Number Sets and Real Numbers Study Notes

Natural numbers (N): the smallest set of numbers used for counting; start at 1 and go to infinity. Notation: N = {1,2,3,\dots}
Whole numbers (W): natural numbers with 0 added; W = {0,1,2,3,\dots}
Integers (Z): whole numbers plus negatives; Z = {\dots,-2,-1,0,1,2,\dots}
Sets get larger as you move from N to W to Z.

Rational numbers (Q): numbers that can be written as a fraction \dfrac{p}{q} where p,q \in \mathbb{Z} and q \neq 0.
Every natural number, whole number, and integer is rational (examples):
- $5 = \dfrac{5}{1}$, so 5 is rational.
- Any integer can be written as a fraction with 1 as the denominator, e.g., -3 = \dfrac{-3}{1}.
Therefore, the sets nest: N \subseteq W \subseteq Z \subseteq Q (all are subsets of the real numbers).
Irrational numbers (I): numbers that are not rational; not expressible as a fraction of integers. Definition on the real line: I = \mathbb{R} \setminus \mathbb{Q}.
The opposite of rational is irrational. Examples of irrational numbers:
- \pi (pi), which is approximately 3.14159\ldots and has nonterminating, nonrepeating decimal expansion.
- \sqrt{2}, the square root of 2, which is irrational.
- \sqrt{a} where a is not a perfect square.
Note: √(-a) with a > 0 is not a real number; it is imaginary/complex (not in \mathbb{R}). In the real-number context, it is considered undefined.
The speaker noted that terms like the square root of a negative number are often described as irrational in casual talk, but rigorously they are not real numbers; they belong to the complex numbers (involving the imaginary unit i = \sqrt{-1}).

Real numbers (R): all numbers on the real number line; includes both rational and irrational numbers.
Relationship: N \subseteq W \subseteq Z \subseteq Q \subseteq R and R = Q \cup I where I = \mathbb{R} \setminus \mathbb{Q}.
Important distinctions:
- Every real number is either rational or irrational (mutually exclusive within \mathbb{R}).
- Not every real number is undefined or imaginary; undefined is not a number, and imaginary numbers lie outside \mathbb{R}.
Pi ((\pi)) is irrational and thus an element of I\subseteq R.
Zero is rational (e.g., 0 = \dfrac{0}{1}) and is a real number.

An expression with zero denominator is undefined; example: \dfrac{5}{0} is undefined.
In standard real-number arithmetic, division by zero is not allowed, hence undefined, not a number in \mathbb{R}.

A number can belong to multiple categories (e.g., a real number can be rational or irrational depending on its representation):
- If a real number is rational, it lies in \mathbb{Q}; if it’s irrational, it lies in I.
- However, a number cannot be both rational and irrational at the same time by definition.
Pi is irrational; a plotting or numeric label like 3.14 is just an approximation of an irrational number.
The square root of a negative number is not a real number; in the real-number system it is undefined. In the complex system, it is represented by an imaginary number (involving i = \sqrt{-1}).
Absolute value basics:
- For any real number x, |x| \ge 0 and |x| = x if x \ge 0, while |x| = -x if x < 0.
- Example: |-12| = 12; hence -\,|a| is the negative of the absolute value of a.
- Example calculation: evaluate the inner expression first, then apply the outer absolute value if present.

Real-number line placement:
- Negative 3.5 lies between -4 and -3 on the real line.
- Negative 3.0 is simply -3.
- Absolute value basics: | -12 | = 12; the value is nonnegative regardless of the sign inside.
Practice with nested expressions (inner to outer):
- Example 1: Evaluate -\left| -11 - | -1 | \right|
- Inner absolute: | -1 | = 1
- Then inside: -11 - 1 = -12
- Outer absolute: | -12 | = 12
- Then the leading minus: -12
- Example 2: Evaluate -\left| 4 - 4 \right|
- Inner subtraction: 4 - 4 = 0
- Absolute: |0| = 0
- Negate: -0 = 0
- Note: There is no concept of a negative zero in standard real arithmetic; the result is simply 0.
A reminder: handle inner parts first when evaluating nested expressions.

Example: -\dfrac{1}{9} (negative one-ninth)
- It is not an integer, but it is a rational number (and a real number).
Example: \dfrac{5}{0}
- Undefined (cannot have zero in the denominator).
Example: \pi
- Falls under irrational numbers; has nonterminating, nonrepeating decimal expansion.
Example: \sqrt{2}
- Irrational (not expressible as a fraction of integers).
Example: 4\sqrt{2}
- Since \sqrt{2} is irrational and 4 is rational and nonzero, their product is irrational. Therefore, 4\sqrt{2} is irrational and a real number. (Note: In the transcript there was a moment of confusion about this classification; the correct classification is irrational and real.)
Example: \sqrt{-a} for a > 0
- Not a real number; belongs to complex numbers (involving the imaginary unit i).

Real-number line hierarchy (inclusion):
- N \subseteq W \subseteq Z \subseteq Q \subseteq R
- Real numbers are either rational or irrational: \mathbb{R} = \mathbb{Q} \cup I with I = \mathbb{R} \setminus \mathbb{Q}.
Undefined vs. imaginary:
- Undefined arises from expressions like \dfrac{a}{0}.
- Imaginary/complex numbers arise from square roots of negative numbers, e.g., \sqrt{-1} = i, which are not real numbers.
Multiplication with irrationals: products of nonzero rationals with irrationals (e.g., 4\sqrt{2}) are irrational.
When evaluating expressions, proceed inner to outer and use the definitions above to classify the result (rational/irrational/real/undefined).