Math 100: Fundamentals of Mathematics

ch2 Set theorywhat is set theory?

set theory is about "collections of stuff"

ex) suppose you want to talk about your toys teddy bear, legos, dolls, action figures, cars, nintendo, interactive books

how can we talk about this group of stuff?

the set of Stevie's toys = { teddy bear, legos, dolls, action figures, cars, nintendo, interactive books }

---- those curly brackets '{' and '}' indicate a set---- life is easier if we give the set a name

...call it 'Stevie's toys' or call it S so, S = { teddy bear, legos, dolls, action figures, cars, nintendo, interactive books }

Q: does order matter in a set? ...NO, because we still have the same 'stuff '

Jimmy lives next doorhis toys are Xbox, crayons, trains, cars, skates, basketball, legosnow, describe this using set notation J = {Xbox, crayons, trains, cars, skates, basketball, legos} this is the set of Jimmy's toys

are these sets the same? in other words, does S = J ? ...no

Janelle has toys L = { skates, basketball, legos, cars, trains, Xbox, crayons}recall: S = { teddy bear, legos, dolls, action figures, cars, nintendo, interactive books }

compare S, J, L ... aha! J = L

what if Janelle buys a Barbie doll? does L = J ?what is the relationship between L and J ?

how do we write that in notation?

means "is contained in" (or "is a subset of")

every "item" is called an element

is nintendo one of Stevie's toys? how do we write that? this says: n "is an element of" S -OR- n is included in S

is nintendo one of Jimmy's toys?how do we write that? this says: n "is not an element of" J -OR- n is not included in J

ex) what is F, the set of all fun toys? cant write it down, its not well-defined, two people might disagree

notation for sets:words: "Jimmy's toys"list notation: { Xbox, crayons, blah blah blah}set builder notation: { x | x is one of Jimmy's toys }

read this as "x such that x is one of Jimmy's toys"

ex) describe the set in different notationswords: "positive even numbers"list: { 2,4,6,8,10,... }set builder notation: { x | x is a positive even number }

ex) T = {3,6,9,12,15,18,...}words: positive multiples of 3

is 36 ϵ T ?...yeswhat about 5 ?

no ... how do we write that?

consider A = {6,12,15}how can you compare A and T ?

do you use ϵ or c ?

we use ϵ to make statements about elementswe use c to make statements about sets

ex) all odd integerslist: { ...-3,-1,1,3,5,7... }

what about { ...,7,9,11,... }this is correct, although the other way is more helpful by reminding

usthat the set includes negative numbers

n(A) is the number of elements in the set A ex) A = {1,3,5,7} ... what is n(A) ?

hw questions 2.1

#26 { x | x is a negative multiple of 6}list: { -6, -12, -18, -24, -30, -36, ... } OR { ...-30, -24, -18, -12, -6 }

#41 A = {0,1,2,3,4,5,6,7} n(A) = 8

#43 A = {2,4,6,8,10,...,1000} n(A) = 500 its half of 1000 since you only have the even numbers

#12 the set of counting numbers between 4 and 14between: do not include the endsfrom: include the ends

#71 9 {6,3,4,8} ...true

#73 {k,c,r,a} = {k,c,a,r} ...true

2.2 Venn diagrams and subsetsor, pretty pictures of sets

note that 2,4 are in set B but not in set A

two different descriptions of subsets:

subset: A is a subset of B if every element of A is also an element of B

ex) A Bex) A A

proper subset: a subset that is not the set itself

ex)

ex) B c Uex) A c U

what does this remind you of?3 < 43 ≤ 43 ≤ 33 < 3 ...thats not true, is it

'little bar' means it can be equalno little bar means it cant be equal

ex) A = { a,c,e }which of the following is a subset of A? proper subset of A?D = {a}E = {b,f}F = {a,c,e}

now write each of those in notation:

what if a set has nothing in it?

Chris. poor broke Chris has no toys.the set of Chris's toys is the empty set

C = { } OR

what are the subsets of C?{ }

what are the proper subsets of C?...there arent any

take the set {a}... what are all the subsets? how many?

{a}, { }take the set {a,b} ... what are all the subsets? how many?

{a,b} {a} {b} { }take the set {a,b,c} ... what are all the subsets? how many?

{a,b,c}{a,b}{c,b}{a,c}{a} {b} {c}{ }notice the pattern: n(A)=1 gives you 2 subsetsn(A)=2 gives you 4 subsetsn(A)=3 gives you 8 subsets ...son(A)=4 gives you 16 subsetsnow,N(A)=n gives you subsets

notice that you are multiplying by 2 each time1 element, 22 elements (2)(2)3 elements (2)(2)(2) etc

if n(A) = n, the number of proper subsets is 2n - 1

note that the empty set is always a subset

ex)U = counting numbers from 1 to 10A = {2,3,5}what is everything not in A?

{1,4,6,7,8,9,10}this is called the complement

notation: A' (also A )

B = {3,4,6,7,8}find: n(B)what is B' ?is A c B ?

how many subsets does B have?

draw the Venn diagram with A,B

they have 3 in common, so 3 goes in the 'overlap'this is called the intersection of A and B

2.3 Set Operationsor, how do you "add" sets?

ex) you try to collect all 9 game pieces given out by BK to collect a prize.Yahnique gets: 2,3,5,7,8Jasmine gets: 2,4,5,7Yahnique has 5 cards, Jasmine has 4 cardsdo they have all 9 cards necessary ?their cards put together are: 2,3,4,5,7,8

together, they have 6 (different) cardswhere did 6 come from?

