math 20 oct exam
bisection method steps
Step 1. check the prime factorization to make sure its not a perfect square.
Step 2. identify the two closest perfect squares, and then write them as equivalent to their square roots.
Step 3. find the midpoint between the to closest square roots, then compute its square
Step 4. determine which half of the interval contains your og number
Repeat steps 3 and 4, use ~ for final answer
set theory
Definition: a set is a collection of objects. The objects in the set are called elements
What kind of objects? Numbers, letters, symbols, names, people, other sets , ... ANYTHING!
order does not matter
x∈S, x∉S
x is contained in S, x is not contained in S
⊂, ⊆
proper subset (has less elements than og set), subset less or equal to
union
The union of sets A and B is the set of all objects contained in A or B (or both)
ex. If A: {a, b, c} and B= {d, e, f}, then the union of A and B is AUB = {a,b, c, d, e, f}
order doesnt matter
intersection
The intersection of sets A and B is the set of all objects contained in A and B. Intersection is denoted by A∩B
ex. A={a,b,c,d,e} B={d,e,f,g,h} A∩B={d,e}
order doesnt matter
complement
Let A be a set contained in a universe U The complement of A is. the set of all plements in U but not in A. The complement of A is denoted by A' or Ac
ex. u={1,2,3,4,5,6}. A={2,4,6}. A'={1,3,5}
order doesnt matter.
B\A
the complement of A with respect to be is everything in B but not in A.
ex. let u= {1,2,3,4,5,...} B= {1,2,3,...,100} A= {2,4,6,...,100}
B'= {101,102,103,...}
A'= {1,3,5,...,97,99,101,102,103,...}
B\A= {1,3,5,97,99}
natural numbers
The set of natural numbers, sometimes called the counting numbers, consists of 1,2,3,4...
denoted by ