Math Problem Statement

Classify each number below as a rational number or an irrational number.

Solution

Let's classify each number in the table as either rational or irrational:

  1. 64-\sqrt{64}: 64=8\sqrt{64} = 8, so 64=8-\sqrt{64} = -8. Since 8-8 is a whole number, it's a rational number.

  2. 25112-25 \frac{1}{12}: This represents the mixed number 25112-25 \frac{1}{12}, which can be written as 30112\frac{-301}{12}. Since it’s a ratio of two integers, it is a rational number.

  3. 10\sqrt{10}: 10\sqrt{10} is a non-perfect square and its decimal representation is non-repeating and non-terminating. Therefore, it is an irrational number.

  4. 1712-\frac{17}{12}: This is already in fractional form. Since it is a ratio of two integers, it is a rational number.

  5. 5π-5\pi: Since π\pi is an irrational number, multiplying it by any non-zero number (like 5-5) still results in an irrational number.

Here’s the completed classification:

ExpressionRationalIrrational
64-\sqrt{64}
25112-25 \frac{1}{12}
10\sqrt{10}
1712-\frac{17}{12}
5π-5\pi

Let me know if you need more details or have any questions!

Related Questions:

  1. How do you determine if a square root is rational or irrational?
  2. What makes a number rational?
  3. Can a negative number be irrational?
  4. How does the multiplication of rational and irrational numbers behave?
  5. Why is π\pi classified as an irrational number?

Tip:

Remember, a rational number can always be written as a fraction of two integers, while an irrational number cannot be expressed this way, and its decimal form is non-repeating and non-terminating.

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Math Problem Analysis

Mathematical Concepts

Number Theory
Rational Numbers
Irrational Numbers
Square Roots
Pi

Formulas

A rational number can be expressed as the ratio of two integers.
An irrational number cannot be written as a simple fraction.
Square roots of non-perfect squares are irrational.

Theorems

A number is rational if it can be written as a fraction of two integers.
Pi (π) is irrational.

Suitable Grade Level

Grades 6-8