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

45 = 3.385742 …

o Integer before decimal (3) must < base 346710 → 3_____4

o Subtract that integer and multiply by that place value

�3467

45 − 3� × 45 = 395

o Repeat for the other place values 39544 = 1.542968 …

346710 → 31____4 13943 = 2.171875

346710 → 312___4 1142 < 1

346710 → 3120__4 1141 = 2.75

346710 → 31202_4 3

40 = 3

346710 → 3120234

Polynomial Algebra 𝑥𝑥3 − 5𝑥𝑥2 + 7𝑥𝑥 − 3

4 terms the entire line is a polynomial

Grabbies: 3(𝑥𝑥2 + 2𝑥𝑥 + 1)

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

Binomial Theorem o (𝑎𝑎 + 𝑏𝑏)3 = (𝑎𝑎 + 𝑏𝑏)(𝑎𝑎 + 𝑏𝑏)(𝑎𝑎 + 𝑏𝑏) = 𝑎𝑎3 + 3𝑎𝑎2𝑏𝑏 + 3𝑎𝑎𝑏𝑏2 + 𝑏𝑏3 o For the ith term of (𝑎𝑎 + 𝑏𝑏)𝑛𝑛 :

𝐶𝐶𝑖𝑖−1 𝑎𝑎𝑛𝑛−(𝑖𝑖−1) 𝑏𝑏𝑛𝑛(𝑖𝑖−1)

o Example: (𝑥𝑥 + 2)5 = 𝑥𝑥5 + 5𝑥𝑥4(2) + 10𝑥𝑥3(2)2 + 10𝑥𝑥2(2)3 + 5𝑥𝑥(2)4 + 25 = 𝑥𝑥5 + 10𝑥𝑥4 + 40𝑥𝑥3 + 80𝑥𝑥2 + 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

Rationals (can be written as fractions) Cyclical permutations

o Arrangement where order matters

o 𝑥𝑥7 produces a repeating decimal that is a cyclical permutation

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���� 1000𝑥𝑥 = 218. 18���� (1) 10𝑥𝑥 = 2. 18���� (2)

(1) − (2): 1000𝑥𝑥 − 10𝑥𝑥 = 216 990𝑥𝑥990

=216990

𝑥𝑥 =1255

∴ 0.218���� =1255

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.

The shaded region below is 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)

S

A B

S C