The Rational Number System - Sixth Grade Mathematics

The rational number system unifies whole numbers, integers, fractions, and decimals into a single cohesive mathematical continuum, establishing that any quantity expressible as a ratio of two integers is rational.

The Formal Definition of Rational Numbers

A rational number is any number that can be expressed as the quotient or ratio of two integers in the form a/b, where a and b are integers and the denominator b is not equal to zero. The formal symbol for the set of rational numbers is Q (derived from the word "Quotient").

Formal Algebraic Definition:
           Q = { a / b  |  a, b are integers and b is not 0 }

Deconstructing the Word "Rational"

The root word of rational is ratio. A ratio is a comparison of two numbers via division. Every whole number, terminating decimal, and repeating decimal is secretly a ratio of two integers.

The Nested Hierarchy of Number Sets

The real number system contains nested subsets, where each successive set encompasses all previous sets.

+-------------------------------------------------------------------------+
|                  THE NESTED HIERARCHY OF NUMBERS                        |
|                                                                         |
|   +-----------------------------------------------------------------+   |
|   | RATIONAL NUMBERS (Q): Fractions, Terminating & Repeating Decs   |   |
|   | Examples: -3/4, 2.75, -0.333..., 18/5, -8.2                     |   |
|   |   +---------------------------------------------------------+   |   |
|   |   | INTEGERS (Z): Whole numbers and their opposites         |   |   |
|   |   | Examples: -5, -12, -450                                 |   |   |
|   |   |   +-------------------------------------------------+   |   |   |
|   |   |   | WHOLE NUMBERS (W): Zero and Natural numbers     |   |   |   |
|   |   |   | Examples: 0                                     |   |   |   |
|   |   |   |   +-----------------------------------------+   |   |   |   |
|   |   |   |   | NATURAL NUMBERS (N): Counting numbers   |   |   |   |   |
|   |   |   |   | Examples: 1, 2, 3, 4, 15, 100           |   |   |   |   |
|   |   |   |   +-----------------------------------------+   |   |   |   |
|   |   |   +-------------------------------------------------+   |   |   |
|   |   +---------------------------------------------------------+   |   |
|   +-----------------------------------------------------------------+   |
+-------------------------------------------------------------------------+

Proving that Common Numbers are Rational

To prove a number is rational, show that it can be rewritten as a ratio of two integers (a/b): Whole numbers and integers: The integer -7 can be written as -7/1. The whole number 0 can be written as 0/1. Mixed numbers: The mixed number 3 and 1/4 can be written as 13/4. Terminating decimals: The decimal 0.45 can be written as 45/100 (which reduces to 9/20). Repeating decimals: The repeating decimal 0.333... can be written as 1/3.

Representing Rational Numbers on a Number Line

Rational numbers occupy all the spaces between consecutive integers on a number line.

-2              -1.5            -1              -0.5             0
|-------+-------|-------+-------|-------+-------|-------+-------|
                ^                               ^
             -1 and 1/2                       -1/2

Locating Negative Rational Numbers

Plotting negative fractions and decimals requires moving left from zero. To plot -1.75: locate -1. Because -1.75 is between -1 and -2, continue moving left toward -2 by 0.75 (three-fourths) of that unit interval. A common error is placing -1.75 between 0 and -1; remember that as numbers become more negative, they move farther to the left.

Exercises

Exercise 1. Show that each of the following numbers is a rational number by writing it in the form a/b, where a and b are integers: (a) -9; (b) 4.6; (c) -2 and 3/8; (d) 0.

Exercise 2. Classify each number into all the number sets to which it belongs (Natural, Whole, Integer, Rational): (a) -15; (b) 28; (c) -3/4; (d) 0.

Exercise 3. Plot the rational numbers -2.25, -1/2, 0.75, and -1 and 3/4 on a number line, and state which number has the greatest absolute value.

Solutions

Solution to Exercise 1. For (a) -9: write as -9/1. For (b) 4.6: write as 4 and 6/10 = 46/10 (or 23/5). For (c) -2 and 3/8: convert mixed number to improper fraction: - (2 * 8 + 3)/8 = -19/8 (or -19/8). For (d) 0: write as 0/1. Since each can be written as an integer divided by a non-zero integer, all four are rational numbers.

Solution to Exercise 2. For (a) -15: Integer and Rational. For (b) 28: Natural, Whole, Integer, and Rational. For (c) -3/4: Rational only (it is not a whole number or integer). For (d) 0: Whole, Integer, and Rational.

Solution to Exercise 3. On a number line, -2.25 lies 2.25 units to the left of 0 (between -2 and -3). The fraction -1/2 = -0.5 lies halfway between 0 and -1. The decimal 0.75 lies three-fourths of the way between 0 and +1. The mixed number -1 and 3/4 = -1.75 lies between -1 and -2. To determine which has the greatest absolute value, find their distances from 0: |-2.25| = 2.25; |-0.5| = 0.5; |0.75| = 0.75; |-1.75| = 1.75. The number -2.25 has the greatest absolute value (2.25).