Basic Math & Pre-Algebra For Dummies, 2nd Edition (2014)
Part IV. Picturing and Measuring - Graphs, Measures, Stats, and Sets
For find out how to use probability to calculate the odds in dice games, go to www.dummies.com/extras/basicmathandprealgebra.
In this part…
· Represent very large and very small numbers with scientific notation
· Weigh and measure with both the English and metric systems
· Understand basic geometry, including points, lines, and angles, plus basic shapes and solids
· Present math info visually, using bar graphs, pie charts, line graphs, and the xy-graph
· Solve word problems involving measurement and geometry
· Answer real-world questions with statistics and probability
· Get familiar with some basic set theory, including union and intersection
Chapter 14. A Perfect Ten: Condensing Numbers with Scientific Notation
In This Chapter
Knowing how to express powers of ten in exponential form
Appreciating how and why scientific notation works
Understanding order of magnitude
Multiplying numbers in scientific notation
Scientists often work with very small or very large measurements — the distance to the next galaxy, the size of an atom, the mass of the Earth, or the number of bacteria cells growing in last week's leftover Chinese takeout. To save on time and space — and to make calculations easier — people developed a sort of shorthand called scientific notation.
Scientific notation uses a sequence of numbers known as the powers of ten, which I introduce in Chapter 2:
· 1 10 100 1,000 10,000 100,000 1,000,000 10,000,000 ...
Each number in the sequence is 10 times more than the preceding number.
Powers of ten are easy to work with, especially when you're multiplying and dividing, because you can just add or drop zeros or move the decimal point. They're also easy to represent in exponential form (as I show you in Chapter 4):
·
Scientific notation is a handy system for writing very large and very small numbers without writing a bunch of 0s. It uses both decimals and exponents (so if you need a little brushing up on decimals, flip to Chapter 11). In this chapter, I introduce you to this powerful method of writing numbers. I also explain the order of magnitude of a number. Finally, I show you how to multiply numbers written in scientific notation.
First Things First: Using Powers of Ten as Exponents
Scientific notation uses powers of ten expressed as exponents, so you need a little background before you can jump in. In this section, I round out your knowledge of exponents, which I first introduce in Chapter 4.
Counting zeros and writing exponents
Numbers starting with a 1 and followed by only 0s (such 10, 100, 1,000, 10,000, and so forth) are called powers of ten, and they're easy to represent as exponents. Powers of ten are the result of multiplying 10 times itself any number of times.
To represent a number that's a power of 10 as an exponential number, count the zeros and raise 10 to that exponent. For example, 1,000 has three zeros, so 1,000 = 10^{3} (10^{3} means to take 10 times itself three times, so it equals 10 × 10 × 10). Table 14-1 shows a list of some powers of ten.
Table 14-1 Powers of Ten Expressed as Exponents
Number |
Exponent |
1 |
10^{0} |
10 |
10^{1} |
100 |
10^{2} |
1,000 |
10^{3} |
10,000 |
10^{4} |
100,000 |
10^{5} |
1,000,000 |
10^{6} |
When you know this trick, representing a lot of large numbers as powers of ten is easy — just count the 0s! For example, the number 1 trillion — 1,000,000,000,000 — is a 1 with twelve 0s after it, so
This trick may not seem like a big deal, but the higher the numbers get, the more space you save by using exponents. For example, a really big number is a googol, which is 1 followed by a hundred 0s. You can write this:
· 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
As you can see, a number of this size is practically unmanageable. You can save yourself some trouble and write 10^{100}.
A 10 raised to a negative number is also a power of ten.
You can also represent decimals using negative exponents. For example,
Although the idea of negative exponents may seem strange, it makes sense when you think about it alongside what you know about positive exponents. For example, to find the value of 10^{7}, start with 1 and make it larger by moving the decimal point seven spaces to the right:
Similarly, to find the value of 10^{–7}, start with 1 and make it smaller by moving the decimal point seven spaces to the left:
Negative powers of 10 always have one fewer 0 between the 1 and the decimal point than the power indicates. In this example, notice that 10^{–7} has six 0s between them.
As with very large numbers, using exponents to represent very small decimals makes practical sense. For example,
As you can see, this decimal is easy to work with in its exponential form but almost impossible to read otherwise.
Adding exponents to multiply
An advantage of using the exponential form to represent powers of ten is that this form is a cinch to multiply. To multiply two powers of ten in exponential form, add their exponents. Here are a few examples:
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Here, I simply multiply these numbers:
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Here's what I'm multiplying: 100,000,000,000,000 × 1,000,000,000,000,000 = 100,000,000,000,000,000,000,000,000,000
You can verify that this multiplication is correct by counting the 0s.
·
Here I'm multiplying a googol by 1 (any number raised to an exponent of 0 equals 1), so the result is a googol.
In each of these cases, you can think of multiplying powers of ten as adding extra 0s to the number.
