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 ď ś đ?&#x2018;Ľđ?&#x2018;Ľ 3 â&#x2C6;&#x2019; 5đ?&#x2018;Ľđ?&#x2018;Ľ 2 + 7đ?&#x2018;Ľđ?&#x2018;Ľ â&#x2C6;&#x2019; 3

4 terms the entire line is a polynomial ď ś Grabbies: 3(đ?&#x2018;Ľđ?&#x2018;Ľ 2 + 2đ?&#x2018;Ľđ?&#x2018;Ľ + 1)

ď ś Elephants (adding like terms): 6đ?&#x2018;&#x2019;đ?&#x2018;&#x2019; + 4đ?&#x2018;&#x201D;đ?&#x2018;&#x201D; + 2đ?&#x2018;&#x2019;đ?&#x2018;&#x2019; + đ?&#x2018;&#x201D;đ?&#x2018;&#x201D;

ď ś Binomial Theorem o (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;?đ?&#x2018;?)3 = (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;?đ?&#x2018;?)(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;?đ?&#x2018;?)(đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;?đ?&#x2018;?) = đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;3 + 3đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;2 đ?&#x2018;?đ?&#x2018;? + 3đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;?đ?&#x2018;? 2 + đ?&#x2018;?đ?&#x2018;? 3 o For the ith term of (đ?&#x2018;&#x17D;đ?&#x2018;&#x17D; + đ?&#x2018;?đ?&#x2018;?)đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; : đ?&#x2018;&#x203A;đ?&#x2018;&#x203A; đ??śđ??śđ?&#x2018;&#x2013;đ?&#x2018;&#x2013;â&#x2C6;&#x2019;1

đ?&#x2018;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;â&#x2C6;&#x2019;(đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;â&#x2C6;&#x2019;1) đ?&#x2018;?đ?&#x2018;? (đ?&#x2018;&#x2013;đ?&#x2018;&#x2013;â&#x2C6;&#x2019;1)

Example: (đ?&#x2018;Ľđ?&#x2018;Ľ + 2)5 = đ?&#x2018;Ľđ?&#x2018;Ľ 5 + 5đ?&#x2018;Ľđ?&#x2018;Ľ 4 (2) + 10đ?&#x2018;Ľđ?&#x2018;Ľ 3 (2)2 + 10đ?&#x2018;Ľđ?&#x2018;Ľ 2 (2)3 + 5đ?&#x2018;Ľđ?&#x2018;Ľ(2)4 + 25 = đ?&#x2018;Ľđ?&#x2018;Ľ 5 + 10đ?&#x2018;Ľđ?&#x2018;Ľ 4 + 40đ?&#x2018;Ľđ?&#x2018;Ľ 3 + 80đ?&#x2018;Ľđ?&#x2018;Ľ 2 + 80đ?&#x2018;Ľđ?&#x2018;Ľ + 32 o Always substitute (separate step) then simplify ď ś 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) ď ś Cyclical permutations o Arrangement where order matters đ?&#x2018;Ľđ?&#x2018;Ľ o produces a repeating decimal that is a cyclical permutation 7

ď ś Repeating decimals (terminology) o Period: the part that repeats o Long form: written out to show at least three repetitions (e.g. 2.181818 â&#x20AC;Ś) ďż˝ďż˝ďż˝) o Short form: written to indicate period and length (e.g. 2. ďż˝18 ď ś Any fraction with a denominator composed of 2s and 5s will not be a repeating decimal ď ś All terminating decimals and repeating decimals are rational numbers ď ś Non-terminating AND non-repeating decimals are NOT fractions => irrational o E.g. Ď&#x20AC; (no pattern) o E.g. 0.01011011101111â&#x20AC;Ś (with pattern) ď ś 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

ďż˝ďż˝ďż˝ďż˝ Let đ?&#x2018;Ľđ?&#x2018;Ľ = 0.218 (1) â&#x2C6;&#x2019; (2):

ďż˝ďż˝ďż˝ďż˝ (1) 1000đ?&#x2018;Ľđ?&#x2018;Ľ = 218. 18 ďż˝ ďż˝ďż˝ďż˝ (2) 10đ?&#x2018;Ľđ?&#x2018;Ľ = 2. 18 1000đ?&#x2018;Ľđ?&#x2018;Ľ â&#x2C6;&#x2019; 10đ?&#x2018;Ľđ?&#x2018;Ľ = 216 990đ?&#x2018;Ľđ?&#x2018;Ľ 216 = 990 990 12 đ?&#x2018;Ľđ?&#x2018;Ľ = 55 12 ďż˝ďż˝ďż˝ďż˝ = â&#x2C6;´ 0.218 55

ďż˝) The set of all rational numbers (â&#x201E;&#x161;) combined with the set of all irrational numbers (â&#x201E;&#x161; produces the set of all real numbers (all numbers which may be written in decimal form) (â&#x201E;?).

