Teaching Rational Numbers: Decimals, Fractions, and More

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 ratios (this tells you where the term rational 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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