Algebra

• The natural numbers are the numbers that we count with: 1, 2, 3, 4, 5, 6,...
• The whole numbers are the numbers we count with and zero: 0, 1, 2, 3, 4, 5, 6,...
• The integers are the numbers we count with, their negatives, and zero: -3, -2, -1, 0, 1, 2, 3,...
• The rational numbers are all numbers that can be expressed as a ratio of two integers. They can be positive or negative.
• Fractions may be proper (less than one; Ex: 1/3) or improper (more than one; Ex: 21/17)
• All integers are rational
• The real numbers can be represented as points on the number line
• All natural numbers are real, but the real number line has many points that are "between" rational numbers which is called irrational numbers
• The imaginary numbers are square roots of negative numbers.
• The complex numbers are all possible sums of real and imaginary numbers. All reals and imaginary numbers are complex.
• The Fundamental Theorem of Algebra says that every polynomial of degree n has exactly n complex roots
• A set is a collection--finite or infinite--of things called members or elements.
• To denote a set, we enclose the elements in braces. The notation a ∈ N means that a is in N, or a "is an element of" N
• Empty set or null set (Ø or {}): The set without any elements. Beware that the set {0} is still a set with one element.
• Union of two sets (A ∪ B): The set of all elements that are in both set of A and B
• Intersection of two sets (A ∩ B): The set of all the elements that are both in A and B. Two sets with no elements in common are disjoint; their intersection is the empty set
• Complement of a set (Ā, A', or Aᶜ): The set of all elements that are not in A
• Subset (A ⊂ C): This means that A is a subset of C if all elements of A are also elements of C
• Proper Subset (A ⊆ C): If A is a proper subset of C then it has some elements but not all of them
• Equality of Sets (A = B): When every element of A is also an element of B and vice versa
• A Venn Diagram is a visual way to represent the relationship between two or more sets
• If A = {1, 2, 3} and B = {2, 4, 6}, what is the union (A ∪ B) of the two sets?

  A ∪ B = {1, 2, 3, 4, 6}
• If A = {1, 2, 3} and B = {2, 4, 6}, what is the intersection (A ∩ B) of the two sets?

  A ∩ B = {2}
• If there is a set {1, 2, 3, 4, 5, 6, 7} and A = {1, 2, 3, 4}, then what is the complement (A')?

  A' = {4, 5, 6, 7}
• What type of set is when C = {1, 2, 3, 4, 5, 6, 7} and A = {1, 2, 3}?

  Subset
• Real numbers satisfy 11 properties: 5 for addition, 5 matching ones for multiplication, and 1 that connects addition and multiplication
• Commutative Property (+): a + b = b + a
• Associative Property (+): (a + b) + c = a + (b + c)
• Identities exist (+): 0 is a real number. a + 0 = 0 + a = a. What is the additive identity?

  0
• Inverses exist (+): -a is a real number. a + (-a) = (-a) + a = 0. Also, -(-a) = a
• Closure (+): a + b is a real number
• Distributive Property (+ and ×): a × (b + c) = a × b + a × c
• Commutative Property (×) = a × b = b × a
• Associative Property (×): a × (b × c) = (a × b) × c
• Identities exist (×): 1 is a real number. a × 1 = 1 × a = a. What is the multiplicative identity?

  1
• Closure (×): a × b is a real number
• Laws of Exponents

  Multiplication Rule: aˣ × aʸ = aˣ⁺ʸ
• Laws of Exponents

  Division Rule: aˣ ÷ aʸ = aˣ ⁻ ʸ
• Laws of Exponents

  Power Rule: (aˣ)ʸ = aˣʸ
• Laws of Exponents

  Product Rule: (ab)ˣ = aˣbˣ