Rational numbers sound like they should be very sensible numbers. In fact, they are. Rational numbers are simply numbers that can be written as fractions or **ratio****s** (this tells you where the term **ratio**nal comes from). The hierarchy of real numbers looks something like this:

**Real Numbers: **Any number that can name a position on a number line is a **real number**. Every position on a number line can be named by a real number in some form.

An important property of real numbers is the Density Property. It says that between any two real numbers, there is always another real number.

**Rational Numbers: **Any number that can be written in fraction form is a **rational number**. This includes integers, terminating decimals, and repeating decimals as well as fractions.

- An
**integer**can be written as a fraction simply by giving it a denominator of one, so any integer is a rational number.

6 = (6/1); 4 = (4/1); 0 = (0/1) - A
**terminating decimal**can be written as a fraction simply by writing it the way you say it: 3.75 =*three and seventy-five hundredths*= 3(75/100), then adding if needed to produce a fraction: (300/100) + (75/100) = (375/100). So, any terminating decimal is a rational number. - A
**repeating decimal**can be written as a fraction using algebraic methods, so any repeating decimal is a rational number.

**Integers:** The counting numbers (1, 2, 3, ...), their opposites (-1, -2, -3, ...), and zero are integers. A common error for students in grade 7 is to assume that the integers account for all (or only) negative numbers.

**Whole Numbers:** Zero and the positive integers are the whole numbers.

**Natural Numbers: **Also called the counting numbers, this set includes all of the whole numbers except zero (1, 2, 3, ...)

**Irrational Numbers: **Any real number that cannot be written in fraction form is an **irrational number**. These numbers include the non-terminating, non-repeating decimals (pi, 0.45445544455544445555..., √2, etc.). Any square root that is not a perfect root is an irrational number. For example, √1 and √4 are rational because √1 = 1 and √4 = 2, but √2 and √3 are irrational; there are no perfect squares between √1 and √4. All four of these numbers do name points on the number line, but they cannot be written as fractions. When a decimal or fractional approximation for an irrational number is used to compute (as in finding the area of a circle), the answer is always approximate and should clearly indicate this.

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