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# The Real Number System

The Real Number System. These are all the numbers that we know and use. . IRRATIONAL NUMBERS. RATIONAL NUMBERS. REAL NUMBERS. INTEGERS. WHOLE NUMBERS. Rational numbers organized into more specific categories…. Real numbers are split into two major categories…. NATURAL NUMBERS.

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## The Real Number System

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1. The Real Number System These are all the numbers that we know and use.

2. IRRATIONAL NUMBERS RATIONAL NUMBERS REAL NUMBERS INTEGERS WHOLE NUMBERS Rational numbers organized into more specific categories… Real numbers are split into two major categories… NATURAL NUMBERS Each category can be considered a set of numbers.

3. Note: There are sets that are nested inside other sets for a reason. If an entire set is contained inside another set, then all members of the set must belong to both sets. REAL NUMBERS IRRATIONAL NUMBERS RATIONAL NUMBERS INTEGERS WHOLE NUMBERS NATURAL NUMBERS So all whole numbers are integers, rational numbers, and real numbers.

4. When classifying numbers, you need to consider the definitions of each set of numbers. • Rational numbers are any real numbers that can be written as fractions. (Any rational number can also be written as a repeating or terminating decimal.) • Irrational numbers are any real numbers that are not rational. An example of an irrational number would be pi or the square root of a number that does not give you a repeating or terminating decimal like the square root of 2. • Integers are positive and negative numbers without fractional parts and zero. • Whole numbers are all positive integers and zero. • Natural numbers are all positive integers. These can also be called counting numbers.

5. Any number that falls in the natural numbers would also be a whole number, an integer, a rational number and a real number… IRRATIONAL NUMBERS RATIONAL NUMBERS REAL NUMBERS REAL NUMBERS INTEGERS WHOLE NUMBERS Rational numbers organized into more specific categories… NATURAL NUMBERS Like the number 4 is… Real numbers are split into two major categories… 4

6. REAL NUMBERS IRRATIONAL NUMBERS RATIONAL NUMBERS What about the number -12? REAL NUMBERS INTEGERS WHOLE NUMBERS -12 Rational numbers organized into more specific categories… NATURAL NUMBERS Real numbers are split into two major categories… -12 is an integer, a rational number and a real number, but not a whole or natural number.

7. Example #1: Place the following numbers in the diagram to determine what kind of numbers they are. REAL NUMBERS IRRATIONAL NUMBERS RATIONAL NUMBERS REAL NUMBERS √‾ 5 ½ INTEGERS WHOLE NUMBERS Rational numbers organized into more specific categories… NATURAL NUMBERS 5.7 Real numbers are split into two major categories… -45 π 0 √‾ 9

8. There are standard symbols that can be used to represent the following sets of numbers: • Real Numbers: • Rational Numbers: • Integers: • Natural Numbers: ℝ ℚ ℤ ℕ

9. We will use: W for whole numbers I for irrational numbers. This will make things easier…

10. Listing the Sets of Rational Numbers: These are all infinite sets of numbers. Integers: Z = {… , -3, -2, -1, 0, 1, 2, 3, …} Whole Numbers: W = {0, 1, 2, 3, …} Natural Numbers: N = {1, 2, 3, …} Just another way to look at these…

11. Example 2: Write the symbol of all the sets to which the following numbers belongs. 0: ¾: -8: 6½: -4.3: 9.2: ℚ W ℤ ℝ Don’t be afraid to use the diagram to help with this! ℚ ℝ I ℚ ℤ ℝ ℚ ℝ ℚ ℝ ℝ ℚ ℝ ℚ ℤ ℚ ℝ W ℕ W ℤ ℚ ℝ ℕ I ℝ

12. Example #3: Which of the following is an irrational number? Why? =12 =9 4.79583152331271 =25 This is irrational because 23 is a prime number and if you take the square root of a prime number you will get a non-repeating decimal.

13. Example #4: Which of the following are not natural numbers? Why? =8 -3 is not a natural number because no negative numbers are in the natural numbers. 2/3 is not a natural number because no fractions are natural numbers. Remember we sometimes call natural numbers counting numbers we do not count objects using negative or fractional numbers.

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