5 + 4 - 3 = 63 cards "dont count" because we already have them

notation:what are the sets? how many in each?Y = {2,3,5,7,8} n(Y) = 5J = {2,4,5,7} n(J) = 4

"both together" = union- the elements in either setnotation: for sets A,B, the union is A U B

ex) Y U J =

"overlap" = intersection- the elements in both setsnotation: for sets A,B the intersection is A ∩ B

ex) Y ∩ J =

"everything else" = complement- the elements not in a certain setnotation: for a set A, the complement is A'

ex) Y' =

"difference" = differenceY - J = {2,3,5,7,8} - {2,4,5,7} = { 3,8}

what about 4? with sets, you cant take away something you didnt start withn(Y) = 5n(J) = 4n(Y - J) = 2

= 5 - 3= n(Y) - n(Y ∩ J)

You Do: draw the diagram and find the elements ... recall Y={2,3,5,7,8} J={2,4,5,7}ex) Y' U J ex) Y ∩ J'

hey! same as Y - J

hey! its the same as(A U B)'

draw the Venn diagram for:ex) A' U B' ex) B' - A

lets get fancy - a Venn diagram with 3 sets!

answer is everything (in A,B,C) shaded, cuz its the union

draw and shade the Venn diagram for:ex) A U B U C ex) A ∩ B ∩ C

heres a tough onedraw and shade the Venn diagram for:ex) (A U B) ∩ C'

remember:do your operation - union, intersection, etc - with the pieces that you have in the problem

hw 2.3 questions

hw 2.3 questions

U = {a,b,c,d,e,f,g}X = {a,c,e,g}Y = {a,b,c}Z = {b,c,d,e,f}

elements which are in A together with elements which are both outside B and outside C

55) X = {1,3,5}Y = {1,2,3}

a) X u Y = {1,2,3,5}b) Y u X = {1,2,3,5}c) conjecture: X u Y = Y u X

conjecture: a statement that you think is true

27) X (X - Y) = {a,c,e,g} {e,g} = {e,g}

how many elements are in A x B?... n(A x B) = 9

cuz its (3)(3)

is (2,a) ϵ A x B ?

note that order mattersthere is a difference between A x B and B x A

Cartesian product

ex) a point (x,y) where x, y are real numbers"�"

notation: x ϵ � ... y ϵ � (x, y) ϵ � x � "R cross R"

ex) A = {a,b,c}B = {1,2,3}

name one element of A x B

A x B = { (a,1) (a,2) (a,3) (b,1) (b,2) (b,3) (c,1) (c,2) (c,3) }

2.4 Cardinal Numbers ... or, counting with sets

ex) survey about which performers a group of people like 33 like Tim32 like Celine28 like Britney11 like Tim and Celine15 like Tim and Britney14 like Celine and Britney5 like all7 like none

Q: how many people are in the survey?

...we have to do this in a certain order - or it will be too hard

what does '6' represent?..people who like Tim and Celine but not Britney

what does 9 represent?...people who like Celine and Britney but not Tim

notice that you have to fill in 'all' and 'none' firstthen the 'pair overlaps', etc

ex) 13 people like spring12 people like summer5 people like both0 people like neither

how many people like spring or summer?

in general,n(A U B) = n(A) + n(B) - n(A ∩ B)

ex) 24 people like red, 19 like blue, 12 like bothhow many like red or blue?solve this (a) with the formula (b) with the diagram

hw 2.4 questions

28)

ch 3 Logic

3.1 statements

what is a statement (in logic)?... something that is either true or false

ex) "this is a math class"ex) "it is raining"not a statement:

- do the dishes!

what is a compound statement ?- one or more statements, joined with connectives

and whats a connective?"not"

ex) "it is not raining""and"

ex) "i am running late and i am in trouble""or"

ex) "today is tuesday, or i have more time to do my homework""if/then"

ex) if you beat me, then pigs must be flying

abstract statement notation- in algebra, we like to use x,y,z as our variables to represent numbers- in logic, we like to use p,q,r as our variables to represent statements

- in algebra, we have operations such as + (plus) - (minus)- in logic, we have connectives such as

NOT ~AND ˄OR ˅IF/THEN →

ex) "Barack Obama is president and he lives in DC" ... rewrite in notationwhat are the statements? what are the connectives?

ex) i will watch the game or i will leave

ex) let p be "i like shoes" ... let q be "i like the circus" ... rewrite in words: p ˅ ~q

ex) let p be "Palin was president", let q be "i would die" ... rewrite: p → q

what is a quantifier ?- it expresses how many items have a certain property"all" (or "every")

ex) all bankers are richsame as: every banker is rich

"some" (or "at least one")ex) some TV channels are in HDsame as: at least one channel is in HD

"none"ex) no giraffes are short

how do you negate a quantifier?ex) what is the opposite of "all dogs go to heaven" ? ...discuss

remember that this is all about what makes a statement true or false. so, what precise fact would make the statement false?

ex) what is the opposite of "some people have brown eyes" ?

Statement Negationall dosome do/at least one isnone do/each one doesnt

hw questions 3.1

#33 y > 12the negation is:

#59 A,B#60 A,B#25 "every dog has its day"the negation is: at least one dog does not have its dayOR some dogs do not have their day#43 ~p or q: she does not have green eyes OR he is 48 years old

3.2 truth tables

NOT "opposite"

ex) p: "its cloudy outside" ...T~p: "its not cloudy outside" ...F

ex) p: "its raining outside" ...F~p: "its not raining outside: ...T

OR

p OR q: its cloudy or its rainy ...T OR F ...T

p OR q: its cloudy OR my name is Jason ... T OR T ... T

p OR q: this is a history class OR it is wednesday ... F OR F ... F

AND

p ˄ q: the wall is white AND i teach math ...T ˄ T ... Tp ˄ q: its raining AND its friday ... F ˄ F ... Fp ˄ q: its raining AND i teach math ... F ˄ T ... Fnote: an AND statement is true only when both statements are true

in logic, there are two values: true, false

note: for an OR statement to be true, you only need one

evaluation

ex) what is the value of ~p ˄ q when p is true and q is true

ex) evaluate (p ˅ q ) ˄ ~q when q is false and p is true

you do:ex) evaluate p ˅ (q ˄ ~q) when q is false and p is true

ex) "its not raining OR my name is not Jason" ... T OR F ... T

ex) "it is not cloudy OR (i do not teach math AND it is thursday)" ... F ˅ (F ˄ T) ... F ˅ F ... F

truth tables for compound statements

every step that it takes to build your entire statement, that is a column in your truth table

ex) draw the truth table for: ~(~p ˅ q)

ex) p ˅ ( q ˅ ~p)

"always true (no matter the values of p, q, etc)" this is called a tautology

ex) p ˅ ~p

always true, since if p is true the whole thing is trueif p is false, then ~p is true ... one of them must be true

ex) draw the truth table for ~p ˄ ~q ex) draw the truth table for ~(p ˅ q)

they are THE SAME. when two compound statements have the same truth table, they are called equivalentthis is an important equivalence...it is called DeMorgan's law for negating an OR statement

~(p ˅ q) = ~p ˄ ~qthe other half of Demorgan's law is about the negation of an AND statement

~(p ˄ q) = ~p ˅ ~qhere's a way to remember each law: "negate both, switch the operation"

ex) draw a truth table for: p ˄ (~q ˅ r)

thats a lot more rows! is a there a way to predict how many rows we will have?

hw questions 3.2

3.3 conditional statement (or "if/then statement")

ex) "if it rains, then i will bring my umbrella"ex) "if there is a transit strike, then i wont give a final"there was a transit strike ... so no finalthere was no transit strike, i dont give a final. .......am i a liar?

