In the applied sciences, scientific notation is often employed as a method of notation for ease of writing and reading. When dealing with real world situations, the numbers we get as solutions are rarely whole numbers and scientific notation gives us rules to follow when using ugly numbers that have a lot of decimal places. In order to understand scientific notation, we must also have a solid understanding of exponents.
A lot of times in both Math and Science you deal with really big numbers or sometimes really small numbers. By the way you'll also deal with really big numbers when you're talking about the national debt. Anyway one thing we're going to be working on here, is how to write really really big numbers or really really small number in what we call "scientific notation." The word scientific sounds like it only belongs in Science class but, you use this in Math also. A number in scientific notation is written as a times 10 to the nth power where a is in between 1 and 10, a can be less, greater than or equal to 1 let me try that again. 1 is less than or equal to a that is what I meant to say, or less than 10. So here we go, when you're looking at examples of these problems make sure this a quantity is less than 10 and make sure it's times the number 10 to some exponent.
Less than 10 times 10 to the fifth good, less than 10 greater than or equal to 1 times 10 to a negative exponent that's okay. And here we have 1.0 times 10 to the hundred and fiftieth, big old exponent that's okay just make sure that you're a value works in this inequality. Let's look at some things that are not scientific notation, negative 3.2 times 10 to the fourth. The times 10 to the fourth bit is good but my a value is negative 3.2 it doesn't work. This one is the same reason, here I have an a value that's 34 it's way the heck bigger than 10, so this guy does not count in scientific notation.
One last thing I want to point out to you before you start doing these problems, is what this looks like on your calculator. Have you ever typed something into your calculator and you get like 3.24561 blah blah e10? That's your calculator telling you scientific notation, what that means is 3.24 blah blah times 10 to the tenth exponent. This little number that comes after the e is the exponent on your 10 might be positive, might be negative something like that's what it looks like on your calculator.
You also have a button most of them look like this, so if you wanted to type in there 3.4 times 10 to the fifth on your calculator you will 3.4 then the EE button which means 10 to the 5 and that would result with the answer how this would be written in what we call standard form. There's another thing I want to show you and that's how to move from scientific notation into what we call standard notation or standard form. Here is how it goes, if I have the number 3.4 times 10 to the fifth, what that means is take the decimal place and move it 5 places to the right 1, 2, 3, 4, 5 that's where my new decimal point is going to go, everything else gets filled in as zeros. So this number written in standard notation would be 340,000.
We use a similar process when 10 is raised to a negative exponent. Only instead of moving the decimal place to the right, now you're going to move it the left 8.32 times 10 to the -16 means take this decimal place and move it 16 spaces that way. Can you guys see why this is kind of annoying to write in standard notation? Let me see if I did it, moved it 1, 2, 3, 4, 5, 6, 7, 8, 9, I didn't even do it yet. But you guys get the idea, I'm going to have lots and lots of zeros, zero point lots of them before I get to 832 at the end. And the way you guys can write these out is by putting the decimal where it started from move it to the right if it's a positive number, move it to the left if it's a negative number.
I do want to show you guys how to do this with a negative one, so instead of writing negative 16 I'm going to change that to negative 6 so I don't have to move it so far I think you guys can appreciate why I'm doing that. Okay so let's writing that second one in standard notation, 8.32 here's my point I'm going to move it 6 places to the left now 6 so my answer is going to be 0.00000832 that's my answer written on standard notation 0. 1, 2, 3, 4, 5 832. You're going to see a lot of these in your homework and it gets to be kind of annoying writing out all of the zeros. So just stick with it for your homework tonight and then be thankful that we have scientific notation so that in the real world we don't have to write out something like a hundred and fifty zeros.