BM1.4 – Working with Rational Numbers

Chapter 1, Lesson 4

In this lesson students will:

  • see the definition of a rational number
  • flexibly think of a rational number as a fraction or decimal number
  • interpret the fraction bar as a division operation, and vice versa
  • find an equivalent fraction given a rational number
  • find a missing numerator or denominator given a rational number and partial information
  • perform computations involving addition, subtraction, multiplication & division of rational numbers by hand, using the business calculator, and using Google Sheets

The Definition of Rational Numbers

A rational number is any number of the form

\displaystyle \frac{a}{b} \textrm{ where } a,b \in \mathbb{R} \textrm{ and } b\neq0.

One way to think about this set of numbers is as the set of all positive and negative fractions together with the number zero. For example, the following numbers are all rational numbers

\displaystyle \frac{2}{3},\; -\frac{2}{3},\; \frac{3}{15},\; 5,\; \frac{17}{11},\; 0,\; -\frac{143}{45}.

It is also possible to represent a fraction as a decimal number. There are three classes of decimal numbers:

  1. Terminating decimals
  2. Decimal numbers with repeated digits
  3. Non-terminating and non-repeating decimal numbers.

Terminating decimal numbers have their digits stop. Some terminating decimal numbers with their fraction equivalents are

\displaystyle \frac{1}{2} = 0.5,\; -\frac{1}{5} = -0.2,\; \frac{1}{4} = 0.25 .

Decimal numbers with repeated digits have a noticeable pattern that repeats. Often we use a bar over top of the digits that continually repeat to write down this decimal number in a compact way. Since our business calculator does not have this capability, be mindful of working with these kinds of decimal numbers, as we will often have to take many decimal places of accuracy. Some decimal numbers with repeated digits, and their fraction equivalents are

\displaystyle \frac{1}{3} = 0.333... = 0.\overline{3},\; -\frac{1}{6} = -0.1666... = -0.1\overline{6}.

Non-terminating and non-repeating decimal numbers do not have a noticeable pattern in their digits, and their digits continue on indefinitely. These kinds of numbers do not have a fractional representation. This means that a rational number cannot be a non-terminating, non-repeating decimal number. We will still however work with some of these decimal numbers and we will see how to use the calculator as these situations arise. Some non-terminating and non-repeating decimal numbers include

\displaystyle \pi \approx 3.1415..., \; \textrm{e} \approx 2.7182..., \; \ln(2) \approx 0.6931... \;.

Equivalence Classes of Fractions

Rational numbers are divided into equivalence classes called equivalent fractions. Two fractions are called equivalent if we can scale up the numerator and denominator of one of the fractions by the same scale factor to obtain the other fraction. The fractions two-thirds and eight-twelfths are equivalent since

\displaystyle \frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}.

In this case, the scale factor is four. There are many fractions that are equivalent to a given fractions. For example, if we consider the fraction one-half, we could find several equivalent fractions:

\displaystyle \frac{1}{2} = \frac{2}{4} = \frac{3}{6} = \frac{4}{8} = ... \; .

If we pause a moment to look at all of these equivalent fractions, we realize that there is not fraction in this equivalence class that contains the largest possible denominator (or numerator). There is, however, a fraction that contains the smallest possible denominator (or numerator). The rational number in the equivalence class that contains the smallest denominator (or numerator) is often called the simplest form or reduced form of our equivalence class. In other words, we would say that one-half is the simplest form of three-sixths or four-eighths. There are some instances in which we would not use simplest form; however, it is common practice to try to use simplest form of a rational number when possible.

Arithmetic Properties of Rational Numbers

Rational numbers have several useful properties when it comes to our main operations. For addition and subtraction, recall that this can be done by adding/subtracting the numerators, provided you have similar units (like denominators). For instance

\displaystyle \frac{1}{7} + \frac{2}{7} = \frac{1+2}{7} = \frac{3}{7}

or

\displaystyle \frac{1}{7} - \frac{2}{7} = \frac{1-2}{7} = -\frac{1}{7}.

When you do not have similar denominators, you will have to find equivalent fractions to ensure a common unit (common denominator) so that the addition and subtraction is possible.

Multiplication and division of fractions are a bit easier, since we do not require similar units. Multiplication and division of fractions can be done by mutiplying/dividing the numerators and denominators from left to right. For instance

\displaystyle \frac{1}{7} \times \frac{3}{4} = \frac{1 \times 3}{7 \times 4} = \frac{3}{28}

or

\displaystyle -\frac{6}{8} \div \frac{3}{2} = \frac{-6 \div 3}{8 \div 2} = -\frac{2}{4} = -\frac{1}{2}.

For more complicated arithmetic involving rational numbers, please see the video below.

Working with Rational Numbers using the Business Calculator & Google Sheets

Working with rational numbers on our calculator or on Google Sheets is not terribly difficult, prided that you remember to use brackets as necessary. If an expression looks complex, it is best to place a set of brackets around the numerator and denominator, and treat the fraction bar as a division.

The video below will show you how to use the calculator and Google Sheets to simplify the expression

\displaystyle \frac{1 - (1+0.05)^{-2}}{1 + \frac{0.05}{4}}

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