Basic Math & Pre-Algebra For Dummies, 2nd Edition (2014)
Part II. Getting a Handle on Whole Numbers
You can use prime factorization to find the greatest common factor (GCF) of difficult numbers. To find out how, go to www.dummies.com/extras/basicmathandprealgebra
In this part…
· Add, subtract, multiply, and divide more complex calculations involving negative numbers, inequalities, exponents, square roots, and absolute value
· Build equations and evaluate expressions
· Understand arithmetic word problems
· Employ a few quick tricks to determine whether one number is divisible by another
· Find the factors and multiples of a number, and discover whether a number is prime or composite
· Calculate the greatest common factor (GCF) and the least common multiple (LCM) of a set of numbers
Chapter 4. Putting the Big Four Operations to Work
In This Chapter
Identifying which operations are inverses of each other
Knowing the operations that are commutative, associative, and distributive
Performing the Big Four operations on negative numbers
Using four symbols for inequality
Understanding exponents, roots, and absolute values
When you understand the Big Four operations that I cover in Chapter 3 — adding, subtracting, multiplying, and dividing — you can begin to look at math on a whole new level. In this chapter, you extend your understanding of the Big Four operations and move beyond them. I begin by focusing on four important properties of the Big Four operations: inverse operations, commutative operations, associative operations, and distribution. Then I show you how to perform the Big Four on negative numbers.
I continue by introducing you to some important symbols for inequality. Finally, you're ready to move beyond the Big Four by discovering three more advanced operations: exponents (also called powers), square roots (also called radicals), and absolute values.
Knowing Properties of the Big Four Operations
When you know how to do the Big Four operations — add, subtract, multiply, and divide — you're ready to grasp a few important properties of these important operations. Properties are features of the Big Four operations that always apply, no matter what numbers you're working with.
In this section, I introduce you to four important ideas: inverse operations, commutative operations, associative operations, and the distributive property. Understanding these properties can show you hidden connections among the Big Four operations, save you time when calculating, and get you comfortable working with more-abstract concepts in math.
Inverse operations
Each of the Big Four operations has an inverse — an operation that undoes it. Addition and subtraction are inverse operations because addition undoes subtraction, and vice versa. For example, here are two equations with inverse operations:
· 1 + 2 = 3
· 3 − 2 = 1
In the first equation, you start with 1 and add 2 to it, which gives you 3. In the second equation, you have 3 and take away 2 from it, which brings you back to 1. The main idea here is that you're given a starting number — in this case, 1 — and when you add a number and then subtract the same number, you end up again with the starting number. This shows you that subtraction undoes addition.
Similarly, addition undoes subtraction — that is, if you subtract a number and then add the same number, you end up where you started. For example,
· 184 − 10 = 174
· 174 + 10 = 184
This time, in the first equation, you start with 184 and take away 10 from it, which gives you 174. In the second equation, you have 174 and add 10 to it, which brings you back to 184. In this case, starting with the number 184, when you subtract a number and then add the same number, the addition undoes the subtraction and you end up back at 184.
In the same way, multiplication and division are inverse operations. For example,
This time, you start with the number 4 and multiply it by 5 to get 20. And then you divide 20 by 5 to return to where you started at 4. So division undoes multiplication. Similarly,
Here, you start with 30, divide by 10, and multiply by 10 to end up back at 30. This shows you that multiplication undoes division.
Commutative operations
Addition and multiplication are both commutative operations. Commutative means that you can switch around the order of the numbers without changing the result. This property of addition and multiplication is called the commutative property. Here's an example of how addition is commutative:
· 3 + 5 = 8 is the same as 5 + 3 = 8
If you start out with 5 books and add 3 books, the result is the same as if you start with 3 books and add 5. In each case, you end up with 8 books.
And here's an example of how multiplication is commutative:
If you have 2 children and want to give them each 7 flowers, you need to buy the same number of flowers as someone who has 7 children and wants to give them each 2 flowers. In both cases, someone buys 14 flowers.
In contrast, subtraction and division are non-commutative operations. When you switch the order of the numbers, the result changes.
