way to compare numbers and to tell which is bigger or has a greater value is to find each number’s position on the number line. The number line goes from negatives on the left to positives on the right (see Figure 2-1). Whichever number is farther to the right has the greater value, meaning it’s bigger.
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Figure 2-1: A number line.
Examples
Q. Using the number line in Figure 2-1, determine which is larger, –16 or –10.
A. The number –10 is to the right of –16, so it’s the bigger of the two numbers.
Q. Which is larger, –1.6 or –1.04?
A. The number –1.04 is to the right of –1.6, so it’s larger. A nice way to compare decimals is to write them with the same number of decimal places. So rewrite –1.6 as –1.60; it’s easier to compare to –1.04 in this format.
Comparing Positives and Negatives with Symbols
Although my mom always told me not to compare myself to other people, comparing numbers to other numbers is often useful. And, when you compare numbers, the greater-than sign (>) and less-than sign (<) come in handy, which is why I use them in Table 2-1, where I put some positive- and negative-signed numbers in perspective.
Table 2-1 Comparing Positive and Negative Numbers
Two other signs related to the greater-than and less-than signs are the greater-than-or-equal-to sign (≥) and the less-than-or-equal-to sign (≤).
So, putting the numbers 6, –2, –18, 3, 16, and –11 in order from smallest to biggest gives you: –18, –11, –2, 3, 6, and 16, which are shown as dots on a number line in Figure 2-2.
Figure 2-2: Positive and negative numbers on a number line.
Zeroing in on zero
But what about 0? I keep comparing numbers to see how far they are from 0. Is 0 positive or negative? The answer is that it’s neither. Zero has the unique distinction of being neither positive nor negative. Zero separates the positive numbers from the negative ones – what a job!
Practice Questions
1. Which is larger, –2 or –8?
2. Which has the greater value, –13 or 2?
3. Which is bigger, –0.003 or –0.03?
4. Which is larger,
Practice Answers
1. -2. The following number line shows that the number –2 is to the right of –8. So –2 is bigger than –8. This is written –2 > –8.
2. 2. The number 2 is to the right of –13. So 2 has a greater value than –13. This is written 2 > –13.
3. – 0.003. The following number line shows that the number –0.003 is to the right of –0.03, which means –0.003 is bigger than –0.03. You can also rewrite –0.03 as –0.030 for easier comparison. The original statement is written –0.003 > –0.03.
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Operations in algebra are nothing like operations in hospitals. Well, you get to dissect things in both, but dissecting numbers is a whole lot easier (and a lot less messy) than dissecting things in a hospital.
Algebra is just a way of generalizing arithmetic, so the operations and rules used in arithmetic work the same for algebra. Some new operations do crop up in algebra, though, just to make things more interesting than adding, subtracting, multiplying, and dividing. I introduce three of those new operations after explaining the difference between a binary operation and a nonbinary operation.
Breaking into binary operations
Bi means two. A bicycle has two wheels. A bigamist has two spouses. A binary operation involves two numbers. Addition, subtraction, multiplication, and division are all binary operations because you need two numbers to perform them. You can add 3 + 4, but you can’t add 3 + if there’s nothing after the plus sign. You need another number.
Introducing nonbinary operations
A nonbinary operation needs just one number to accomplish what it does. A nonbinary operation performs a task and spits out the answer. Square roots are nonbinary operations. You find
Getting it absolutely right with absolute value
One of the most frequently used nonbinary operations is the one that finds the absolute value of a number – its value without a sign. The absolute value tells you how far a number is from 0. It doesn’t pay any attention to whether the number is less than or greater than 0; it just determines how far it is from 0.
Remember: The symbol for absolute value is two vertical bars:
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Here are some examples of the absolute-value operation:
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