Even though it is more commonly known as 3. Actually it is 3. It would keep going and going and going without any real repetition or pattern. In other words, it would be a non terminating, non repeating decimal, which again, can not be written as a rational number, 1 integer over another integer. That would include natural numbers, whole numbers and integers. Example 4: Graph the set on a number line.
In this problem, we have a -. Since it is halfway between these two numbers, I would place the dot halfway between. The other numbers are integers that are already marked clearly on the graph. Natural numbers? Note that simplifies to be 5, which is a natural number. Whole numbers? Rational numbers? Irrational numbers?
They are non-repeating, non-terminating decimals. Real numbers? Example 6: Place a or to make the statement true. Example 7: Place a or to make the statement true. Example 8: Place a or to make the statement true. Example 9: Place a or to make the statement true. Example Determine if the statement is true or false? In fact, there are no elements in N that are in I. Absolute Value Most people know that when you take the absolute value of ANY number other than 0 the answer is positive.
But, do you know WHY? Well, let me tell you why! Distance is always going to be positive unless it is 0 whether the number you are taking the absolute value of is positive or negative.
The following are illustrations of what absolute value means using the numbers 3 and Example Find the absolute value. I came up with 7, how about you? Example Find the absolute value. Let's talk it through. First of all, if we just concentrate on -2 , we would get 2.
That means we are going to take the opposite of what we get for the absolute value. Putting that together we get -2 for our answer. Note that the absolute value part of the problem was still positive. We just had a negative on the outside of it that made the final answer negative.
Opposites Opposites are two numbers that are on opposite sides of the origin 0 on the number line, but have the same absolute value. In other words, opposites are the same distance away from the origin, but in opposite directions. The opposite of x is the number - x. Keep in mind that the opposite of 0 is 0. When you see a negative sign in front of an expression, you can think of it as taking the opposite of it.
For example, if you had - -2 , you can think of it as the opposite of Example Write the opposite of 1. The opposite of 1. Example Write the opposite of The opposite of -3 is 3 , since both of these numbers have the same absolute value but are on opposite sides of the origin on the number line. Practice Problems. At the link you will find the answer as well as any steps that went into finding that answer.
Practice Problems 1a - 1c: List the elements of each set. Practice Problem 2a: Graph the set on a number line. Practice Problems 4a - 4c : Place or to make each statement true. The sequence is arithmetic because there is a common difference. The common difference is 4. Analysis of the Solution The graph of each of these sequences is shown in Figure 1. Figure 1. Try It 1 Is the given sequence arithmetic?
Try It 2 Is the given sequence arithmetic? How To: Given the first term and the common difference of an arithmetic sequence, find the first several terms. Add the common difference to the first term to find the second term.
Add the common difference to the second term to find the third term. Continue until all of the desired terms are identified. Write the terms separated by commas within brackets. Analysis of the Solution As expected, the graph of the sequence consists of points on a line as shown in Figure 2. Figure 2. As you saw, if two or more fractions have the same denominator, you can compare them by looking at their numerators.
The larger the numerator, the larger the fraction. On the previous page, we compared fractions that have the same bottom numbers , or denominators. But you know that fractions can have any number as a denominator. What happens when you need to compare fractions with different bottom numbers?
It's difficult to tell just by looking at them. After all, 2 is larger than 1 , but the denominators aren't the same. With an illustration, it was easy to compare these fractions, but how could we have done it without the picture? Click through the slideshow to learn how to compare fractions with different denominators.
Before we compare them, we need to change both fractions so they have the same denominator , or bottom number. First, we'll find the smallest number that can be divided by both denominators. We call that the lowest common denominator. Using a multiplication table makes this easy. All of the numbers on the 8 row can be divided evenly by 8. We can use the multiplication table again. All of the numbers in the 6 row can be divided evenly by 6.
Let's compare the two rows. It looks like there are a few numbers that can be divided evenly by both 6 and 8. Now we're going to change our fractions so they both have the same denominator: In order to change the denominator to Since we multiplied the denominator by 3 , we'll also multiply the numerator, or top number, by 3. Since any number times 1 is equal to itself We also changed its denominator to Our old denominator was 6.
To get 24 , we multiplied 6 by 4. Now that the denominators are the same, we can compare the two fractions by looking at their numerators. If you did the math or even just looked at the picture, you might have been able to tell that they're equal.
One-half is easier to say than four-eighths , and for most people it's also easier to understand.
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