Scientific Notation Review
Scientific Notation was developed in order to easily represent numbers that are either very large or very small. The Andromeda Galaxy (the closest one to our Milky Way galaxy) contains at least 200,000,000,000 stars. The number of stars in the Andromeda Galaxy can be written as:
2.0 x 100,000,000,000
It is that large number, 100,000,000,000 which cause the problem. But that is just a multiple of ten. In fact it is ten times itself eleven times:
10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 100,000,000,000
A more convenient way of writing 100,000,000,000 is 1011. The small number to the right of the ten is called the "exponent," or the "power of ten." It represents the number of zeros that follow the 1.
Though we think of zero as having no value, zeroes can make a number much bigger or smaller. Think about the difference between 10 dollars and 100 dollars. Any one who has balanced a checkbook knows that one zero can make a big difference in the value of the number. In the same way, 0.1 (one-tenth) of the US military budget is much more than 0.01 (one-hundredth) of the budget. (Though either one is probably more money than most of us will ever see in our checkbooks!)
So we would write 200,000,000,000 in scientific notation as: 2.0 x 1011
This number is read as follows: "two point zero times ten to the eleventh." There are some rules for expressing numbers in scientific notation:
Rule #1: There are two parts to the number, separated by a multiplication sign. The first is called the coefficient (2.0) and the second is called the power of ten.
Rule #2: The first number will ALWAYS start with a digit that is between 1 and 9, and will NEVER be zero.
Rule #3: The decimal point will ALWAYS follow the first digit.
How Does Scientific Notation Work?
As we said above, the exponent refers to the number of zeros that follow the 1. So:
101 = 10;
102 = 100;
103 = 1,000, and so on.
Similarly, 100 = 1, since the zero exponent means that no zeros follow the 1.
Negative exponents indicate negative powers of 10, which are expressed as fractions with 1 in the numerator (on top) and the power of 10 in the denominator (on the bottom). So:
10-1 = 1/10;
10-2 = 1/100;
10-3 = 1/1,000, and so on.
This allows us to express other small numbers this way. For example:
2.5 x 10-3 =
2.5 x 1/1,000 =
0.0025
Every number can be expressed in Scientific Notation. In our first example, 200,000,000,000 should be written as 2.0 x 1011. In theory, it can be written as 20 x 1010, but by convention (Rule #3) the number is usually written as 2.0 x 1011 so that the lead number is less than 10, followed by as many decimal places as necessary.
It is easy to see that all the variations above are just different ways to represent the same number:
200,000,000,000 =
20 x 1010 (20 x 10,000,000,000)
2.0 x 1011 (2.0 x 100,000,000,000)
.2 x 1012 (.2 x 1,000,000,000,000)
This illustrates another way to think about Scientific Notation: the exponent will tell you how the decimal point moves; a positive exponent moves the decimal point to the right, and a negative one moves it to the left. So for example:
4.0 x 102 = 400 (2 places to the right of 4);
While 4.0 x 10-2 = 0.04 (2 places to the left of 4).
Note that Scientific Notation is also sometimes expressed using the symbol E (for exponent), meaning “times ten to the”, as in 4 E 2 (meaning 4.0 x 10 raised to the power of 2). Similarly 4 E -2 means 4 times 10 raised to -2, or = 4 x 10-2 = 0.04. This method of expression makes it easier to type in scientific notation and is the standard method used in calculators.
Multiplication:
When multiplying numbers expressed in scientific notation, the exponents can simply be added together. This is because the exponent represents the number of zeros following the one. So:
101 x 102 = 10 x 100 = 1,000 = 103
Checking that we see: 101 x 102 = 101+2 = 103
Similarly 101 x 10-3 = 101-3 = 10-2 = .01
Again when we check we see that: 10 x 1/1000 = 1/100 = .01
Look at another example: (4.0 x 105) x (3.0 x 10-1).
The 4 and the 3 are multiplied, giving 12, but the exponents 5 and -1 are added, so the answer is:
12 x 104, or 1.2 x 105.
Let's check: (4 x 105) x (3 x 10-1) =
400,00 x 0.3 =
123,000 =
1.2 x 105.
Interesting note: another way to see that 100 = 1 is as follows:
101 x 10-1 = 101-1 = 100.
It is also: 10 x 1/10 = 1.
So 100 = 1
Division:
Let's look at a simple example:
(6.0 x 108)
------------
(3.0 x 105)
To solve this problem, first divide the COEFFICIENTS: 6 by the 3, to get 2. The exponent on the power of 10 in the denominator (the bottom) is then moved to the numerator (the top), reversing its sign and then adding the two numbers together. (Remember that little trick from your old math classes?) So we move the 105 to the numerator with a negative exponent, which then looks like this:
2 x 108+(-5)
All that's left now is to reduce this expression by adding the exponents. So the answer is:
2.0 x 103 or 2,000