Algebra Lesson: Rational Numbers

What are Rational Numbers?

First, a few terms:

Terminating Decimal: A decimal that ends, having a finite number of digits after the decimal point. Sample: 3/4 = 0.75

Repeating Decimal: A decimal that doesn't end; it shows a repeating pattern of digits after the decimal point. Sample: 1/3 = 0.3333...

An Example Problem

For the examples below, show that the quotient of two integers is either a terminating or repeating decimal.

  1. 3/4 = 0.75 (this is a terminating decimal)
  2. 5/27 = 0.185. If you continue the division process, you will repeat 185 without bound. So, 5/27 = repeating decimal.

Example

Show that 1.3456 is the quotient of two integers.

Steps:

  1. 1) Express the decimal as a fraction over the proper place value.
  2. 2) Include the whole number in the numerator to match denominator place value.
  3. 3) Reduce to the lowest terms.
$$ 1.3456 $$ $$ = 1 + \frac{3456}{10000} $$ $$ = \frac{10000}{10000} + \frac{3456}{10000} $$ $$ = \frac{13,456}{10,000} $$ $$ = \frac{841}{625} $$

Another example:

Show that 0.252525... (repeating) is the quotient of two integers.

Steps:

  1. 1) Let N = decimal digit repeated at least 3 times. This is our first equation.
  2. 2) Let 100N = represent the hundreths place of 0.25 times N. This is our second equation. We will need to multiply 0.25 by 100 to find our whole number (25).

So far we have this:

$$ N = 0.252525... \text{ (our first equation)} $$ $$ 100N = 25.252525... \text{ (our second equation)} $$
  1. 3) Subtract equation 1 from 2 this way:
$$ 100N - N = (25.252525...) - (0.252525...) $$ $$ 99N = 25.000 $$
  1. 4) Divide both sides by the coefficient to find the value of N.
  2. 5) Reduce fraction if needed.
$$ N = 25/99 $$

NOTE: To find the whole number in samples like equation 2: If two digits repeat, multiply by 100, if one digit repeats, multiply by 10, if 3 digits repeat, multiply by 1,000, etc.

Example:

Is the number \(\sqrt{2} + 3.8\) a rational or irrational number?

Well, what is a rational number and what is an irrational number?

Rational Number:

A number r is rational if it can be written as a fraction r = p/q where both p and q are integers.

What is an irrational number?

The number \(\sqrt{5}\) by itself is not rational and is called irrational. This is not a definition of irrational numbers. In math, it's not quite true that what is not rational is irrational. Irrationality is a term reserved for a very special kind of number.

In the example above, when adding a rational and irrational number, the result will be the sum of a nonrepeating and nonterminating decimal. So, \(\sqrt{2} + 3.8\) = irrational.

Let's look at some more examples.

Express 4.555... as a quotient of two numbers.

$$ N = 4.555... $$ $$ 10N = 45.555... $$ $$ 10N - N = (45.555...)-(4.555...) $$ $$ 9N = 41.000 $$ $$ N = \frac{41}{9} $$

Express 0.7575... as a quotient of two numbers.

$$ N = 0.757575... $$ $$ 100N = 75.757575... $$ $$ 100N-N=(75.757575...)-(0.757575...) $$ $$ 99N = 75.000 $$ $$ N = 75/99 $$ $$ N = 25/33 $$

For this next question, there is no repetend.

$$ 0.64 = x $$ $$ 0.64*100 = 100*x $$ $$ 64 = 100*x $$ $$ x = \frac{64}{100} = \frac{16}{25} $$

Note: A rational number is a number that can be expressed as a terminating or a repeating decimal. An irrational number is a number that can be expressed as a nonterminating, nonrepeating decimal.

By Mr. Feliz (c) 2005