Math Problem Statement

Answer the questions in the picture

Please classify the numbers in the table below by checking (√) all rows that apply to each number.

Solution

In order to classify the numbers in the table based on the number systems, let's review the definitions of each system:

  1. Counting Numbers (ℕ): These are the set of positive whole numbers starting from 1, i.e., {1, 2, 3, ...}.
  2. Whole Numbers (ℕ₀): These are the set of counting numbers including zero, i.e., {0, 1, 2, 3, ...}.
  3. Integers (ℤ): These include all positive and negative whole numbers, including zero, i.e., {..., -3, -2, -1, 0, 1, 2, 3, ...}.
  4. Rationals (ℚ): These are numbers that can be expressed as the quotient of two integers (a/b, where a and b are integers and b ≠ 0).
  5. Irrationals (ℚ'): These are numbers that cannot be expressed as the quotient of two integers. They have non-terminating and non-repeating decimal expansions.
  6. Reals (ℝ): These include all rational and irrational numbers, covering both positive and negative numbers, fractions, and irrational numbers.
  7. Imaginary (i): These are numbers of the form a + bi, where a and b are real numbers and i represents the square root of -1.
  8. Complex (ℂ): These are numbers of the form a + bi, where a and b are real numbers.

Classifying the Numbers:

  • 0: Whole number, Integer, Rational, Real.
  • 1: Counting number, Whole number, Integer, Rational, Real.
  • -100: Integer, Rational, Real.
  • 5^(1/2): Rational, Real (It's the square root of 5, which is irrational but is considered real).
  • 1 - π: Irrational (π is irrational and subtracting a rational number from it keeps it irrational), Real.
  • 50: Counting number, Whole number, Integer, Rational, Real.
  • 10 e: Real (e is irrational, and multiplying by a rational number keeps it real).
  • 1 - 2i: Complex, Imaginary.
  • 100 i: Complex, Imaginary.
  • 13.13: Rational, Real.
  • 3.33...: Rational (repeating decimal), Real.
  • 1/6: Rational, Real.
  • 2.221221122111...: Rational (a repeating decimal), Real.

This is the classification for each number based on their properties across number systems.

Would you like further details or clarification on any of the points?

Here are some related questions for exploration:

  1. What is the difference between rational and irrational numbers?
  2. How do you identify if a number is irrational based on its decimal expansion?
  3. Can every irrational number be classified as a real number?
  4. How do imaginary and complex numbers differ?
  5. What is the significance of e in real number calculations?

Tip: A good way to determine whether a decimal is rational or irrational is to check if it repeats or terminates. Repeating and terminating decimals are rational.

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Math Problem Analysis

Mathematical Concepts

Number Systems
Counting Numbers
Whole Numbers
Integers
Rational Numbers
Irrational Numbers
Real Numbers
Imaginary Numbers
Complex Numbers

Formulas

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Theorems

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Suitable Grade Level

Grades 7-9