🦭 Number Sets (ℕ, ℤ, ℚ, ℝ)
Learn about positive numbers, negative numbers, and zero. Practice filtering sets with detailed step-by-step explanations.
Positive & Negative Numbers
Positive & Negative Numbers
Every real number belongs to exactly one of three categories:
• Positive Numbers: Numbers strictly greater than (). Examples: .
• Negative Numbers: Numbers strictly less than (). Examples: .
• Zero (): Zero is neither positive nor negative.
This partition can be represented using set notation as:
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: (i.e., ).
• Non-Negative Numbers: All numbers that are not negative. This is the union of positive numbers and zero: (i.e., ).
Complex Numbers:
Complex numbers with a non-zero imaginary part (like , , , ) 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!
Frequently Asked Questions
What is a set and what are its elements?
A set is a collection of distinct objects or numbers called elements. We write a set by listing its elements inside curly braces, like A = {1, 2, 3}.
What do the symbols ∈ and ∉ mean?
The symbol ∈ means “belongs to” or “is an element of” (e.g. 2 ∈ {1, 2, 3} is true). The symbol ∉ means “does not belong to” (e.g. 4 ∉ {1, 2, 3} is true).
What is the union of two sets?
The union A ∪ B contains all elements that appear in A, in B, or in both. No duplicates are listed. Example: {1, 2} ∪ {2, 3} = {1, 2, 3}.
What is the intersection of two sets?
The intersection A ∩ B contains only elements that appear in both A and B. Example: {1, 2, 3} ∩ {2, 3, 4} = {2, 3}.
What are Integers (ℤ)?
Integers include all whole numbers — positive, negative, and zero: ℤ = {…, -3, -2, -1, 0, 1, 2, 3, …}. Every natural number is also an integer, but not every integer is natural.
What is the difference between real and complex numbers?
Real numbers include all rational and irrational numbers. Complex numbers include all real numbers as well as numbers containing the imaginary unit $i$ (where $i^2 = -1$), allowing us to solve roots of negative numbers.
Is zero positive or negative?
Zero is neither positive nor negative. It is the boundary between them.