It is probably one of the first topics you will learn in early physics courses. As such, it is crucial that you understand it and are able to use it. Honestly, without scientific notation, physics problems get INCREDIBLY difficult, and if you don’t get it at first, you should really put in extra effort to figure it out. You won’t regret it… and it really isn’t all that hard. It just sounds crazy.
Scientific Notation is merely a short-hand way of expressing really large or really small numbers. It doesn’t sound all that important, but I will show you how convenient it is.
Let’s think about the measure of time. The length of time it takes the Earth to revolve once around the Sun is a year. We are all familiar with this. We also know that there are 12 months in a year, and 365 days (usually) in a year. Let’s go further though… What about hours? Minutes? Seconds? In a year with 365 days, there are:
365 days * 24 hours/day = 8760 hours
8760 hours * 60 minutes/hour = 525,600 minutes
525,600 minutes * 60 seconds/minute = 31,536,000 seconds
The numbers get pretty big. Why don’t you think about how many seconds there are in 100 years: 3,153,600,000 seconds!
Obviously, you don’t want to have to write down THAT number over and over again, and any numbers you calculate from it, in your equations to solve problems. This is where Scientific Notation comes in.
Scientific Notation basically takes the first non-zero numbers and multiplies that by some factor of 10. Each position in a multi-digit number is represented by 1 power of 10. You have ones, tens, hundreds, thousands, ten thousands, and so on…. and on the opposite side of the decimal, you have tenths, hundredths, thousandths, etc.
So, you can see that:
10 can be written as 1 x 101 (one times ten to the power of 1)
100 can be written as 1 x 102 (one times ten to the power of 2)
1000 can be written as 1 x 103 (one times ten to the power of 3)
and so on…
(If you also recall you rules for exponents, you can see that this pattern continues both up and down… 1 is 100)
Scientific Notation is just expressing things as powers of 10. All you have to do is move the decimal place x number of places so that you have one digit before it, and then multiply your number by 10 to the power of the x places you moved the decimal.
36 = 3.6 x 101… because the decimal place moves 1 to the left
189 = 1.89 x 102… because the decimal place moves 2 to the left
5389 = 5.389 x 103… because the decimal place moves 3 to the left
What about for bigger numbers?
22000 = 2.2 x 104… move the decimal 4 places to the left. Also note that you don’t need to record zeroes if they are the last digits.
On the other hand:
22001 = 2.2001 x 104… same power, but the zeroes cannot be ignored, because they set the position for the final digit, the 1.
Let’s look back at our example of time:
8760 hours = 8.76 x 103 hours
525,600 minutes = 5.256 x 105 minutes
31,536,000 seconds = 3.1536 x 107 seconds
3,153,600,000 seconds = 3.1536 x 109 seconds
Now you can appreciate that if you have to rewrite 3.1536 x 109 seconds several times in a calculation, it is simpler to keep track of… and more importantly, there is far less of a chance of accidentally leaving out some of the zeroes and completely getting the wrong answer.
I won’t go into too much detail for extremely small numbers, since it is essentially the same concepts as I have described above. The thing to remember here is that you move the decimal place to the RIGHT this time, and give a negative power of 10. I will, however, leave you with a few examples to work through so that you can hopefully understand it more completely.
0.1 is equal to 1 x 10^(–1) (one times ten to the power of negative 1)
0.01 is equal to 1 x 10^(–2) (one times ten to the power of negative 2)
0.001 is equal to 1 x 10^(–3) (one times ten to the power of negative 3)
and so on…
0.53 = 5.3 x 10^(–1)
0.0687 = 6.87 x 10^(–2)
0.0000873 = 8.73 x 10^(–5)
0.0000000070067 = 7.0067 x 10^(–9)
Another helpful tip is that if your starting number is between 1 and 0 (ie. it is SMALL), it gets a negative power. If it is greater than 1 (ie. it is BIG) it gets a positive power. Positives for big, negatives for small.
Scientific Notation for small quantities is equally as handy as for large numbers. Consider that a virus may be as small as 1 x 10^(–7) m, or that a proton has a mass of 1.7 x 10^(–27) kg. Also imagine what a chemist’s life would be like without Scientific Notation. They would have to write out this proton mass every time they need it in a calculation. It would probably get very tiring, very quickly, and probably with several errors, if they had to write out every time 1700000000000000000000000000 kg. (Of course, with that many digits, one would hope this chemist would include a little bit more precision)
As always, let me know if you found this helpful or would like some more clarification!