Physics For Beginners page 09
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- Gravity (continued)
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6. Gravity (continued)
- Scientific Notation: A Necessary Evil and Another Detour
First, let me apologize right off the bat. I had stated we were going to learn a "little" more about big G, the mysterious gravitational constant. We will have to wait until the next section for that because, once again, we have to take another detour. In the next section, we will have to deal with some pretty big numbers. For example, the mass of the earth in kilograms is a very large number. In order to deal with these numbers, I am going to introduce the scientific notation. Don't let this big name scare you. There is nothing to be afraid of. The scientific notation is just a way for us to keep track of really big and really small numbers. I won't go into too much detail about this topic. I just want to introduce enough of it so we are comfortable working with very large and small numbers. Finally, it goes without saying that if you already know about the scientific notation, please disregard this section.
- Sliding to the Right
The weird title of this section should make more sense as you read on. And, no, it doesn't have anything to do with a Will Smith song, though I could really watch M.I.B. (which, by the way, still hasn't been released on DVD as of 2/1/2000) again right about now. Ok, well, enough about Will Smith.
To introduce this section, I am going to do some multiplying, but first we will need a number. Since I am the one typing, I'm going to use my favorite number, 1.37. Feel free to choose your own number because the following discussion will be true for any number. It won't have any bearing on this discussion.
Next, we will multiply our special number (1.37) by several numbers. We will first multiply 1.37 by 10. After that, we will multiply 1.37 by 100. After that, we will multiply 1.37 by 1000. You should be getting the picture by now. Basically, the numbers we are multiplying with 1.37 are numbers that all start with a 1 at the beginning followed by some zeroes. For example: 10, 100, 1000, 10000,.....
First, let's multiply 1.37 by 10. This is fairly trivial, but pay attention to how the decimal point in 1.37 moves compared to the number of zeroes in 10.
1.37 x 10 = 13.7
If you were paying attention, you will notice that the decimal point moved to the right by one place, which is exactly the same number of zeroes in the number 10. In the picture below, I have done this explicitly so it is easier to see.
Okay, well, let's do another example. Let's multiply 1.37 by 100. Once again, pay attention to the way the decimal point moves compared to the number of zeroes in 100.
1.37 x 100 = 137.0
If you were paying attention, you should notice that the decimal point moved to the right by two places, which is exactly the same number of zeroes in 100. Once again, I've included a picture below to illustrate this point.
Well, it should be fairly obvious what's going on by now. Every time we multiply 1.37 by a number beginning with 1 followed by some zeroes, the result is that the decimal point in 1.37 moves to the right by the same number of zeroes in the number we multiplied with 1.37. The main thing to keep in mind is that this only works for numbers that start with 1 followed by some zeroes.
For example, if we multiply 1.37 by 1000, the result is that we move the decimal point in 1.37 three places to the right because there are three zeroes in 1000.
1.37 x 1000 = 1370.0
However, this would not be true if we multiplied 1.37 by 1001 because the numbers to the right of the first 1 are not all zeroes.
In summary, multiplying numbers by 10, 100, 1000, 10000... is very easy. The result of multiplying by these numbers can be easily obtained by moving the decimal point to the right by the number of zeroes following the 1 out front. For example, the result of 5.7 x 10000 is 57000 which is easily obtained by moving the decimal point in 5.7 to the right by four places because there are 4 zeroes in 10000. As you continue reading, keep this little fact in mind.
At this point, you might be asking why we are doing something so simple as multiplying things by 10, 100, 1000,.... Well, the reason we are doing this is because this is the basis of the scientific notation.
So, without further ado, let me introduce the scientific notation. While you are reading this, keep in mind that its purpose is just to keep track of very big or very small numbers. I'll begin by giving you one example.
Example 1: What number does 1.37E+3 represent?
Ok, let me first explain the above notation. If you will notice, there is my favorite number (1.37) out front. After my favorite number, there is a capital E followed by a +3.
What does the E+3 mean? Well, the E+3 is telling us to multiply 1.37 by a number with 1 on left followed by three zeroes. In other words, it is telling us to multiply 1.37 by 1000. The number "3" in E+3 is what tells us how many zeroes follow the beginning 1 in 1000. I have written this explicitly out below.
1.37E+3 = 1.37 x 1000
Well, now we know that 1.37E+3 is just a fancy way of writing 1.37 x 1000, we can figure out the answer easily because we know how to multiply by numbers like 1000. As you will recall from above, all we have to do is move the decimal point in 1.37 to the right by three places because there are three zeroes in 1000. I have written the answer explicitly below.
