Number Sets (ℕ, ℤ, ℚ, ℝ): Positive & Negative Numbers

Real Numbers and Sign:
Every real number $x \in \mathbb{R}$ belongs to exactly one of three categories:
• Positive Numbers: Numbers strictly greater than $0$ ($x > 0$). Examples: $1, 2, \sqrt{4}, \pi$.
• Negative Numbers: Numbers strictly less than $0$ ($x < 0$). Examples: $-1, -2, -\sqrt{9}, -\pi$.
• Zero ($0$): Zero is neither positive nor negative.

This partition can be represented using set notation as:

Non-Positive and Non-Negative Numbers:
Sometimes we group zero with the positive or negative numbers:
• Non-Positive Numbers: All numbers that are not positive. This is the union of negative numbers and zero: $\text{Negative Numbers} \cup \{0\}$ (i.e., $x \le 0$).
• Non-Negative Numbers: All numbers that are not negative. This is the union of positive numbers and zero: $\text{Positive Numbers} \cup \{0\}$ (i.e., $x \ge 0$).

Complex Numbers:
Complex numbers with a non-zero imaginary part (like $i$, $-i$, $2i$, $1+i$) are not real numbers. They do not lie on the real number line, and therefore they cannot be ordered. They are neither positive, negative, nor zero!