...no, thats the truththere was no transit strike, i do give a final. ....am i a liar?

...no, thats the truththere was a transit strike, and i gave a final

.....liar!- the only way for a conditional to be false

notation: p → q "if p, then q"

truth table:

only way for a conditional to be false: first part true, but second part false [dont carry through on promise]ex) if the Colts win, then i'll pay you $5

only on the hook if the Colts win. if they lose, i can do whatever i want

the conditional is false in only one case.- what other statement that we know is false in only one case?

- how can we set up an equivalent statement? (this is a tough question that requires some thought)

convert Colts problem to "OR"p: the Colts winq: i'll pay you $5

~p ˅ q: the Colts dont win OR i pay you $5

negation of conditional (if/then):

~(p → q) = p ˄ ~qhey!~(~p ˅ q) = p ˄ ~qgreat!:p → q is the same as ~p ˅ q,so their negations should be the same also...and they are!

ex) make the truth table for:(p → ~q) ˅ ~q

note that:"p is true" is the same as "p""q is false" is the same as "~q"when we write "p" by itself, that is like saying it is true

note that: if/then is only F in one case. so when its F in the first row, you know that all the other times its T

3.4 more conditional

related forms:consider "if there is a transit strike, then you go home (no final)" p →

q

- converse: q → pex) "if you go home, then there is a transit strike"

- inverse: ~p → ~qex) if there is no transit strike, then you dont go home"

- contrapositive: ~q → ~pex) "if you dont go home, then there is no transit strike"

question: is the original statement equivalent to its inverse?to figure it out, draw a truth table

this is not the same as the original statement"the fallacy of the inverse"even though the inverse sounds like it should be right, notice that the original statement tells you nothing about what happens when there is no transit strikealso, "the fallacy of the converse"ex) if i own a Mac then i own a computer

converse: if i own a computer, then i own a Mac ....clearly notinverse: if i dont own a Mac, then i dont own a computer ......clearly

notwhat about the contrapositive?

if i dont own a computer, then i dont own a Macanother logic function: biconditionalmeaning: p and q happen at the same timenotation: p ↔ q "p if and only if q"interpretation: p → q and q → p ... its an if/then statement going both ways

"if and only if"notation: ↔ex) my name is Jason Samuels if and only if my SSN is ***-**-****ex) today is thursday if and only if tomorrow is friday

when is it true?when both statements have same value (both true or both false)

applications: circuitson/off just like true/falsewant: statement is T exactly when circuit is ON

(ON means that a current can get from start to finish)

statement is F exactly when circuit is OFF

for each circuit, when would it be on? what is the matching logic statement?

need both to be closed to have a signal

hw questions 3.3,3.4

3.3-14 if p is true, then ~p → (q ˅ r) is trueyes, because ~p is false, so the if/then statement must be true

3.3-60

3.4-1 if beauty were a minute, then you would be an hour b->hconverse: h->b if you are an hour, then beauty is a minute inverse: ~b -> ~h if beauty is not a minute, then you are not an hourcontrapositive: ~h -> ~b if you are not an hour, then beauty is not a minute

3.3#94) (~p ˄ ~q) ˄ ~r ... draw the circuit

if loving you is wrong, i dont want to be right

other ways to say "if p then q" p → q ex) "if it rains, then i will bring my umbrella"

q, if pex) i will bring my umbrella if it rains ...still "if p", just putting it last

p implies qex) its raining implies that i am bringing my umbrella

ex) if its a dog, then its a mammal

all p are qex) all dogs are mammals

p is qex) a dog is a mammal+++++ex) all baseball players are athletes...convert to "if/then"if you are a baseball player, then you are an athletehow can we draw the diagram of this?

these are also called Euler diagrams

ex) all kids go to schooleveryone in school has a favorite teacher

what can you conclude?

.....all kids have a favorite teacherbecause K is completely inside T

on every cold day, i wear my winter coatit is a cold day

so, what is the conclusion?

.....i am wearing my winter coat

what about:on a cold day, i wear my winter coatit is not a cold day

what is the conclusion?...no conclusion! outisde C circle, might be inside W circle, or might be outside

do it again, using logic notation

C → W "if its a cold day, then i wear my winter coat"C "its a cold day"W "i wear my winter coat"

this is called the law of detachment, or modus ponensp → qpconclusion?qin this case, q is a valid conclusion because it is supported by what you know

if its a cold day, then i wear my winter coatit is not a cold day

C -> W~Cif you try to conclude ~W, you are using the inverse....but you cant do that"fallacy of the inverse"

ex) i own a computersome computers are laptopsconclusion.....i own a laptop

is that a valid conclusion?no....it is not a valid conclusion from the things we know

here is an example of a valid argument:i own a computerall computers are desktops or laptopsi do not own a desktopconclusion....i own a laptop

other ways to make a valid conclusion:

ex) if its a cold day, then i wear my winter coati am not wearing my winter coat

conclude?....it is not a cold day

why does this make sense?- Euler diagram...outside W is outside C- logic notation: C → W is the same as ~W → ~C (contrapositive)

this form of conclusion is called the law of contraposition, or modus tollensp → q~qconclude....~p

ex) i have a nickel in my pocket or i have a quarter in my pocketi do not have a nickel

conclusion?.....i have a quarter

this is the law of disjunctive inferencen ˅ q~nconclusion....q

laws of inference:

- law of detachment (modus ponens)p → qp...q

- law of contraposition (modus tollens)p → q~q.....~p

- law of disjunctive syllogismp ˅ q~p....q

- law of transitivityp → qq → r.... p → r

other useful laws:pq... p ˄ q

p ˄ q... p

note: for a full list, http://en.wikipedia.org/wiki/List_of_rules_of_inference

if its a weekend, then im not going to work. i'll go to work or watch tv. its a weekend....what can you conclude?

hw questions

3.5#21 Little Rock is northeast of texarkanalittle rock is northeast of austinconclusion: texarkana is northeast of austin

see if a conclusion is valid using a truth table

n ˅ q is the same as: [(n ˅ q) ˄ ~n] → q~nconclusion: q

"it is always true that if n ˅ q then q~n

n q ~n n ˅ q (n ˅ q) ˄ ~n [(n ˅ q) ˄ ~n] → q

see if this is a valid argument using a truth table: ex) p → q; p → ~q. therefore, ~p

ex) the law of transitivity

ex) p→q, q therefore p

see if a verbal argument is valid OR make a valid conclusion

ex) if i am a man, then i am proud. if i am proud, then i will be humbled. therefore, if i am a man, then i will be humbled.

ex) my computer is working or it is in the repair shop. if my friend comes over, he fixes my computer. if he fixes my computer, it is not in the repair shop. my friend comes over. what is a valid conclusion that uses all parts of the argument?

...therefore, my computer is working

ex) if that tree is infested with pine bark beetles, then it will die. people plant trees on Arbor Day and it will not die. therefore, if people plant trees on Arbor Day, then that tree is not infested with pine bark beetles. is that a valid argument? (hint: use a truth table)

hw 3.5 questions

18) all birds fly; all planes fly. therefore, a bird is not a plane.

28)

note: problems with "some" are tricky, because that doesnt tell you exactly how to draw the circles. you have to think about what is possible.

hw 3.6 questions

18)

Inductive reasoning- from specific cases to general pattern

Deductive reasoning- use general pattern to solve specific cases

many ways to use a general rule:ex) every day in february is a winter day

my birthday is february 28conclusion: ...my birthday is on a winter day

compare:ex) today is a summer day

tomorrow is a summer dayso is the day after that, and the day after, and the day after that,

and the day after that, and the day after that.conclusion: every day is a summer day

ch1: Problem Solving

1.1: inductive reasoning v. deductive reasoning

lets play a game: take your age. double it. add 1. square it (i'll wait). subtract 1. divide by 4. subtract your age.now, tell me the number.

inductive reasoning:ex) 37 x 3 = 111

37 x 6 = 22237 x 9 = 33337 x 12 = 444

guess:37 x 15 = 37 x 18 = 37 x 21 =

you found the pattern

ex) find the pattern:1,1,2,3,5,8,13,21,....guess the next number...34,55,89,144,...add the last two numbers to get the next numberthis is a famous sequence [mathematical list]the Fibonacci sequencenote: count the petals in each spiral on a sunflower, or leaves in each spiral on a pinecone, they are numbers from the Fibonacci sequence ... we will study this sequence and its applications later in the semester

the number of regions in a circle with n points connected is:

didnt see that coming, did you?

ex) circles and chordsif you put points on a circle and connect them with chords, how many regions do you get?

1.2 examples of inductive reasoning...with number patterns

guess the next number:ex) 2,9,16,23,30,...

ex) 6 10 16 24 34 46 ...

ex) heres a twist: make a sequence using this method

6 _ _ _ _ _

8

4 4 4 4 4

ex) find the next number in the sequence,using the method of common difference

method of common difference

surprising sums

ex) 1+3 = 41+3+5= 91+3+5+7= 161+3+5+7+9= 251+3+5+7+9+11=36

rule? : adding first n positive consecutive odd integers =

what is the sum of the first n positive integers: 1+2+3+...+n

Figurate numbers

triangular numbers

Tn = n(n+1)2

square numbers

Sn = n2

pentagon numbers

Pn = n(3n-1)2

hw 1.2 questions

1.3 strategies for problem solving

ex) what is the ones digit of 21000 ?

what about the ones digit of 23001

ex) what is the units digit of 51776 ?

four steps to solving a problem (from Polya)1. understand the problem2. make a plan3. carry out the plan4. look back and check

common strategies:- look for a pattern (just used that)

- trial and error

ex) draw the figure without lifting your pen or retracing any lines

- break down into cases

ex) how many rectangles are there in each picture?

for more strategies, see 1.3 p20

same question, if you have 20 black, 20 white, 10 red

SudokuKenKen / MathDoku

the distance from NYC to Miami is 1386 miles.if you drive 70mph, how long will the trip take...approximately?

note: real answer is a little less, since 1386<1400

reading a graph

a) what was the profit in 1996?b) what was the profit in 1995?c) what was the increase in profit from 1998 to 1999?d) between which two consecutive years was the greatest increase in profit (over 1995-2000)?e) which is the only year when profit did not increase from the previous year?

a) which group had a higher average math SAT score in 2003?b) in how many years did males have a higher average math SAT score?c) in how many years did the male average math SAT score beat 500?

hw questions

1.3#9 place 1,2,...15 in the boxes so consecutive boxes add to perfect squares

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _

hw questions 1.3

(12)(21) palindrome(32)(27)(31)(51)(53)

how fast is the car driving?

note: this is deductive reasoning i.e. if it is a different palindrome, the last digit has to be different...etc

strategy: find a patterninductive reasoning

helpful related question: what is the largest number of socks i can pull *without* getting a pair? W B .....2 socksnext sock *must* give you a pair, so the answer is three

ex) socks: 2 black, 2 white, 2 red, 2 green, 2 bluehow many do you have to pull to make sure you get a pair?Bk,W,R,G,Bl...next one must give a pair

strategy: solve a similar problem

how many squares?

strategy: break down into cases

deductive reasoning: every step is determined by everything you already know

Every row, column, box must contain each digit 1...9you are given a bunch of numbers to start

hw questions 1.3

hw questions 1.4

bonus super hard extra credit

note that, in the triangle below, every number 1-6 is used once, and every number is the difference of the two numbers above it. this is a 'difference triangle'. make a difference triangle for the numbers 1-10.