1.37E+3 = 1.37 x 1000 = 1370.0
To repeat, 1.37E+3 is just a compact way of writing 1.37 x 1000. They mean the same thing. Once we knew that 1.37E+3 was the same thing as 1.37 x 1000, we easily figured out the answer was 1370.0 because all we had to do was move the decimal point in 1.37 to the right by three places.
Let's do another example to get some practice with the scientific notation.
Example 2: What number does 1.37E+6 represent?
This time the E+6 is telling us to multiply the 1.37 by 1000000. Remember the number "6" in E+6 tells us how many zeroes follow the 1 in 1000000. Once we know that 1.37E+6 means 1.37 x 1000000, we can obtain the answer because we know how to multiply by numbers like 1000000. All we have to do is move the decimal point in 1.37 to the right by the number of zeroes in 1000000.
1.37E+6 = 1.37 x 1000000 = 1370000.0
Do you see the convenience of using the scientific notation? Instead of writing 1370000.0, I can write the same number much more compactly as 1.37E+6. With these numbers, you might not mind writing those zeroes out, but with much bigger numbers, the scientific notation really comes in handy.
Can you imagine writing out 1.37E+120? How much time would it take you to write out all those zeroes? In addition, you would have to go back and count the number of zeroes to make sure the number was correct. You can see how the scientific notation makes keeping track of very large numbers much easier.
Now, before we move on, let me make one final observation. To do this, let's do another example. This time we'll use a different number. Remember the scientific notation works for all other numbers, not just my favorite number. Plus, I'm getting tired of my favorite number if you can believe that.
Example 3: What number does 2.17E+4 represent?
This time, in addition to multiplying, pay careful attention to how the decimal point in 2.17 moves and compare it to E+4. If you were paying attention earlier, you probably already know the answer to this.
Well, 2.17E+4 tells us to multiply 2.17 by 10000. Remember, the number of zeroes in 10000 is the same as the number in E+4. Since we know everything about multiplying with numbers like 10000, the answer is easily obtained. All we have to do is move the decimal point in 2.17 to the right by four places because there are four zeroes in 10000.
2.17E+4 = 2.17 x 10000 = 21700.0
If you will notice, the decimal point in 2.17 moves to the right by four places. This shouldn't come as too much of a surprise because we discussed this very same thing at the beginning of this section. If you compare the number of times the decimal point moves to the right with the number in E+4, you should notice they are the same.
Therefore, to get the number that 2.17E+4 represents, all we have to do is move the decimal point over to the right by four places. If you go back and look over the first two examples, you will notice that a similar thing happens in those examples.
In general, whenever you encounter a number like 3.14E+25, what it is telling you is that the real number is obtained by moving the decimal point to the right by the positive number that follows the E. In this specific case, the real number is obtained by taking the decimal point in 3.14 and moving it to the right by 25 places. Can you imagine writing this number out in all its glory? It should be obvious by now how convenient the scientific notation is for representing very large numbers.
Example 4: Which is larger? 6.1E+4 or 6.1E+5
First of all, see if you can do this without actually multiplying any numbers. Try to figure out what number 6.1E+4 represents and compare it to the number 6.1E+5 represents without actually doing any multiplication. You should be able to do this easily by moving the decimal points.
Well, 6.1E+4 is actually 61000.0. In addition, 6.1E+5 is actually 610000.0. Therefore, it is obvious that 6.1E+5 is larger. Remember, the positive number that follows the E tells us how many places to move the decimal point to the right.
Another way of seeing that 6.1E+5 is larger than 6.1E+4 is to figure out what the 6.1's are multiplied by.
If you will recall, 6.1E+4 = 6.1 x 10000
(In fact, that is what 6.1E+4 really means. Note that the number of zeroes in 10000 is the same as the number in E+4)Likewise, 6.1E+5 = 6.1 x 100000
(Similarly, that is what 6.1E+5 really means. Note that the number of zeroes in 100000 is the same as the number in E+5)Since the 6.1 in 6.1E+5 is multiplied by the larger number, we know 6.1E+5 is larger than 6.1E+4.
If you understood this last example, then you have a good understanding of this aspect of the scientific notation. If you are still having some troubles, take some time now and go back to review this section before moving on to the next section.
- Sliding to the Left
We started out the last section by taking my favorite number and multiplying it by 10, 100, 1000,... So, let's start out this section by taking my favorite number and dividing it by 10, 100, 1000,....
