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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}
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: { Xbox, crayons, blah blah blah}set builder: { 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 ?
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 1.1
#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 b,d 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 ...uh, no
'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?2: {a}, { }take the set {a,b} ... what are all the subsets? how many?4: {a,b} {a} {b} { }take the set {a,b,c} ... what are all the subsets? how many?8: {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 elementsex) 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'
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 tihnk 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
truth tables for trickier compound statementsex) "its not raining OR my name is not Jason"
... T OR F ... Tex) "it is not cloudy OR (i do not teach math AND it is thursday)"
... F ˅ (F ˄ T) ... F ˅ F ... F
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 the 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
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, 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", 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
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
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)
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
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 a cold day, i wear my fake fur coatit is a cold day
so, what is the conclusion?
.....i am wearing my fake fur coat
what about:on a cold day, i wear my fake fur coatit is not a cold day
what is the conclusion?...no conclusion! outisde CD circle, might be inside FFC circle, or might be outside
do it again, using logic notation
CD → FFC "if its a cold day, then i wear my fake fur coat"CD "its a cold day"FFC "i wear my fake fur 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
on a cold day, i wear my fake fur coatit is not a cold day
CD -> FFC~CDif you try to conclude ~FFC, 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 fake fur coati am not wearing my fake fur coat
conclude?....it is not a cold day
why does this make sense?- Euler diagram...outside FFC is outside CD- logic notation: CD → FFC is the same as ~FFC → ~CD (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
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) a ˅ b; a → c; b → c. therefore, c
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?
...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. (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 units digit of 21000 ?
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
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 1.4
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
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
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 twoalso called:binary databitsmachine language
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: 111001011000111100001
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
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?...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 ... no
which numbers do we need to check?- only primes- we could stop after checking 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
its 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 even
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 ?
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
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 (p40)
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 balls you must put in 5 boxes. each box must have a ball. 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) 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
"P" = permutation(also known as "arrangement")
= 726,485,760
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...buth 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) 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?
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 january and 23 days in february to work. how many ways are there to do this?
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