numeration and mathematical systems (ch4) [selected topics]

our number system: Hindu-Arabic

* place value0 "nothing"the difference between

220

a way to multiply by hand (for us, the usual way):

lattice method from India and Persia

ex) 5326 x 817

converting between bases (4.3)

binary data- on and off- its the signal that makes a computer run ... how?- information!

mathematical background:ex) 527 ... compare: roman numerals "III" = 3but if we write "III" it means "one hundred eleven"

different understanding of writing number...place value

ex) how many sheep are there? ||||||||||||| "tally marks"

|||| |||| |||"grouping"

13this grouping choice is called "base ten"

another grouping choice - base fivefives ones __ __

lets compare these different choicesbase ten ↔ base five

notation: 13ten = 23five "13" in base ten is the same as "23" in base fiveequation: 1·10 + 3·1 = 2·5 + 3·1picture:

note: if no base is written, assume its base ten

how do you count in different bases?base ten:base five:

what is the largest digit in base ten?what is the largest digit in base five?what is the largest digit in base eight?

what happens (in base eight) when you count past 7? ...you go to the next place value column

base eight:

base three:

determining place values in different bases:

base ten:

base five:

base three:

ex) in base five, what is the integer right before 100?

confirm by converting to base ten:

base five: no "50" ... its 100no "500" ... it's 1000

you do: convert 1212three into base ten

converting numbers between bases

convert TO base ten

ex) convert 2021three to base tenwell, what does each digit represent?

ex) convert 4236seven to base ten

ex) convert 8347four to base ten

but wait!! in base four, what digits are there? ....0,1,2,3this number does not exist

ex) convert 10011two to base ten

application: computer programmingex) encode color, e.g. green = "00/ff/00"

bases bigger than ten...how can you have "base sixteen" ?

now, go the other way- a little harder

ex) convert 97 into base five

ex) convert 155 into base four

how does this apply for computer information?on/off = 0,10 = off1 = on

→ this is base two

also called:binary databitsmachine language

note: the text uses a different method. i think this method is clearer and easier to use.

note: how many digits do you need to encode a letter:in base ten? ...twoin base two? ...seven

convert 110010two to base ten

convert 45 to base two

representing text (letters)

ex) CAT →base ten → 03 01 20in reverse: 030120 → 03/01/20 → C A T

ex) convert to letters using base ten: 0805121215

computers use the following system:A → Z is 65 →90 ... in base two thats 1000001 → 1011010a → z is 97 → 122 ... in base two thats 1100001 → 1111010

ex) convert "C" to binary codeone way: letter → base ten → base two

C 67 1000011another way: count from 1000001

ex) convert into a message: 110001111000011110010

hw questions, 4.3

60) encode "X"X -> (convert to #) -> 88 -> (convert to base2) ->

note: this conversion is FROM base

62) 1001000/1000101/1001100/1010000have: binary code (base2)do: convert to base10then: convert to letter

13) find the smallest and largest 4-digit number (and base10 equivalent) in base threesmallest = 1000 -> (convert to base10) ->largest = 2222 -> (convert to base10) ->

review - ch4

NUMBERS

ch6 - real numbers- what are they- how do we do arithmetic- decimals, percents, fractions

ch5 - number theory- factorization (gcf, lcm)- modular arithmetic (clock arithmetic)- Fibonacci sequence

6.1 Real Numbers

....what are they?

whole numbers: 0,1,2,3,4,.....integers: ....,-3,-2,-1,0,1,2,3.....fractions: an integer divided by an integer (also called "rational numbers")

different ways to think of fractions:- a part of something- can be also written as a decimal- a "division" e.g. 8 divided by 4- a number over a number

2 is a whole number, but you can think of it as a fraction

lets write numbers in order!

note: sometimes useful to use fractionsex) group of three, one is male ..."1/3 is male"sometimes not usefulex) put amounts in order, see number line above...........for that, decimals work better

one more "type" of (real) number:irrational numberex) ....you cannot write it as a fraction (rational number)definition: any real number you cannot write as a rational number OR when you write it as a decimal, it keeps going without stopping and without repeating

some examples:ex) -4 thats rational (in fact its an integer)ex) 1/3 = .3333333.... thats rationalex) 4.83736867265... thats irrational (continues forever, no repeating)

real numbers: rational and irrational numbers, taken together

signs and absolute value

ex) - (-4) = 4

absolute value...make it positive (full definition: distance from 0)ex) | 5 | = 5ex) | -8 | = 8definition:| x | = { x if x > 0

{ -x if x < 0 ...because that makes it positive

ex) | 7 - 9 | = | -2 | = 2....first solve whats insidenot: | 7 - 9| = 7 + 9

6.2 operations on integers

ex) 7 + 9 = 16ex) 9 + 7 = 16ex) 8 - 3 = 5ex) 3 - 8 = -5ex) -3 + 8 = 5 8 - 3 = 5 --> 8 + (-3) = 5compare:ex) 5 - 4 = 1ex) 5(-4) = -20ex) 5 + (-4) = 1 note: must know what parentheses mean in a given situation

- multiplication- do it first

mult: (3)(5)do me first: 3 - (2+5)both: 7(6-2)neither: 5 + (-4)

signs and addition/subtractionex) 9+7 = 16ex) 8 - 3 = 5ex) 3 - 8 = -5ex) -8 - 11 = -19 [lose some, then lose some more...size gets bigger, stays negative]ex) -12 + 34 = 22 [for addition/subtraction, keep sign of the bigger number]

signs and multiplication/divisionex) (3)(2) = 6ex) (-12)(8) = -96ex) (7)(-4) = -28ex) (-5)(-6) = 30ex)

ex) (2)(-4)(5)(-3)(-5) = - 600[switch the sign every time you multiply/divide by a negative][OR, cross off negative's in pairs]

signs and parentheses togetherex) 4 - (5 - x)

= 4 + (-1)(5 - x)= 4 + -5 + x ...this is what it means to 'distribute the

negative'

6.3 operations on rational numbers (fractions)

addition, subtraction

ex) addition/subtraction with fractions: same denominator, so add/subtract numerators

these mean exactly the same thing

multiplication/divisionstrangely, this is easier

keep the fraction balanced:what you do to top you must also do to bottom (similar to equations)

different rules to calculate by hand for addition/subtraction vs.

note: calculators are bad at handling this, if you are not careful they will ÷2 then ÷5, which is wrong

reduce first, then multiply

dividing by a number is the same as multiplying by its inverse (reciprocal)

decimals

.34

.01 is of a dollar

fractions and decimals

convert:ex) .3 =

ex) .79 =

ex) .5237 =

ex) 374 =100

note that 5237 ≈ 51000

estimating is helpful and important!

percents

"per" = "cent" =

"46 percent" = 46 ÷ 100 = 46/100 = .46

"5 percent" =

ex) what is 30 percent of 60 ?

ex) 12 is what percent of 96 ?

