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Rational and Irrational Numbers

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Understanding Rational and Irrational Numbers

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Rational and Irrational Numbers

Algebra Foundations

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You will identify and approximate irrational numbers by distinguishing them from rational numbers through decimal expansions and geometric reasoning.

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Now You Know

After completing this tutorial, you will be able to complete the following:

  • Define a rational number.
  • Define an irrational number.
  • Represent rational numbers on the number line.
  • Approximate irrational numbers given their location on the number line.
  • Classify numbers as rational or irrational.

Everything You'll Have Covered

Most people learn first learn about numbers by counting, which is to say they first learn the counting numbers 1, 2, 3, and so forth. These numbers are appropriately called natural numbers. They are useful for counting whole quantities, such as sheep in a field or cookies in a jar. However, natural numbers are not capable of describing situations of debt (for example, I owe you a sheep). Describing debt in any useful way requires integers, which is the set of all negative or nonnegative whole numbers. Integers result from subtracting and adding natural numbers. In order to describe fractional quantities such as two thirds of a cookie, rational numbers are used; these result from taking the ratios of integers. At this point, one might feel that there are no additional numbers, that we can describe the world by rational numbers alone. Intuitively, it does seem that any quantity should be measurable by using integers or fractions. Surprisingly, this is not the case.

Consider a 1 centimeter by 1 centimeter square.

A diagonal of this square spans the distance between two opposite corners, and its length can be determined from the Pythagorean theorem, which states that in a right triangle with leg lengths a and b, and hypotenuse c.

We now make an extraordinary claim: c is not a rational number. This claim is extraordinary because it states that no ordinary fraction can describe the length of c. It is impossible to measure c with perfect accuracy using an ordinary ruler that divides centimeters into halves, thirds, and so forth.

The claim that c is irrational is proved by assuming that c does in fact have a fractional representation and deriving a logical contradiction.

Assume c can be represented by a fraction containing integers. The fraction could then be put into its simplest terms, meaning that the denominator and numerator have no common factors.

Tutorial Details

Approximate Time 25 Minutes
Pre-requisite Concepts Students should understand the concepts of fractions and number lines.
Course Algebra Foundations
Type of Tutorial Concept Development
Key Vocabulary decimal number, irrational, irrational number