point to the right in a number, you are multiplying it by 10. Conversely, every time you move the decimal point to the left you are dividing by 10. If you move more than one place you are multiplying (or dividing) by powers of 10. If you move it 2 places to the right you are multiplying the number by 10² = 100. A number in scientific notation is of the form:
a x 10ⁿ
the coefficient a is a decimal number between 1.0 and 10 and
n is an integer that says how many places to move the decimal, either right (+) or left (-).

So, that distance of the Earth from the sun, 93,000,000 miles, becomes 9.3 x 10⁷
because you move the decimal point to the right 7 places to get the decimal number 93,000,000
That size of an atom, 0.00000003 centimeter, becomes 3.0 x 10^-7 because you move the decimal point 7 places to the left to get its decimal representation 0.00000003

To figure out the power of 10, think
"how many places do I move the decimal point, and which way?"
Here is a table of some common numbers and their representation in both decimals and scientific notation:

Number	Decimal form	Scientific notation
Light-year: the distance light travels in a year	5,878,000,000,000 miles	5.878 x 10¹²
Number of cells in the human body	100,000,000,000,000	1.0 x10¹⁴
Mass of a dust particle	0.000000000753 kg	7.53 x 10^-10
Age of a kid in the 6th grade	12	1.2 x 10¹

Doing arithmetic in scientific notation:

Multiply	Multiply the coefficients and add the exponents. If the resulting coefficient is more than or equal to 10 then move the decimal place one place to the left and add 1 to the exponent	Example: 4.2 x 10² x 3.8 x 10³ = (4.2 x 3.8) x 10^{(2 + 3)} = 15.96 x 10⁵ = 1.596 x 10⁶ In decimals: 420 x 3,800 = 1,596,000
Divide	Divide the coefficients and subtract the second exponent from the first one If the resulting coefficient is less than 1 then move the decimal place one place to the right and subtract 1 from the exponent	Example: 4.2 x 10² ÷ 6.0 x 10³ = (4.2 ÷ 6.0) x 10^{(2 - 3)} = 0.7 x 10^{- 1} = 7.0 x 10^-2 In decimals: 420 ÷ 6000 = 0.07
Add or subtract	Both numbers must have the same exponent to add or subtract. Add or subtract from one of the exponents to make it equal to the other one. Move the decimal point in the coefficient of the number whose exponent you modified that many places to the left or right. Add or subtract the coefficients	Example: 1.2 x 10² + 3.1 x 10³ Change 3.1 x 10³ to 31. x 10² and add: = (1.2 + 31.) x 10² = 32.2 x 10² = 3.22 x 10³ In decimals: 120 + 3100 = 3220

Now here are a couple of problems for you:

Divide:	4.6 x 10⁴ ÷ 2.3 x 10² = _____ x 10	In decimals: __________
Add:	4.6 x 10 ³ + 2.3 x 10 ² = ___ x 10 ² + 2.3 x 10 ² = ___ x 10 ^² = ______ x 10 In decimals: ______________	← make the exponents the same ← make the coefficient between 1 and 10