## Use of Scientific Notation

Scientific notation is, essentially, a method for writing really big or really small numbers. It is called scientific notation because these tiny or huge numbers are often found in scientific work -- imagine describing the size of an atom or the mass of the earth!

For example, you might have the number 6000000000000. That’s really big, right? Unfortunately, it isn’t easy to tell exactly *how* big at first glance with all those zeroes stuck on the end. Instead, the number could be written as 6 * 1000000000000. Then you can change the 1000000000000 to an easier to understand number: \(10^12\). Putting it all together, we have \(6 * 10^12\). Now you can compare that number to others, because the \(10^12\) means there are 12 zeroes at the end.

The value of scientific notation becomes clear when you try to multiply or divide these numbers. What is \(50000000 * 3000000\)? You could do this relatively easily by multiplying 3 times 5 and then adding up all the zeroes, but that still takes time, and you could easily miscount all the zeroes. Instead, scientific notation allows us to multiply \(5 * 10^7\) times \(3 *10^6\). You multiply the 5 and the 3 to get 15, and then add the exponents. The answer is \(15*10^13\). However, in order for scientific notation to be completely correct, the number at the beginning must be between 1 and 10. The 15 has to be changed into 1.5, and to make up for this we multiply the whole thing by another factor of 10, giving \(1.5*10^14\).

Scientific notation can also be used for very *small* numbers in much the same way. 0.000005 is written as \(5*10^-6\), because you use negative exponents on the 10 when the number is very small. Remember, negative exponents do not make the number negative, but just very small. Try multiplying \(.00009 * .00003\). The numbers in scientific notation are \(9*10^-5\) times \(3*10^-5\). The answer is computed the same way as before, yielding \(27*10^-10\), or \(2.7*10^-9\). Here are a few more examples to illustrate the principles of scientific notation:

Here's multiplication examples:

$$ (5*10^8)(4*10^15) = 20 * 10^23 = 2*10^24 $$ $$ (5*10^-8)(4*10^15) = 20 * 10^7 = 2*10^8 $$... and division:

$$ \frac{9*10^6}{3*10^4} = 3*10^2 \text{ (when dividing, you subtract the exponents) } $$ $$ \frac{8*10^-5}{2*10^-3} = 4*10^-2 $$Addition and subtraction:

$$ 5*10^7 - 4*10^7 = 1*10^7 $$ $$ 2*10^3 + 3*10^3 = 5*10^3 $$ $$ 2*10^4 + 2*10^3 = 2*10^4+0.2*10^4 = 2.2*10^4 $$