6.5 convert between fraction--decimal--percent

note: more than "1" is more than "100%"

decimal-->fractionlook at place value of last decimal digit, thats what you divide by

theres more on fractions and repeating decimals in the text...it is extra credit material

when do we prefer decimals?ex) in one class, 20 out of 23 students passedin another class 15 out of 17 students passedwhich class has the higher pass rate?

which is "better", fraction or decimal?...it depends on what you are trying to do

when do we prefer fractions?ex) a class has 7 students, 3 femalewhat part is female3/7 OR 42.857..%

ch6 hw questions

ch6 homework questions

ch5 some topics from number theory

+ prime and composite numbers+ "clock" arithmetic+ fibonacci numbers, and more

note: prime numbers are important in encryption and decryption (e.g. sending your credit card number over the internet in code)

what is a prime number?...a whole number {1,2,3...} which has exactly two factors, 1 and itselfex) is 5 prime?...yes, 5=(5)(1) ...and thats itex) is 8 prime?..no, because 8=(2)(4) ...or you could say that 2 is a factor of 8 [or 2 divides 8 evenly]ex) is 25 prime?...no, because 25=(5)(5), so 5 is a factor

what is a composite number?...any number with a factor that is not 1 or itself - every other whole number (greater than 1)

1 is the multiplicative identity (which is a "unit")

ex) is 57 prime?...no, 57=(3)(19)but that was a little trickyex is 119 prime?...this requires work

how do you figure out if a number is prime?...check every number to see if it is a factor

ex) is 83 prime?check 2 ... no [not even]check 3 ... nocheck 4 ...unnecessary, because 2 does not go into it, so 4 cant eithercheck 5 ... no [doesnt end in 0 or 5]check 6 ... unnecessarycheck 7 ... nocheck 11 ... no [for 2-digit number, digits would have to be the same].....so, 83 is prime

which numbers do we need to check?- only primes- we could stop after checking - for 83 - the number 10...why? imagine we are writing down all the factors of a number

you only need to check for "small" factors, you dont need to check for "big" factors

where is the cutoff between "small" and "big" factors?ex)

so the square root represents the switching point between "small" and "big" factors ... so we dont need to check once we pass the square root

big help....suppose you wanted to know if 141 was primeinstead of checking 140 numbers, you only have to check 1-11[since 141 is less than 12and since we are only need to check primes, we check:2,3,5,7,11 ... and thats it

suppose you wanted to find all the primes up to a certain number, say 60how could you do that?

before (on "83") we checked only prime numbers...dont need to check composite numbers, because, for example, if 3 does not "go in" then 9 does not "go in"

1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 5051 52 53 54 55 56 57 58 59 60

but 60 is between 7 and 8so the last number i need to check is 7[did you notice that all the multiples of 11 were already crossed off?]

what do we have?....all the prime numbers from 1-60

this is called: the Sieve of Eratostheneswhen we shake our sieve,the composites fall out, the primes stay in

divisibility tests

ex) is 38326376 divisible by 3?...no

ex) 5692184, is it divisible by 3?

how did i predict the remainder?

lets do an easier oneex) is 396528282 divisible by 2?...yes, because its evenex) is 4842876285 divisible by 2?...no, and the remainder is 1

what is the "trick" for 3?....a number is divisible by 3 when the sum of the digits is divisible by 3...in fact, the remainder of the number divided by 3 is the same as

the remainder of (the sum of the digits) divided by 3

ex) 59 ÷ 3 = 19 r25+9 = 1414 ÷ 3 = 4 r2

ex) is 4209 divisible by 3?..yes, because 4+2+0+9 = 15 ... 3 "goes into" 15 [3 is a factor of 15]

The Divisibility Tests

2 is a factor if: the units digit is 0,2,4,6,83 is a factor if: 3 divides the sum of the digits4 is a factor if: 4 divides the last two digits

ex) does 4 divide 3528? ..you know 3500 is a multiple of 4, so can multiples of 4 get us from there to 28? ...yes

5 is a factor if: the units digit is 0,56 is a factor if: both 2 and 3 are factors

ex) 49476, is 6 a factor? ...2 goes in, 3 goes in (sum of digits is 30)so yes, 6 is a factor

7 ...8 is a factor if: 8 divided the last three digits

note that: 8 goes into 1000ex) 489376, is 8 a factor?does 8 "go into" 376? ...8 goes into 200, 8 goes into 160

so 8 goes into 360...368...376...yes9 is a factor if: 9 divides the sum of the digits

ex) is 9 a factor of 63534658946 ?

note that when you divide the number by 9, the remainder will be 5

ex) does 3 go into 4572981ex) does 6 go into 474762972

ex) does 4 go into 3877264ex) does 8 go into 9872646528746200

bonus: why does the divisibility test for 9 work??

hw questions 5.1

#45 what is the divisibility test for 6? ....if its divisible by both 2 and 3guess the divisibility test for 15.....if its divisible by both 3 and 5extra: we could make up the divisibility rule for....? 35 .... div by 5 and 7

21 ... div by 3 and 7#23 123456789 [a 9-digit number]is it divisible by (a) 2? (b) 3? (c) 4? 5? 6? 8? 9?2? ...no, ends in odd number3? ...yes, 1+2+3+4+5+6+7+8+9 = 45, divisible by 3

[note: 93737665869483366589569038576086, sum digits=735, take sum again]4? ...no, not divisible by 25? ...no doesnt end in 0,56? ...no, not divisible by 28? ...no9? ...yes, sum digits =45, 9 goes into 45extra: div by 15? ....no, not divisible by 5

#27 name two primes which are consecutive numbers.2,3are there any others? ...nowhy not?e.g. 5,6 ... 11,12 ... 13,14 ....one number will always be even

#41 is 8,493,969 divisible by 11 ?8-4+9-3+9-6+9 = 22 ... so yes

"clock" arithmetic [5.4]

ex) if its 8 oclock, what time will it be in 6 hours?....2 oclock...after you reach 12, the next number is 1

ex) if its 7 oclock, what time will it be in 14 hours?

