Unit 1 Review (mpm2dg)

Numbers, Algebra, Set Theory Base Conversion  Subscript indicates base 321034 (base 4)  Show place value to convert to base 10 321034 = 3 × 40 + 0 × 41 + 1 × 42 + 2 × 43 + 3 × 44 321034 = 91510  When foreign base ≥ 10, capital letters are used base 14: 0 1 2 3 4 5 6 7 8 9 A B C D 10 11 12 13 14 15 16 17 18 19 1A 1B 1C 1D 20 … 9C 9D A0 A1 A2 … DD 100 … 199 19A 19B …  To convert from base 10: o Divide by the largest multiple of the new base 346710 → ______4 3467 = 3.385742 … 45 o Integer before decimal (3) must < base 346710 → 3_____4 o Subtract that integer and multiply by that place value 3467 � 5 − 3� × 45 = 395 4 o Repeat for the other place values 395 = 1.542968 … 44 346710 → 31____4 139 = 2.171875 43 346710 → 312___4 11 <1 42 346710 → 3120__4 11 = 2.75 41 346710 → 31202_4 3 =3 40 346710 → 3120234

Polynomial Algebra  𝑥𝑥 3 − 5𝑥𝑥 2 + 7𝑥𝑥 − 3

4 terms the entire line is a polynomial  Grabbies: 3(𝑥𝑥 2 + 2𝑥𝑥 + 1)

 Elephants (adding like terms): 6𝑒𝑒 + 4𝑔𝑔 + 2𝑒𝑒 + 𝑔𝑔

 Binomial Theorem o (𝑎𝑎 + 𝑏𝑏)3 = (𝑎𝑎 + 𝑏𝑏)(𝑎𝑎 + 𝑏𝑏)(𝑎𝑎 + 𝑏𝑏) = 𝑎𝑎3 + 3𝑎𝑎2 𝑏𝑏 + 3𝑎𝑎𝑏𝑏 2 + 𝑏𝑏 3 o For the ith term of (𝑎𝑎 + 𝑏𝑏)𝑛𝑛 : 𝑛𝑛 𝐶𝐶𝑖𝑖−1

𝑎𝑎𝑛𝑛−(𝑖𝑖−1) 𝑏𝑏 (𝑖𝑖−1)

Example: (𝑥𝑥 + 2)5 = 𝑥𝑥 5 + 5𝑥𝑥 4 (2) + 10𝑥𝑥 3 (2)2 + 10𝑥𝑥 2 (2)3 + 5𝑥𝑥(2)4 + 25 = 𝑥𝑥 5 + 10𝑥𝑥 4 + 40𝑥𝑥 3 + 80𝑥𝑥 2 + 80𝑥𝑥 + 32 o Always substitute (separate step) then simplify  I Hate Fractions o Multiply both sides by integer to eliminate all fractions o When multiplying/dividing both sides of an inequality by a neg. number, switch sign o

Rationals (can be written as fractions)  Cyclical permutations o Arrangement where order matters 𝑥𝑥 o produces a repeating decimal that is a cyclical permutation 7

 Repeating decimals (terminology) o Period: the part that repeats o Long form: written out to show at least three repetitions (e.g. 2.181818 …) ���) o Short form: written to indicate period and length (e.g. 2. �18  Any fraction with a denominator composed of 2s and 5s will not be a repeating decimal  All terminating decimals and repeating decimals are rational numbers  Non-terminating AND non-repeating decimals are NOT fractions => irrational o E.g. π (no pattern) o E.g. 0.01011011101111… (with pattern)  Converting repeating decimals to fractions o Let x equal the number (coursepack says in long form) o Multiply x so that the decimal is after the first period o Multiply x so that the decimal is before the first period o Number equations and subtract o Isolate x o Write a concluding statement that answers the original question

o

���� Let 𝑥𝑥 = 0.218 (1) − (2):

���� (1) 1000𝑥𝑥 = 218. 18 � ��� (2) 10𝑥𝑥 = 2. 18 1000𝑥𝑥 − 10𝑥𝑥 = 216 990𝑥𝑥 216 = 990 990 12 𝑥𝑥 = 55 12 ���� = ∴ 0.218 55

�) The set of all rational numbers (ℚ) combined with the set of all irrational numbers (ℚ produces the set of all real numbers (all numbers which may be written in decimal form) (ℝ).

Set Operators  Addition is a binary operator on Real numbers (two inputs, one answer)  Squaring is a unary operator on Real numbers (one input, one answer)  Union ∪ and intersection ∩ are binary set operators

{1,2,3} ∪ {3,4,5} = {1,2,3,4,5} Union combines two sets (produces a larger set)

{1,2,3} ∩ {3,4,5} = {3} ≠ 3 Intersection identifies elements common to both sets (produces a set)

 Set-builder notation o {𝑥𝑥|𝑥𝑥 > 2, 𝑥𝑥 ∈ I} (the set of all x such that x is greater than 2 and x is an element of the set of integers) o Always provide simplest answer o {𝑥𝑥|𝑥𝑥 > 2 and 𝑥𝑥 < 4 or 𝑥𝑥 < 6, 𝑥𝑥 ∈ ℝ} (wrong) AND cannot be used with an OR without brackets o , implies brackets on both sides o OR is like union, AND is like intersection  Complement (unary operator) (′) The COMPLEMENT of a set is all the elements in the UNIVERSE but not in the given set.  Universe (set) (S) The UNIVERSE is the set of all possible elements in a problem. All other sets in the problem must be chosen from only these elements. (it is the context of a set)  Proper subset (comparative) (⊂) Not equal to and one contains fewer elements than the other does. The first is a set chosen entirely from elements contained in the second set. A ⊂ B (A is a proper subset of B)  Subset (comparative) (⊆) Similar to PROPER SUBSET, but includes the possibility of equality.  Cardinality (unary operator) (𝑛𝑛()) The number of elements in a set, and is a real number.

Sets as Venn Diagrams  Sets are ovals -> region inside represents contents  The universe is a rect. containing all ovals A and B are mutually exclusive (disjoint) but not collectively exhaustive.

S

A

B

The shaded region below is C’.

S C

Set Properties  Commutative (order in which it is written doesn’t matter) A ∪ B = B ∪ A, A ∩ B = B ∩ A  Distributive (grabbies) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)  Associative (brackets not necessary when same operation) A ∪ (B ∪ C) = (A ∪ B) ∪ C = A ∪ B ∪ C  DeMorgan’s Laws (A ∪ B)′ = A′ ∩ B′ (A ∩ B)′ = A′ ∪ B′  Universal set S ∪ A = S S ∩ A = A S′ = ∅ A ⊂ S  Null set ∅ ∪ A = A ∅ ∩ A = ∅ ∅′ = S ∅ ⊂ A

Cardinality Formulas 𝑛𝑛(A ∪ B) = 𝑛𝑛(A) + 𝑛𝑛(B) − 𝑛𝑛(A ∩ B)

𝑛𝑛(A ∪ B ∪ C ∪ D) = 𝑛𝑛(A) + 𝑛𝑛(B) + 𝑛𝑛(C) + 𝑛𝑛(D) − 𝑛𝑛(A ∩ B) − 𝑛𝑛(A ∩ C) − 𝑛𝑛(A ∩ D) − 𝑛𝑛(B ∩ C) − 𝑛𝑛(B ∩ D) − 𝑛𝑛(C ∩ D) + 𝑛𝑛(A ∩ B ∩ C) + 𝑛𝑛(A ∩ B ∩ D) + 𝑛𝑛(B ∩ C ∩ D) + 𝑛𝑛(A ∩ C ∩ D) − 𝑛𝑛(A ∩ B ∩ C ∩ D) Pattern: cardinality of unions = + cardinalities of odd objects (1, 3, 5 intersected sets, etc) - cardinalities of even objects (2, 4, 6 intersected sets, etc)