The rules for multiplying powers of ten by adding exponents also apply to negative exponents. For example,
Working with Scientific Notation
Scientific notation is a system for writing very large and very small numbers that makes them easier to work with. Every number can be written in scientific notation as the product of two numbers (two numbers multiplied together):
· A decimal greater than or equal to 1 and less than 10 (see Chapter 11 for more on decimals)
· A power of ten written as an exponent (see the preceding section)
Writing in scientific notation
Here's how to write any number in scientific notation:
1. Write the number as a decimal (if it isn't one already).
Suppose you want to change the number 360,000,000 to scientific notation. First, write it as a decimal:
o 360,000,000.0
2. Move the decimal point just enough places to change this number to a new number that's between 1 and 10.
Move the decimal point to the right or left so that only one nonzero digit comes before the decimal point. Drop any leading or trailing zeros as necessary.
Using 360,000,000.0, only the 3 should come before the decimal point. So move the decimal point eight places to the left, drop the trailing zeros, and get 3.6:
o 360,000,000.0 becomes 3.6.
3. Multiply the new number by 10 raised to the number of places you moved the decimal point in Step 2.
You moved the decimal point eight places, so multiply the new number by 10^{8}:
o 3.6 × 10^{8}
4. If you moved the decimal point to the right in Step 2, put a minus sign on the exponent.
You moved the decimal point to the left, so you don't have to take any action here. Thus, 360,000,000 in scientific notation is 3.6 × 10^{8}.
Changing a decimal to scientific notation basically follows the same process. For example, suppose you want to change the number 0.00006113 to scientific notation:
1. Write 0.00006113 as a decimal (this step's easy because it's already a decimal):
o 0.00006113
2. To change 0.00006113 to a new number between 1 and 10, move the decimal point five places to the right and drop the leading zeros:
o 6.113
3. Because you moved the decimal point five places to the right, multiply the new number by 10^{-5}:
So 0.00006113 in scientific notation is .
When you get used to writing numbers in scientific notation, you can do it all in one step. Here are a few examples:
Seeing why scientific notation works
When you understand how scientific notation works, you're in a better position to understand why it works. Suppose you're working with the number 4,500. First of all, you can multiply any number by 1 without changing it, so here's a valid equation:
Because 4,500 ends in a 0, it's divisible by 10 (see Chapter 7 for info on divisibility). So you can factor out a 10 as follows:
Also, because 4,500 ends in two 0s, it's divisible by 100, so you can factor out 100:
In each case, you drop another 0 after the 45 and place it after the 1. At this point, you have no more 0s to drop, but you can continue the pattern by moving the decimal point one place to the left:
What you've been doing from the beginning is moving the decimal point one place to the left and multiplying by 10. But you can just as easily move the decimal point one place to the right and multiply by 0.1, two places right by multiplying by 0.01, and three places right by multiplying by 0.001:
As you can see, you have total flexibility to express 4,500 as a decimal multiplied by a power of ten. As it happens, in scientific notation, the decimal must be between 1 and 10, so the following form is the equation of choice:
The final step is to change 1,000 to exponential form. Just count the 0s in 1,000 and write that number as the exponent on the 10:
The net effect is that you moved the decimal point three places to the left and raised 10 to an exponent of 3. You can see how this idea can work for any number, no matter how large or small.
Understanding order of magnitude
A good question to ask is why scientific notation always uses a decimal between 1 and 10. The answer has to do with order of magnitude. Order of magnitude is a simple way to keep track of roughly how large a number is so you can compare numbers more easily. The order of magnitude of a number is its exponent in scientific notation. For example,
Every number starting with 10 but less than 100 has an order of magnitude of 1. Every number starting with 100 but less than 1,000 has an order of magnitude of 2.
Multiplying with scientific notation
Multiplying numbers that are in scientific notation is fairly simple because multiplying powers of ten is easy, as you see earlier in this chapter in “Adding exponents to multiply.” Here's how to multiply two numbers that are in scientific notation:
1. Multiply the two decimal parts of the numbers.
Suppose you want to multiply the following:
Multiplication is commutative (see Chapter 4), so you can change the order of the numbers without changing the result. And because of the associative property, you can also change how you group the numbers. Therefore, you can rewrite this problem as
Multiply what's in the first set of parentheses — — to find the decimal part of the solution:
2. Multiply the two exponential parts by adding their exponents.
Now multiply :
3. Write the answer as the product of the numbers you found in Steps 1 and 2.
4. If the decimal part of the solution is 10 or greater, move the decimal point one place to the left and add 1 to the exponent.
Because 8.6 is less than 10, you don't have to move the decimal point again, so the answer is 8.6 × 10^{12}.
Note: This number equals 8,600,000,000,000.
This method works even when one or both of the exponents are negative numbers. For example, if you follow the preceding series of steps, you find that (6.02 × 10^{23})(9 × 10^{–28}) = 5.418 × 10^{–4}. Note: In decimal form, this number equals 0.0005418.