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Describe all the sets and subsets the number 2/3 belong

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The number \( \frac{2}{3} \) can belong to different sets and subsets based on various mathematical contexts. Here’s a breakdown:

  1. Natural Numbers (N): The set of natural numbers includes positive integers starting from 1 (i.e., {1, 2, 3, ...}). Since \( \frac{2}{3} \) is not a natural number, it does not belong to this set.

  2. Whole Numbers (W): Whole numbers include all natural numbers and zero (i.e., {0, 1, 2, 3, ...}). \( \frac{2}{3} \) is not a whole number, so it does not belong to this set either.

  3. Integers (Z): The set of integers includes positive and negative whole numbers, as well as zero (i.e., {..., -2, -1, 0, 1, 2, ...}). Since \( \frac{2}{3} \) is not an integer, it does not belong to this set.

  4. Rational Numbers (Q): The set of rational numbers consists of all numbers that can be expressed as the quotient of two integers, where the denominator is not zero. Since \( \frac{2}{3} \) can be expressed in this form, it belongs to the set of rational numbers.

  5. Real Numbers (R): The set of real numbers includes all rational and irrational numbers. Since \( \frac{2}{3} \) is a rational number, it also belongs to the set of real numbers.

  6. Complex Numbers (C): The set of complex numbers includes all numbers of the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit. Since \( \frac{2}{3} \) can be expressed as \( \frac{2}{3} + 0i \), it belongs to the set of complex numbers.

Summary of Sets Containing \( \frac{2}{3} \):

  • Rational Numbers (Q)
  • Real Numbers (R)
  • Complex Numbers (C)

Subsets:

  1. Subsets of Rational Numbers:

    • Decimals (e.g., \( \frac{2}{3} = 0.666...\))
    • Proper Fractions (as \( \frac{2}{3} \) is less than 1)

  2. Subsets of Real Numbers:

    • Irrational Numbers (that would not include \( \frac{2}{3} \))
    • Decimal Numbers (including both rational and irrational decimals)
    • Positive Real Numbers (since \( \frac{2}{3} > 0 \))

In summary, \( \frac{2}{3} \) belongs to the sets: rational numbers, real numbers, and complex numbers, with several relevant subsets in each category.