12hrs brings you back to 7oclock, so just add 2....9 oclock

ex) at 9oclock, you are scheduled to take three 3-hour tests consecutively. at what time do you finish?(3)(3) =99+9=1818-12 = 6oclock

ex) starting at 10oclock you take three 5-hour tests. at what time do you finish?(3)(5) = 15 = 12+310+3 = 13 --> 1

10+(3)(5) = 10 + 15 = 2525-12-12 = 1better: 25-24 = 1

so far: using "clock", wrap around after 12next: we will "wrap around" after.......

whatever we want

in clock arithmetic, wrap around every 12in modular arithmetic, wrap around every ...whatever we want

terminology: clock...."mod 12"ex) counting in mod 12:0,1,2,3,4,5,6,7,8,9,10,11,0,1,2,3if you have a number, "reduce" it by subtracting 12

ex) using mod 12, 14 = 2 (mod 12)OR 14 ≡ 2 (mod 12)

notation: ≡ "equals (using mod)" or "congruent"note: sometimes we just use "=" (because we are lazy)

ex) counting in mod 5: 0,1,2,3,4,0,1,2,3,4,0,1..."reduce" any number by subtracting 5

ex) 17 ≡ __ (mod 5)arithmeticex) 8 + 6 ≡ 2 (mod 12)ex) 2+4 = ? (mod 5)

2+4=6, but mod 5 that "reduces" to 1ex) 4+4 = ? (mod 5)

4+4 = 8 ... 8-5 = 3ex) (4)(3) = ? (mod 5)

(4)(3) = 12 ... 12-5=7 ... 7-5=2(4)(3) = 2 (mod 5)

you do:ex) (4)(5) = ? (mod 7)

ex) 3+6 = ? (mod 8)

ex) (3)(2) = ? (mod 4)

note: consider 20 (mod 7)20 7 = 2 r 6so we subtracted two 7's, and the answer was 6"mod" is the same as remainder

extra credit note: since "mod" gives you the remainder, what does that tell you when converting bases?

subtraction:ex) 5-2 = ? (mod 8)

answer=3ex) 4-7 (mod 8) 0,1,2,3,4,5,6,7,0,1,2,3,4...

4-7 = -3 ... what does that mean??"negative" means count back from 0count back 3 from 0 ... answer = 5could also get that by adding 8

hmmm.....-3= -3+8 = 5 (mod 8)also equal to: (mod 8) 5=13=21=29=....if you keep adding 8, you wont change the remainderalso equal to (mod 8) 5 = -3 = -11 = -19 = -27 = .... [can also subtract 8]you do:ex) 2 - 5 (mod 6) = ?

ex) 1 - 8 (mod 9) = ?

divisionex) 12 ÷ 2 = 6

why? 2·6 = 12

ex) 6 ÷ 5 (mod 7) = ?how do we answer this? its the same as asking:(?)(5) = 6 (mod 7)

it turns out that (4)(5) = 6 (mod 7) [check it]so:also we know that:

in mod arithmetic:We must think of division as the opposite of multiplicationwe cannot think of division as grouping

to do division, you must use multiplication chart

use the table to answer these questions:

ex) 4 ÷ 3 = ? (mod 5)

ex) 2 ÷ 2 = ? (mod 5)

ex) 1 ÷ 3 = ? (mod 5)

5. Fibonacci numbers and the Golden Ratio

the Fibonacci sequence:1,1,2,3,5,8,13,21,34,55,89,144,233,.....

notation for Fibonacci numbers: is the nth Fibonacci numberwe set F1=1, F2=1, and to find every other Fibonacci number,we say Fn + Fn+1 = Fn+2

in other words, 1+1=2, 1+2=3, 2+3=5, etc

...and now...

the rabbit problem:

which of these is the prettiest rectangle?

this question has been asked in many studies and overwhelmingly people always answer the rectangle which can be described in the following way:

small rectangle is in same proportion as large rectangleratio of the length of sides is called the Golden Ratio

how can we solve for the Golden Ratio exactly?

where does the Fibonacci sequence appear?

ex) bee family treehow do bees reproduce?a male bee hatches from an unfertilized egg, so it has a mother but no fathera female bee hatches from a fertilized egg, so it has a mother and a fatherso, how many ancestors does a male bee have 2 generations ago? 3 generations? n generations?

let M=male, F=female

ex) pinecones

ex) flowers - seeds and petals

spirals of 13 and 21 spirals of 8 and 13 spirals of 21 and 34 spirals of 3 and 5

where does the Golden Ratio appear?

ex) human bodyideal ratio of height to navel height

ex) pentagram

- the Stradivarius violin

- the Mona Lisa

- the Parthenonthe Parthenon was designed by Phidias,

to celebrate Pericles saving Athens during the Trojan War, in the 5th century BC. The first letter in Phidias' name is φ ("phi"), and this letter is used to represent the Golden Ratio

- the Cathedral of Chartresbuilt in France in the 13th century

- the Unites d'Habitation, designed by Le Corbusier in France

The Fibonacci sequence and the Golden Ratio

what is the ratio of consecutive Fibonacci numbers?

another amazing connection between the Fibonacci numbers and the Golden Ratio:to find a Fibonacci number, you have to find the ones before it, right?wrong! there is a formula for any single Fibonacci number. and, amazingly, it involves the Golden Ratio:Fn = where φ is the Golden Ratio

so with numbers this is: Fn =

check it:ex) F5 =

Amazing Fibonacci facts

what is the sum of the first n Fibonacci numbers?sum = 1+1+2+3+5+8+13+21+...+Fn

in other words, sum = F1 + F2 + F3 + ... + Fn

we know that 1+2=3, 2+3=5, etcso notice that 1=3-2, 2=5-3, etcnow the sum is: sum = (2-1)+(3-2)+(5-3)+(8-5)+ ... + (Fn+2 - Fn+1)

and then a miracle happens: everything cancels, except:sum = Fn+2 - 1

what is the sum of the squares of the first n Fibonacci numbers?sum = 12+12+22+32+52+82+132+212+ ... +Fn2 we can answer this question by using a very clever picture

so, F12 + F22 + F32 +F42 + F52 + F62 + ... + Fn2 = Fn·Fn+1

hw questions ch5

5.5 #28 obtain pythagorean triples using F: 1,2,3,5first: (1)(5)second: (2)(3)x(2)third: 2 + 3

5.5#6 what is the approximate value of the Golden Ratio?1.61803...

