Math Problem Statement

Classify each number as being a natural number (N), whole number (W), integer (I), rational number (Q), and/or irrational number (Q′).

ⓐ 36−−√

ⓑ 83

Solution

Let's classify each number according to the following categories:

  • Natural Numbers (N): These are positive counting numbers: 1, 2, 3, etc.
  • Whole Numbers (W): These include all natural numbers and zero: 0, 1, 2, 3, etc.
  • Integers (I): These include all whole numbers and their negatives: ..., -2, -1, 0, 1, 2, etc.
  • Rational Numbers (Q): These are numbers that can be expressed as a fraction ab\frac{a}{b}, where aa and bb are integers, and b0b \neq 0.
  • Irrational Numbers (Q′): These are numbers that cannot be expressed as a fraction of two integers (e.g., π\pi, 2\sqrt{2}).

Now, let's classify the numbers:

36\sqrt{36}

  • 36=6\sqrt{36} = 6
  • Natural Number (N): Yes, 6 is a natural number.
  • Whole Number (W): Yes, 6 is a whole number.
  • Integer (I): Yes, 6 is an integer.
  • Rational Number (Q): Yes, 6 can be expressed as 61\frac{6}{1}, so it's rational.
  • Irrational Number (Q′): No, 36\sqrt{36} is not irrational because it simplifies to a rational number (6).

So, 36\sqrt{36} is a natural number, whole number, integer, and rational number.

ⓑ 83

  • 83 is a positive integer.
  • Natural Number (N): Yes, 83 is a natural number.
  • Whole Number (W): Yes, 83 is a whole number.
  • Integer (I): Yes, 83 is an integer.
  • Rational Number (Q): Yes, 83 can be expressed as 831\frac{83}{1}, so it's rational.
  • Irrational Number (Q′): No, 83 is not irrational.

So, 83 is a natural number, whole number, integer, and rational number.

Summary:

  • 36\sqrt{36}: N, W, I, Q
  • ⓑ 83: N, W, I, Q

Would you like more details or have any questions?

Here are 5 related questions:

  1. Can an irrational number also be a rational number?
  2. Is π\pi a rational or irrational number?
  3. How do you determine if a square root is rational or irrational?
  4. Is zero considered a natural number or just a whole number?
  5. What is the smallest positive integer?

Tip: A square root of a perfect square is always a rational number.

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Math Problem Analysis

Mathematical Concepts

Number Classification
Square Roots
Rational and Irrational Numbers

Formulas

√n for square roots
Fraction form for rational numbers (a/b where b ≠ 0)

Theorems

The square root of a perfect square is a rational number
A number is rational if it can be written as a fraction of two integers

Suitable Grade Level

Grades 6-8