Here's an example of how subtraction is non-commutative:
Subtraction is non-commutative, so if you have $6 and spend $4, the result is not the same as if you have $4 and spend $6. In the first case, you still have $2 left over. In the second case, you owe $2. In other words, switching the numbers turns the result into a negative number. (I discuss negative numbers later in this chapter.)
And here's an example of how division is non-commutative:
For example, when you have five dog biscuits to divide between two dogs, each dog gets two biscuits and you have one biscuit left over. But when you switch the numbers and try to divide two biscuits among five dogs, you don't have enough biscuits to go around, so each dog gets none and you have two left over.
Associative operations
Addition and multiplication are both associative operations, which means that you can group them differently without changing the result. This property of addition and multiplication is also called the associative property.Here's an example of how addition is associative. Suppose you want to add 3 + 6 + 2. You can calculate in two ways:
In the first case, I start by adding 3 + 6 and then add 2. In the second case, I start by adding 6 + 2 and then add 3. Either way, the sum is 11.
And here's an example of how multiplication is associative. Suppose you want to multiply . You can calculate in two ways:
In the first case, I start by multiplying 5 × 2 and then multiply by 4. In the second case, I start by multiplying 2 × 4 and then multiply by 5. Either way, the product is 40. In contrast, subtraction and division are non-associative operations. This means that grouping them in different ways changes the result.
Don't confuse the commutative property with the associative property. The commutative property tells you that it's okay to switch two numbers that you're adding or multiplying. The associative property tells you that it's okay to regroup three (or more) numbers using parentheses.
Taken together, the commutative and associative properties allow you to completely rearrange and regroup a string of numbers that you're adding or multiplying without changing the result. You'll find the freedom to rearrange expressions as you like to be very useful as you move on to algebra in Part V.
Distribution to lighten the load
If you've ever tried to carry a heavy bag of groceries, you may have found that distributing the contents into two smaller bags is helpful. This same concept also works for multiplication.
In math, distribution (also called the distributive property of multiplication over addition) allows you to split a large multiplication problem into two smaller ones and add the results to get the answer.
For example, suppose you want to multiply these two numbers:
· 17 × 101
You can go ahead and just multiply them, but distribution provides a different way to think about the problem that you may find easier. Because 101 = 100 + 1, you can split this problem into two easier problems, as follows:
You take the number outside the parentheses, multiply it by each number inside the parentheses one at a time, and then add the products. At this point, you may be able to calculate the two multiplications in your head and then add them up easily:
· = 1,700 + 17 = 1,717
Distribution becomes even more useful when you get to algebra in Part V.
Doing Big Four Operations with Negative Numbers
In Chapter 1, I show you how to use the number line to understand how negative numbers work. In this section, I give you a closer look at how to perform the Big Four operations with negative numbers. Negative numbers result when you subtract a larger number from a smaller one. For example,
· 5 − 8 = −3
In real-world applications, negative numbers represent debt. For example, if you have only five chairs to sell but a customer pays for eight of them, you owe her three more chairs. Even though you may have trouble picturing −3 chairs, you still need to account for this debt, and negative numbers are the right tool for the job.
Addition and subtraction with negative numbers
The great secret to adding and subtracting negative numbers is to turn every problem into a series of ups and downs on the number line. When you know how to do this, you find that all these problems are quite simple.
So in this section, I explain how to add and subtract negative numbers on the number line. Don't worry about memorizing every little bit of this procedure. Instead, just follow along so you get a sense of how negative numbers fit onto the number line. (If you need a quick refresher on how the number line works, see Chapter 1.)
Starting with a negative number
When you're adding and subtracting on the number line, starting with a negative number isn't much different from starting with a positive number. For example, suppose you want to calculate −3 + 4. Using the up and down rules, you start at −3 and go up 4:
·
So −3 + 4 = 1.
Similarly, suppose you want to calculate −2 − 5. Again, the up and down rules help you out. You're subtracting, so move to the left: start at −2, down 5:
So −2 − 5 = −7.