Set Operators ď ś Addition is a binary operator on Real numbers (two inputs, one answer) ď ś Squaring is a unary operator on Real numbers (one input, one answer) ď ś Union â&#x2C6;Ş and intersection â&#x2C6;Š are binary set operators

{1,2,3} â&#x2C6;Ş {3,4,5} = {1,2,3,4,5} Union combines two sets (produces a larger set)

{1,2,3} â&#x2C6;Š {3,4,5} = {3} â&#x2030; 3 Intersection identifies elements common to both sets (produces a set)

ď ś Set-builder notation o {đ?&#x2018;Ľđ?&#x2018;Ľ|đ?&#x2018;Ľđ?&#x2018;Ľ > 2, đ?&#x2018;Ľđ?&#x2018;Ľ â&#x2C6;&#x2C6; 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 {đ?&#x2018;Ľđ?&#x2018;Ľ|đ?&#x2018;Ľđ?&#x2018;Ľ > 2 and đ?&#x2018;Ľđ?&#x2018;Ľ < 4 or đ?&#x2018;Ľđ?&#x2018;Ľ < 6, đ?&#x2018;Ľđ?&#x2018;Ľ â&#x2C6;&#x2C6; â&#x201E;?} (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 ď ś Complement (unary operator) (â&#x20AC;˛) The COMPLEMENT of a set is all the elements in the UNIVERSE but not in the given set. ď ś 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) ď ś Proper subset (comparative) (â&#x160;&#x201A;) 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 â&#x160;&#x201A; B (A is a proper subset of B) ď ś Subset (comparative) (â&#x160;&#x2020;) Similar to PROPER SUBSET, but includes the possibility of equality. ď ś Cardinality (unary operator) (đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;()) The number of elements in a set, and is a real number.

Sets as Venn Diagrams ď ś Sets are ovals -> region inside represents contents ď ś 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 Câ&#x20AC;&#x2122;.

S C

Set Properties ď ś Commutative (order in which it is written doesnâ&#x20AC;&#x2122;t matter) A â&#x2C6;Ş B = B â&#x2C6;Ş A, A â&#x2C6;Š B = B â&#x2C6;Š A ď ś Distributive (grabbies) A â&#x2C6;Ş (B â&#x2C6;Š C) = (A â&#x2C6;Ş B) â&#x2C6;Š (A â&#x2C6;Ş C) ď ś Associative (brackets not necessary when same operation) A â&#x2C6;Ş (B â&#x2C6;Ş C) = (A â&#x2C6;Ş B) â&#x2C6;Ş C = A â&#x2C6;Ş B â&#x2C6;Ş C ď ś DeMorganâ&#x20AC;&#x2122;s Laws (A â&#x2C6;Ş B)â&#x20AC;˛ = Aâ&#x20AC;˛ â&#x2C6;Š Bâ&#x20AC;˛ (A â&#x2C6;Š B)â&#x20AC;˛ = Aâ&#x20AC;˛ â&#x2C6;Ş Bâ&#x20AC;˛ ď ś Universal set S â&#x2C6;Ş A = S S â&#x2C6;Š A = A Sâ&#x20AC;˛ = â&#x2C6;&#x2026; A â&#x160;&#x201A; S ď ś Null set â&#x2C6;&#x2026; â&#x2C6;Ş A = A â&#x2C6;&#x2026; â&#x2C6;Š A = â&#x2C6;&#x2026; â&#x2C6;&#x2026;â&#x20AC;˛ = S â&#x2C6;&#x2026; â&#x160;&#x201A; A

Cardinality Formulas đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;(A â&#x2C6;Ş B) = đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;(A) + đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;(B) â&#x2C6;&#x2019; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;(A â&#x2C6;Š B)

đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;(A â&#x2C6;Ş B â&#x2C6;Ş C â&#x2C6;Ş D) = đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;(A) + đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;(B) + đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;(C) + đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;(D) â&#x2C6;&#x2019; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;(A â&#x2C6;Š B) â&#x2C6;&#x2019; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;(A â&#x2C6;Š C) â&#x2C6;&#x2019; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;(A â&#x2C6;Š D) â&#x2C6;&#x2019; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;(B â&#x2C6;Š C) â&#x2C6;&#x2019; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;(B â&#x2C6;Š D) â&#x2C6;&#x2019; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;(C â&#x2C6;Š D) + đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;(A â&#x2C6;Š B â&#x2C6;Š C) + đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;(A â&#x2C6;Š B â&#x2C6;Š D) + đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;(B â&#x2C6;Š C â&#x2C6;Š D) + đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;(A â&#x2C6;Š C â&#x2C6;Š D) â&#x2C6;&#x2019; đ?&#x2018;&#x203A;đ?&#x2018;&#x203A;(A â&#x2C6;Š B â&#x2C6;Š C â&#x2C6;Š 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)

Unit 1 Review (MPM2DG)

A review of number systems, preliminary algebra, and set theory.