Some numbers are simply too big or too small to be written down, they contain to many zeros. Nobody would dream of writing the number of stars in our universe as 100,000,000,000 or the diameter of an atom as 0.0000000005 meters. You would have to count the zeros to figure out what the number stands for (100 billion stars and 50 nm). Just by looking at a number once we would like to know how big that number it and what the value of the number is. These two things are different but are both accomplished by the way of writing the numbers in what is called “scientific notation”.
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Scientific notation does this by breaking up the number into two parts, one that specifies the magnitude and anther that gives the value. We begin by counting the “zeros” a number has and write it like 10^5 if the number has 5 zeros. One million has six zeros, so we write 10^6. The numbers 5 and 6 are called exponents. For very small numbers the exponent has to be negative. Why? Look to the left: This achieves half of what we want. The exponents tell us immediately the number of digits a number has. This is also called the “order of magnitude”. In the big scheme of things this is the most important part – getting the number of zeros, or the order of magnitude right. If you multiply 35 by 35, which is the “better” answer: 122 or 1000? The calculator gives us 1225. If we were to do this in our heads 1000 would be the better answer. Getting the digits (122) right is a small consolation if you think about it as calculating your salary in dollars. |
Now all we need to find is the value. If 100 is 102 then 125 is written as 1.25´102. The first number tells you what value the number has, while the second gives you the number of zeros.
The most important thing about a number is the magnitude. Scientific notation lets you find the magnitude just by looking at it. Consider this number: 7,210,718,000,00 which is 7.2107´1012 or about 7 trillion. This is by the way our national debt.
There is another benefit of working with scientific notation. A few rules make it very easy for us to work with big numbers without using a calculator. Multiplying and dividing numbers written in scientific notation is very easy. The rules for working w
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Here are two examples where the rules of multiplying and dividing exponents are are applied. In addition I worked out one problem where the exponent itself has an eponent. Read carefully through all the examples.
Keeping these rules in mind will help when solving problems that involve numbers written in scientific notation. Of course, you could use the calculator to do all of this. So, try this one:
If you get a number other than 0.5´109 or half a billion forget the calculator. |
