Scientific Notation Example and Unit Analysis

by Jared Rovny

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    00:00 This problem you see in front of you is a perfect example of where you might want to use scientific notation because the numbers seem so big when you look at them and the computation might seem intimidating, especially in the context of an exam if you’re trying to save time.

    00:15 What we’re gonna do is try to solve this much more quickly and simply using scientific notation.

    00:20 So what I’d like you to try to do is solve this but use scientific notations instead and see if you can speed up the process much more quickly. If you try that, and pause, you can give it another shot. Let's jump in and see what it’s look like.

    00:32 The first thing we’re wanna do if we’re gonna solve this using scientific notation is rewrite each of our three numbers in scientific notation. So first, let’s rewrite this 3200 by again, taking the decimal point from the right and moving it one, two, three spaces over.

    00:47 If this is the case then we have 3.2 times 10 to the third power, so this is the same number just rewritten in scientific notation. Doing the same thing with these other numbers, We see, one, two, three, four, five, six places. So this is 5 times 10 to the sixth power. And then lastly, there’s a very very long number is one two three four five six seven eight nine places.

    01:12 And so this number, which we’re dividing is 2.5 times 10 to the ninth power.

    01:21 To make this computation very easy, what we’re going to do is not swipe over and take each of these prefixes, the 3.2, the 5 and the 2.5 and bring them out front on their own.

    01:34 3.2 times 5, divided by 2.5. And then we can make this much easier just by putting all of our powers of 10 over here on the right so we have 10 to the third times 10 to the sixth, divided by 10 to the ninth.

    01:54 Now the last thing we have to do is be careful to remember all of our rules of multiplying exponents and they look like this. 10 to the three times 10 to the six is 10 to the, just analyzing this part, three plus six over 10 to the nine.

    02:12 And you can see this will be 10 to the six plus three which is nine, divided by 10 to the nine.

    02:17 And so we see that these will cancel becoming one. So this entire term on the right hand side actually simplify down to being 1. Meaning that all we have to do is to analyze these number out front.

    02:29 3.2 times 5 divided by 2.5. A simple way to do this might be to see how many times 2.5 goes into 5, which is twice. Then this whole thing has simplified down to 3.2 times 2, which is 6.4, and this is our final answer, which looks a lot simpler than the original problem worked when we first jump into it. And this shows you that scientific notation is a great way to make any big problem be much more efficient just by thinking about the powers of 10 out on their own.

    03:05 Now we’re going to move to a… unit analysis and here’s a great example of how you can think of unit analysis.

    03:11 This equation you see in front of you is not something that you should try to memorize right now or worry about. The point is that we will in the future, pretty soon, get into an equation, like this one, which is an equation of motion, where this x depends on many different variables.

    03:25 X equals x for the zero plus v zero t and many other things. And what we’re interested in is what if you were in an exam setting and you didn’t remember exactly what the equation was that you have in front of you, whether it had an a times t in it, one half a times t, or whether it’s one half a times t squared.

    03:42 We can use unit analysis to quickly understand and remember which of these it should be without having to look back through any tables or any panicking. Here’s how you do this.

    03:52 If x depending on a times t squared versus x depending on a times t, we can look at the units of these.

    03:57 These units are something that we’ll go over in which you’ll know from a physical basis from the problem and something that we’ll discuss. Specifically, we’re gonna use brackets around the variables like these square brackets around x, square brackets around t, et cetera, to indicate the units of that variable.

    04:14 So brackets around x means the units of x are, and in this case the unit of x are meters.

    04:20 This is indicating a position variable. The units of t are seconds because this is our time variable and that we’ll discuss soon. The units of a are meters per second per second or meters per second squared.

    04:33 If you know these things you can immediately solve what we have above this information which is does x depend on a times t squared or does x depend on a times t. All we have to do to figure this out is analyze the units. If I look at the units of a times t squared, looking at these units below I can see that the units of acceleration are meters per second squared. So if I multiply by t squared, I’m multiplying by seconds squared. So the seconds squared from the numerator from t squared and the seconds squared from the denominator from this variable a will cancel evenly with just meters and that’s what we want, is meters for this variable x. However if I look at this other quantity, a times t, it’s just meters per second squared times seconds and only one of those units of second will cancel, leaving me with incorrect units for x which we know needs to be meters. It will leave me with meters per second.

    05:24 And so we can immediately see that the units of acceleration times time, which is what this a is, acceleration, the units of a times t are incorrect. They’re meters per second. And so just by looking at the units we can see that the units for this variable x, if you’re writing this out in an equation, must be a times t squared.

    About the Lecture

    The lecture Scientific Notation Example and Unit Analysis by Jared Rovny is from the course Methods and Common Calculations. It contains the following chapters:

    • Scientific Notation Example
    • Unit Analysis

    Included Quiz Questions

    1. (6.5 × 1.5/2.3) × 10^8
    2. (6.5 × 1.5/2.3) × 10^6
    3. (6.5 × 1.5/2.3) × 10^4
    4. (65 × 15/23) × 10^8
    5. (6.5 × 1.5/2.3) × 10^7
    1. [F] = [Ma]
    2. [F] = [M²a]
    3. [F] = [M/a]
    4. [F] = [Ma]²
    5. [F] = [a/M]
    1. 1.85 x 10^-3
    2. 0.185 x 10^-3
    3. 18.5 x 10^1
    4. 1.8 x 10^-4
    5. 1.8500 x 10^-2
    1. 2,500,000 × 1800 / 2,700
    2. 2,500,000 × 2700 / 1,800
    3. 25,000,000 / 1,800 × 2700
    4. 2.5^8 × 1,800 × 2700
    5. 2.5^8 × 2700 × 1,800
    1. (3.2 × 10^3)(5 × 10^6)/ (2.5 × 10^9)
    2. (5 × 10^3) (3.25 × 10^6)/ (2.5 × 10^9)
    3. (32 × 10^3) (5 × 10^6)/ (25 × 10^9)
    4. (32 × 10^6) (5 × 10^8)/ (2.5×10^9)
    5. (3.2 × 10^3) (5 × 10^6)/ (25 × 10^7)
    1. [x] = [v][t]
    2. [x] = [v]/[t]
    3. [x] = [v][t]²
    4. [x] = [v]²[t]
    5. [x] = [t]/[v]
    1. [a] = [v]/[t]
    2. [a] = [v]/[t]²
    3. [a] = [v]²/[t]
    4. [a] = [v][t]
    5. [a] = [v][t]²

    Author of lecture Scientific Notation Example and Unit Analysis

     Jared Rovny

    Jared Rovny

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