Once again, pay attention to how the decimal point moves compared to the number of zeroes in 10, 100, 1000,....
First, let's divide 1.37 by 10.
1.37 / 10 = .137
If you will notice, the decimal point has moved to the left by one place, which is exactly the number of zeroes in 10. I have done this explicitly in the figure below.
Next, let's divide my favorite number by 100.
1.37 / 100 = .0137
Once again, the decimal point moves to the left by the same number of zeroes in 100. In this case, since there are two zeroes in 100, the decimal point in 1.37 moves to the left by two places. I have done this explicitly in the figure below.
Therefore, whenever we divide 1.37 by a number with a 1 at the beginning followed by some zeroes, the result is that the decimal point in 1.37 moves to the left by the number of zeroes in the number with 1 at the front followed by some zeroes. For instance, if you divide 1.37 by 1000, the result can be obtained by moving the decimal point in 1.37 to the left by three places because there are three zeroes in 1000.
1.37 / 1000 = .00137
As in the discussion above, this is only true if the number we divide 1.37 by has a 1 at the front followed by some zeroes. Therefore, this will be true for 10, 100, 1000, 10000,... However, this will not be true for numbers like 101.
In summary, we see that dividing numbers by 10, 100, 1000, 10000... is very easy. The result of dividing by these numbers can be easily obtained by moving the decimal point to the left by the number of zeroes following the 1 out front. For example, the result of 9.1 / 100 is 0.091 which is easily obtained by moving the decimal point in 9.1 to the left by two places because the number of zeroes following the 1 in 100 is two. Once again, keep this fact in mind as you continue reading.
Ok, I can already tell that you know where this is leading, but I'll go ahead and say it. In the above discussion regarding the scientific notation, we introduced the scientific notation. If you were careful, you might have noticed that the number following the E was a positive number. In fact, we explicitly wrote the + sign in front of the number which followed the E. If you will recall, the scientific notation was just a way of writing huge numbers compactly. The (+) number following the E told us that the real number could be obtained by moving the decimal point to the right by the (+) number following the E. For instance, 2.0E+5 is just a compact way of writing 200000. Notice how the decimal point in the real number, 200000, can be obtained from 2.0E+5 by moving the decimal point in 2.0 to the right by five places. If you are not comfortable with this discussion, please make sure you understand this section thoroughly before proceeding.
In the above section, the number following the E in the scientific notation was positive. In addition, we were multiplying numbers by such numbers as 10, 100, 1000, 10000... In this section, we started out by dividing numbers by 10, 100, 1000, 10000... Hmmm.... I wonder whether the number following the E in this section will be positive or negative?
Once again, I will introduce the scientific notation by an example.
Example 1: What number does 1.37E-3 represent?
If you will notice, this is very similar to the example 1 in the section above. The only difference is that the number following the E is negative instead of positive. As we will see, that makes all the difference in the world.
In this case, the E-3 following my favorite number (1.37) is telling us to divide my favorite number by a number with 1 in front followed by three zeroes. In other words, it is telling us to divide 1.37 by 1000. The number "3" in E-3 is what tells us how many zeroes follow the 1 in 1000. The (-) sign in E-3 tells us that we have to divide instead of multiply. I have written this out explicitly below.
1.37E-3 = 1.37 / 1000 = 0.00137
While the (+) sign in 1.37E+3 told us to multiply 1.37 by 1000, the (-) sign in 1.37E-3 tells us to divide 1.37 by 1000. That is the only difference. The sign following the E tells us whether to multiply or divide. The number following the sign tells us the number of zeroes that follow the 1 in 1000.
Did you pay attention to how many places the decimal point moved to the left in this example? If you didn't, go back and look at the example and compare the number of places the decimal point in 1.37 moves to the left with the number in E-3.
If you will recall, we moved the decimal point in 1.37 to the left by three places which is exactly the number in E-3. This should come as no surprise because we know that 1.37E-3 really means 1.37 / 1000. And, as we saw earlier, dividing by numbers like 1000 is very easy. In fact, all we have to do is move the decimal point in 1.37 to the left by 3 places.
Ok, well, that wasn't too hard. Let's do a couple more examples to make sure we understand what is going on.
Example 2: What number does 5.7E-4 represent?
There are no tricks here. Try this example on your own first. If you don't get the answer, take a look at the previous example for some help. Pay careful attention to how the decimal point moves in this example compared to the number in E-4.
