Lesson Objectives
• Learn the definition of a Natural Number
• Learn the definition of a Whole Number
• Learn the definition of an Integer
• Learn the definition of a Rational Number
• Learn the definition of an Irrational Number
• Demonstrate the ability to classify any real number

## Classifying Real Numbers

### How can we classify a Real Number?

#### Natural Numbers

When we first start to work with numbers, we generally count on our fingers or count any number of physical objects placed in front of us. To perform these actions, we involve a set of numbers known as the counting numbers or the natural numbers. The natural numbers are used to count a number of objects. This set begins with the number one and increases in increments of one out to positive infinity.
Natural Numbers: {1, 2, 3, 4,...}
Note: The three dots "..." after the 4 in our set are used to indicate that the pattern continues forever. After 4 comes 5, then 6, so on and so forth...
We can also use a number line to visually display our set of natural numbers. The leftmost number is a 1 and then each additional notch on the number line is shown as an increase of 1 or the next natural number. Since we can't list all of the numbers, we draw an arrow to the right to indicate the natural numbers continue out to positive infinity.

#### Whole Numbers

As we move on, we begin to consider the possibility of not having anything to count. This is represented with the number 0. When 0 is added to the set of natural numbers, we end up with the whole numbers. Whole Numbers: {0, 1, 2, 3, 4,...}

#### Integers

In some cases, it is not enough to have only the whole numbers. What if we need to describe a temperature that is less than zero, an altitude that is below sea level, or a negative bank balance? To deal with these situations, we can introduce the integers. These are the whole numbers and their opposites or negatives. To form this set, we just take the whole numbers and then working to the left of zero, we start counting from 1 again, only this time we place a negative symbol in front of the number. Integers: {...-4, -3, -2, -1, 0, 1, 2, 3, 4,...}

#### Rational Numbers

Additionally, we may have a situation in which we don't have a whole amount. Suppose we eat 3 slices of pizza out of a total of 8 slices that are available. Currently, we have more pizza than zero and less than one whole pizza. We can use a fraction such as 5/8 to describe the amount of pizza that is remaining or another fraction 3/8 to describe the amount of pizza that was eaten. This type of number is known as a rational number. A rational number can be written as the quotient of two integers. In decimal form, a rational number will always terminate or repeat the same pattern of digits forever. For example, in decimal form: 9/11 is 0.818181..., where the 81 repeats forever. $$\frac{9}{11}=0. \overline{81}$$ Note the bar over the digits 81, indicate the pattern continues forever... Rational Numbers: {p/q | p and q are integers and q ≠ 0}

#### Irrational Numbers

Lastly, we have the irrational numbers. These are real numbers that don't fall in the category of being a rational number. This means we can't write an irrational number as the quotient of two integers. When we see an irrational number in decimal form, the number will not terminate nor repeat the same pattern forever. The most famous example of an irrational number is pi, which is the ratio of a circle's circumference to its diameter. When we need to work with pi, we generally give an approximation of 3.14159, but after the 9 comes an infinite number of digits.
Irrational Numbers: {x | x is a real number but is not rational}
Together, the rational numbers and irrational numbers make up all of the real numbers. Therefore, if we are dealing with a real number, it must be either rational or irrational. As we work our way up the chain of rational numbers, we can see that a number such as 5, could be classified as a natural number, a whole number, an integer, or a rational number (since 5/1 = 5). So all natural numbers are whole numbers, all whole numbers are integers, and all integers are rational numbers. But reversing this isn't always going to work. For example: -5 is a rational number and an integer, but it's not a whole number or a natural number. 3/4 is a rational number, but not an integer, whole number, or natural number.

#### Skills Check:

Example #1

Which Numbers are Whole Numbers?

A
$$13, 17, -51$$
B
$$3, \sqrt{5}, -2$$
C
$$\frac{1}{4}, 0, -\sqrt{2}$$
D
$$-2, -9, 11$$
E
$$6, \frac{18}{1}, 0$$

Example #2

Which Numbers are Integers?

A
$$55, -3, -\frac{1}{2}$$
B
$$-18, 7, \sqrt{13}$$
C
$$\frac{1}{3}, -22, 18$$
D
$$-44, -\frac{9}{1}, -88$$
E
$$-\frac{6}{11}, \frac{13}{1}, 0$$

Example #3

How can we Classify the Number: $$\sqrt{3}$$

A
Rational
B
Whole Number
C
Integer
D
Irrational
E
Natural Number