Understanding Real Numbers in Algebra

1. Algebra 2 1.1 Real Numbers and Number Operations

2. Key Concepts 1 - 4 There are two main groups of numbers • Real numbers – those numbers used most often in the real world and in algebra

3. Key Concepts 1 - 4 There are two main groups of numbers • Real numbers – those numbers used most often in the real world and in algebra • Imaginary numbers – also called complex numbers, These are used to explore the concepts of mathematics beyond our real world

4. Different Types of Real Numbers • Whole numbers – these are the counting numbers, 0, 1, 2, 3, . . . Notice that they do not include negative numbers

5. Different Types of Real Numbers • Whole numbers • Integers – These are the whole numbers and the negative numbers, . . . ,-3,-2,-1, 0, 1, 2, 3, . . .

6. Different Types of Real Numbers • Whole numbers • Integers • Rational numbers – These are numbers that can be written as ratios of integers. Numbers such as ½ and -4 since -4 can be written as

7. Different Types of Real Numbers • Whole numbers • Integers • Rational numbers • Irrational numbers – Those numbers that cannot be written as ratios (fractions) such as

8. Different Types of Real Numbers • Whole numbers • Integers • Rational numbers • Irrational numbers – Those numbers that cannot be written as ratios (fractions) such as Some will argue that you can write this as a fraction by placing it above 1. That is why we use the term ratio of integers. A ratio is a comparison of one integer to another integer. The Square root of two is a non-repeating, unending decimal number which can not be written since the decimal values go on forever. So we use the symbol shown above for the square root of two.

9. Different Types of Real Numbers These four groups can be found in the chart on page 3 of the textbook • Whole numbers • Integers • Rational numbers • Irrational numbers

10. Different Types of Real Numbers • Whole numbers • Integers • Rational numbers • Irrational numbers Each of these groups are sub groups (or subsets) of the set of real numbers. Some numbers can be found in all of the first three sets listed (such as the number 2). If a number is to be found in the first three sets, they are never part of the fourth, the fourth set stands alone, but all four subsets make up the set of real numbers.

11. Whenever you read “real numbers” in the book, they are referring to a number which can be found in any of the four subsets we just looked at. That means it could be a whole number, a negative number, a ratio, or an irrational number.

12. That also means that an answer of “No real solution” refers to a number which cannot be placed in any of these four groups. A number such as the square root of negative one is not a real number. If my answer is the square root of a negative number then I say, “No real solution.”

13. -6 -6 -4 -2 0 2 4 6 6 The next topic addressed in this section is that of a number line and placing real numbers on the number line. Here are the basic facts about a number line.