Adding a negative number
Suppose you want to calculate −2 + −4. You already know to start at −2, but where do you go from there? Here's the up and down rule for adding a negative number:
Adding a negative number is the same as subtracting a positive number — go down on the number line.
By this rule, −2 + −4 is the same as −2 − 4, so start at −2, down 4:
So −2 + (−4) = −6.
Note: The problem −2 + −4 can also be written as −2 + (−4). Some people prefer to use this convention so that two operation symbols (− and +) aren't side by side. Don't let it trip you up. The problem is the same.
If you rewrite a subtraction problem as an addition problem — for instance, rewriting 3 − 7 as 3 + (−7) — you can use the commutative and associative properties of addition, which I discuss earlier in this chapter. Just remember to keep the negative sign attached to the number when you rearrange: (−7) + 3.
Subtracting a negative number
The last rule you need to know is how to subtract a negative number. For example, suppose you want to calculate 2 − (−3). Here's the up and down rule:
Subtracting a negative number is the same as adding a positive number — go up on the number line.
This rule tells you that 2 − (−3) is the same as 2 + 3, so start at 2, up 3:
So 2 − (−3) = 5.
When subtracting negative numbers, you can think of the two minus signs canceling each other out to create a positive.
Multiplication and division with negative numbers
Multiplication and division with negative numbers is virtually the same as with positive numbers. The presence of one or more minus signs (−) doesn't change the numerical part of the answer. The only question is whether the sign is positive or negative:
Just remember that when you multiply or divide two numbers,
· If the numbers have the same sign, the result is always positive.
· If the numbers have opposite signs, the result is always negative.
For example,
As you can see, the numerical portion of the answer is always 6. The only question is whether the complete answer is 6 or −6. That's where the rule of same or opposite signs comes in.
Another way of thinking of this rule is that the two negatives cancel each other out to make a positive.
Similarly, look at these four division equations:
In this case, the numerical portion of the answer is always 5. When the signs are the same, the result is positive, and when the signs are different, the result is negative.
Understanding Units
Anything that can be counted is a unit. That category is a pretty large one because almost anything that you can name can be counted. You discover more about units of measurement in Chapter 15. For now, just understand that all units can be counted, which means that you can apply the Big Four operations to units.
Adding and subtracting units
Adding and subtracting units isn't very different from adding and subtracting numbers. Just remember that you can add or subtract only when the units are the same. For example,
· 3 chairs + 2 chairs = 5 chairs
4 oranges − 1 orange = 3 oranges
What happens when you try to add or subtract different units? Here's an example:
· 3 chairs + 2 tables = ?
The only way you can complete this addition is to make the units the same:
· 3 pieces of furniture + 2 pieces of furniture = 5 pieces of furniture
Multiplying and dividing units
You can always multiply and divide units by a number. For example, suppose you have four chairs and but find that you need twice as many for a party. Here's how you represent this idea in math:
Similarly, suppose you have 20 cherries and want to split them among four people. Here's how you represent this idea:
But you have to be careful when multiplying or dividing units by units. For example:
Neither of these equations makes any sense. In these cases, multiplying or dividing by units is meaningless.
In many cases, however, multiplying and dividing units is okay. For example, multiplying units of length (such as inches, miles, or meters) results in square units. For example,
You find out more about units of length in Chapter 15. Similarly, here are some examples of when dividing units makes sense:
In these cases, you read the fraction slash (/) as per: slices of pizza per person or miles per hour. You find out more about multiplying and dividing by units in Chapter 15, when I show you how to convert from one unit of measurement to another.
Understanding Inequalities
Sometimes you want to talk about when two quantities are different. These statements are called inequalities. In this section, I discuss six types of inequalities: ≠ (doesn't equal), < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to), and ≈ (approximately equals).
Doesn't equal (≠)
The simplest inequality is ≠, which you use when two quantities are not equal. For example,
You can read ≠ as “doesn't equal” or “is not equal to.” Therefore, read as “two plus two doesn't equal five.”
