Subsets of Real Numbers

Real Number System Diagram (OBSERVE)

  • R: Real Numbers

  • Q: Rational Numbers

  • Q': Irrational Numbers

  • Z: Integers

  • N: Natural Numbers

  • Z+: Positive Integers

  • Z-: Negative Integers

  • Note: OBSERVE the Real Number System relationships among these sets.

Plotting Integers on the Number Line (Location of Signs)

  • When you plot the integers on the number line, positive integers lie to the right of 0 and negative integers lie to the left of 0.

Mini Lesson: Subsets of Real Numbers

  • #1 Show that 8 and -11 are rational numbers. Take Note: A rational number is any number that can be written as a fraction of two integers (p/q), where the denominator is not zero: pq,q≠0\frac{p}{q}, \quad q \neq 0

    • Example: 8=81,−11=−1118 = \frac{8}{1},\quad -11 = \frac{-11}{1}

  • #2 Why is 4/0 not a rational number?
    Take Note: Since 4/0 has a denominator of zero, it is not a rational number—because division by zero is undefined in mathematics: 40\frac{4}{0} is undefined.

  • #3 TRUE or FALSE

    • TRUE: −136∈Q-136 \in \mathbb{Q}

    • TRUE: 152∈Q\frac{15}{2} \in \mathbb{Q}

    • FALSE: 142∉Z\frac{14}{2} \notin \mathbb{Z} (14/2 = 7, which is an integer)

  • #4 Plotting numbers on the number line

    • Plot the following numbers on the number line. (Numbers to plot are not listed in the transcript.)

Examples to Classify (Narrowing Down Subsets)

  • The following numbers illustrate rational vs irrational and other subsets:

    • π\pi (irrational)

    • 34\frac{3}{4} (rational)

    • −7-7 (integer; rational)

    • 5.335.33 (rational; can be expressed as a fraction, e.g., 533100\frac{533}{100})

    • 3.1023…3.1023\ldots (non-terminating, non-repeating decimal would be irrational if it truly continues without repetition; if it's a stated repeating pattern, it would be rational; the transcript presents it as a non-terminating decimal example to discuss decimals in context.)

    • 1212 (integer; rational)

    • 4.64.6 (rational; can be expressed as 4610\frac{46}{10})

Terminology and Relationships

  • Natural numbers: N\mathbb{N}

  • Integers: Z\mathbb{Z}

  • Rational numbers: Q\mathbb{Q}

  • Real numbers: R\mathbb{R}

  • Subset relationships: N⊂Z⊂Q⊂R\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}

Additional Notes and Key Takeaways

  • Remember: a rational number is any number that can be expressed as a fraction of two integers with a nonzero denominator.

  • Irrational numbers cannot be expressed as such a fraction.

  • Distinctions among decimals:

    • Terminating vs non-terminating decimals

    • Repeating vs non-repeating decimals

  • Goals for this unit: classify numbers into the right subsets, understand their properties, and connect these ideas to real-world reasoning and decision making.

Examples for Quick Practice (from the transcript prompts)

  • Classify the following in terms of subsets and provide reasoning: π, 34, −7, 5.33, 12, 4.6, 3.1023…\pi,\ \frac{3}{4},\ -7,\ 5.33,\ 12,\ 4.6,\ 3.1023\ldots

  • Plot numbers on a number line as a follow-up activity (exact numbers to plot would be provided in-class).