Well, as we said, the (-) sign in E-4 just means we have to divide instead of multiply. The number "4" in E-4 just means the number we are dealing with is 10000 because the "4" in E-4 tells us the number we divide by should be 1 followed by four zeroes.
Therefore, 5.7E-4 is really 5.7 / 10000
Now that we know this, the answer is very easy to obtain because we know everything about dividing by numbers like 10000. All we have to do is move the decimal point in 5.7 by four places to the left because there are four zeroes in 10000.
As a result, the answer is 0.00057
If you will notice, we moved the decimal point in 5.7 to the left by four places which is the same as the number in E-4. This was as expected because we knew that 5.7E-4 is just a compact way of writing 5.7 / 10000, and we know how to divide by numbers like 10000.
In general, whenever you see a number like 2.1E-9, to get the number that 2.1E-9 really represents, all we have to do is move the decimal point in 2.1 to the left by 9 places. We know the number of places to move by the number "9" in E-9, and we know to move left because of the (-) sign in E-9.
Example 3: Which number is smaller? 3.2E-3 or 3.2E-5
See if you can do this without actually doing any calculations. You should be able to do this just by moving the decimal points around. Once again, try it on your own before reading further.
Well, the easiest way to see this is to write down what these numbers really are.
3.2E-3 = 0.0032
3.2E-5 = 0.000032
It should be obvious now that 3.2E-5 is smaller than 3.2E-3
Another way of seeing this is to see what numbers the 3.2's are divided by.
If you will recall, 3.2E-3 is the same thing as 3.2 / 1000
(Notice the number of zeroes in 1000 is the same as the number in E-3)Likewise, 3.2E-5 is the same thing as 3.2 / 100000
(Notice the number of zeroes in 100000 is the same as the number in E-5)Since the 3.2 in 3.2E-5 is divided by a larger number, the result is smaller. Therefore, 3.2E-5 is smaller than 3.2E-3. If this is unclear, try dividing the numbers yourself and compare the result. Well, if you understood this last example, then you have a good understanding of the scientific notation. Before moving on, let me just summarize what we learned in this section.
- Summary
It is often a good idea, when dealing with a new idea, to summarize all the main points in one area for easy review. To summarize, I'll just choose a specific number as an example. Remember, however, that this will be true in general. Finally, keep in mind that there is no new mathematics involved with the scientific notation other than multiplication and division. It is just a convenient and compact way to handle large and small numbers.
- 1.23E+4
This is just a fancy way of writing 1.23 x 10000. The (+) sign in E+4 tells us to multiply. The number "4" in E+4 tells us to multiply 1.23 by a number with 1 at the front followed by 4 zeroes. In other words, it tells us to multiply by 10000.Once we know this, the result is easy because we know how to multiply by numbers like 10000. All we have to do is move the decimal point in 1.23 to the right by the number of zeroes in 10000.
In this case, we have to move the decimal point in 1.23 to the right by 4 places because there are 4 zeroes in 10000. Once you are comfortable with this, you can just move the decimal point without having to actually multiply anything. When you see a number like 1.23E+4, you will instinctively know to move the decimal point in 1.23 to the right by 4 places.
- 1.23E+4 = 1.23 x 10000 = 12300
- 1.23E+4 = 1.23 x 10000 = 12300
- 1.23E-4
This is just a fancy way of writing 1.23 / 10000. The (-) sign in E-4 tells us to divide. The number "4" in E-4 tells us to divide 1.23 by a number with 1 at the front followed by 4 zeroes. In other words, it tells us to divide by 10000.Once we know this, the result is easy because we know how to divide by numbers like 10000. All we have to do is move the decimal point in 1.23 to the left by the number of zeroes in 10000.
In this case, we have to move the decimal point in 1.23 to the left by 4 places because there are 4 zeroes in 10000. Once you are comfortable with this, you can just move the decimal point without having to actually divide anything. When you see a number like 1.23E-4, you will instinctively know to move the decimal point in 1.23 to the left by 4 places.
- 1.23E-4 = 1.23 / 10000 = 0.000123
- 1.23E-4 = 1.23 / 10000 = 0.000123
- 1.23E+4
Oh, by the way, that Men In Black dvd still hasn't been released since I started typing. Oh well.
(As an aside, for those of you with a scientific calculator, you can actually enter numbers in scientific notation. For example, to enter a number like 2.5E+25, you would enter 2.5 first. Next, look for a button labeled something like EXP. Press that button. Finally, enter the +25. It is a fairly simple affair. Get some practice now because we'll be dealing with big numbers next time.)
- Sliding to the Right