Less than (<) and greater than (>)
The symbol < means less than. For example, the following statements are true:
· 4 < 5
100 < 1,000
2 + 2 < 5
Similarly, the symbol > means greater than. For example,
· 5 > 4
100 > 99
2 + 2 > 3
The two symbols < and > are similar and easily confused. Here are two simple ways to remember which is which:
· Notice that the < looks sort of like an L. This L should remind you that it means less than.
· Remember that, in any true statement, the large open mouth of the symbol is on the side of the greater amount, and the small point is on the side of the lesser amount.
Less than or equal to (≤) and greater than or equal to (≥)
The symbol ≤ means less than or equal to. For example, the following statements are true:
Similarly, the symbol ≥ means greater than or equal to. For example,
The symbols ≤ and ≥ are called inclusive inequalities because they include (allow) the possibility that both sides are equal. In contrast, the symbols < and > are called exclusive inequalities because they exclude (don't allow) this possibility.
Approximately equals (≈)
In Chapter 2, I show you how rounding numbers makes large numbers easier to work with. In that chapter, I also introduce ≈, which means approximately equals.
For example,
You can also use ≈ when you estimate the answer to a problem:
Moving Beyond the Big Four: Exponents, Square Roots, and Absolute Value
In this section, I introduce you to three new operations that you need as you move on with math: exponents, square roots, and absolute value. As with the Big Four operations, these three operations tweak numbers in various ways.
To tell the truth, these three operations have fewer everyday applications than the Big Four. But you'll be seeing a lot more of them as you progress in your study of math. Fortunately, they aren't difficult, so this is a good time to become familiar with them.
Understanding exponents
Exponents (also called powers) are shorthand for repeated multiplication. For example, 2^{3} means to multiply 2 by itself three times. To do that, use the following notation:
In this example, 2 is the base number and 3 is the exponent. You can read 2^{3} as “2 to the third power” or “2 to the power of 3” (or even “2 cubed,” which has to do with the formula for finding the value of a cube — see Chapter 16 for details).
Here's another example:
· 10^{5} means to multiply 10 by itself five times
That works out like this:
This time, 10 is the base number and 5 is the exponent. Read 10^{5} as “10 to the fifth power” or “10 to the power of 5.”
When the base number is 10, figuring out any exponent is easy. Just write down a 1 and that many 0s after it:
1 with two 0s |
1 with seven 0s |
1 with twenty 0s |
10^{2} = 100 |
10^{7} = 10,000,000 |
10^{20} = 100,000,000,000,000,000,000 |
Exponents with a base number of 10 are important in scientific notation, which I cover in Chapter 14.
The most common exponent is the number 2. When you take any whole number to the power of 2, the result is a square number. (For more information on square numbers, see Chapter 1.) For this reason, taking a number to the power of 2 is called squaring that number. You can read 3^{2} as “three squared,” 4^{2} as “four squared,” and so forth. Here are some squared numbers:
Any number (except 0) raised to the 0 power equals 1. So 1^{0}, 37^{0}, and 999,999^{0} are equivalent, or equal, because they all equal 1.
Discovering your roots
Earlier in this chapter, in “Knowing Properties of the Big Four Operations,” I show you how addition and subtraction are inverse operations. I also show you how multiplication and division are inverse operations. In a similar way, roots are the inverse operation of exponents.
The most common root is the square root. A square root undoes an exponent of 2. For example,
You can read the symbol either as “the square root of” or as “radical.” So read as either “the square root of 9” or “radical 9.”
As you can see, when you take the square root of any square number, the result is the number that you multiplied by itself to get that square number in the first place. For example, to find , you ask the question, “What number when multiplied by itself equals 100?” The answer here is 10 because
You probably won't use square roots much until you get to algebra, but at that point, they become handy.
Figuring out absolute value
The absolute value of a number is the positive value of that number. It tells you how far away from 0 a number is on the number line. The symbol for absolute value is a set of vertical bars.
Taking the absolute value of a positive number doesn't change that number's value. For example,
· |3| = 3
|12| = 12
|145| = 145
However, taking the absolute value of a negative number changes it to a positive number:
· |−5| = 5
|−10| = 10
|−212| = 212
Finally, the absolute value of 0 is simply 0:
· |0| = 0