5.4#63 Chi NOrl SFRobin [21-day] 1,2,85,12 6,18,19Christine [30-day] 23,29,30 5,6,17 8,10,15,20,25

when will they both be in Chicago? (over 60 days)Robin: 1, 2, 8 Christine: 23,29,30 in common: 23,29

22,23,29 53,59,6043,44,50

how do you set up this problem in math notation?

counting [ch11]

ex) suppose Jason wakes up and gets dressed. he has 3 different shirts to wear, and 2 different pants. how many ways can he get dressed (how many outfits)?

(3)(2) = 6 can we always do this?...in this case, yes

each box represents one outfitthere are (2)(3) boxes, so there are 6 outfits

first basic counting techniqueif you want to do a couple of different things,if there are "a" ways to do the first thing,

and "b" ways to do the secondand "c" ways to do the third....

then the total number of possibilities is:(a)(b)(c).....

ex) suppose a family has 3 kids, boys or girls. how many possibilities are there? what are they? (not in amount, but the order they are born in)

BBBBBGBGGBGBGGGGGBGBBGBG

two possibilities for each child ... this is a truth table! (in disguise)

OR make a tree

ex) suppose i get dressedi can choose from 3 caps, 4 shirts, 2 pants, 6 socks, 2 shoeshow many different outfits are possible?= (3)(4)(2)(6)(2)= 288

ex) state license platein many states, the plate is 3 letters followed by 3 numbershow many possible plates are there?

total possibilities = (26)(26)(26)(10)(10)(10) = 17,576,000

second counting technique:

ex) you have 6 cans of paint of different colors and 4 walls to paint. how many ways are there to paint the room?

think of each wall as an event

here, once you use a can of paint,it CAN be used again

With Replacement

ex) suppose you have 5 different balls you must put in 5 different boxes. one ball in each box. how many ways are there to do that?

NOT: (5)(5)(5)(5)(5) ..why? .....because after you put a ball in first box, it cant go in any other box

here, once you use a ball it CANNOT be used again

Without Replacement

ex) suppose you have to visit 7 stores (once each). in how many different orders can you visit the stores?

notation: 7! "7 factorial"

ex) suppose 8 horses run in a race. how many different "top three" finishes are possible (win-place-show)?

notation: 8P3 "8 P 3" or "P(8,3)" or "8 permute 3"

sort of like writing out 8!,but only writing the first three numbers

note that 7! = 7P7

ex) when a baseball manager makes a lineup, he has 14 hitters to put in the batting order, positions 1 through 9. how many ways are there to do that?

"P" = permutation(also known as "arrangement")

= 726,485,760

ex) find P(9,3)

ex) if you have 12 pairs of shoes...how many ways are there to wear them in one week?what if you dont wear the same pair twice?

how many ways are there to select a committee of 3 from a group of five?Yahnique, Ruth, Valinda, Samantha, Dewitte.g.: Yahnique, Ruth Valindaor: Yahnique Ruth Samanthaor: Samantha Yahnique Ruth...but the last two are the same, since all positions are equal...so, order does not matter

this counts the last two committees as two different committees ... but we dont want that

how many times does (5)(4)(3) count this one committee?how many different ways to write down those three names?

every 3-person committee has been counted (3)(2)(1) times...so we need to divide by (3)(2)(1) = 6

notation: 5C3 = 5·4·3 "5 C 3" or "5 choose 3" or "5 combination 3" 3·2·1

an easy way to think of 5C3: 5! (in the denominator, 2+3=5)3!·2!

is that the same thing?

so 5C3 does all the work for you

what is one difference between these types of problems?ORDER MATTERS vs. ORDER DOES NOT MATTER

ex) find 8C6

ex) find 10C2

ex) how many ways are there to select, from 30 students, 6 of them to go on a trip

= 593775ex) how many ways can you select, from 30 baseball teams, 8 to go to the playoffs?

ex) how many 4-digit lottery numbers are there?

ex) how many ways can you elect 5 out of 20 soldiers to go fight?

you do:ex) how many 4 digit lottery numbers are there in base 6?

ex) how many ways can you elect a colonel and sergeant from 9 candidates?

ex) how many ways are there to select 5 starting pitchers from 11 pitchers?

now...how many ways are there to make a batting order AND select starting pitchers?

ex) for your work schedule, you need to select 21 days in april and 23 days in february to work. how many ways are there to do this?

1406106702000

trickier:ex) suppose you have a string of three lights, which can be on or off. a) how many possibilities are there?

b) how many possibilities are there if you cannot have two consecutive lights off?

here we have a restriction- we cannot multiply, because the number of possibilities at each step changes (could be 1 or 2)- helpful to make a tree (or list)

note: just like making subsets (e.g. from a group of toys, how many ways can you make a subset of toys)each toy represents a "step" with two possibilities: you either take the toy, or you dont

ex) doll, truck, scooter, ball

11.4 Pascal's Triangle

lets make a triangle of numbers, using a simple rule:start with a 1enter numbers down to the left and down to the right of every number, and each number you enter is the sum of the two numbers above it (above left and above right)this gives:

1 1 1 1 2 1

1 3 3 1

thats very nice ... so what?

now lets calculate every combination number, nCr

0C0 = 1C0 = 1C1 =2C0 = 2C1 = 2C2 =3C0 = 3C1 = 3C2 = 3C3 =

so what does this tell us, and how can we use it?

if you want to find 7C5 for example, it lives in Pascal's Triangle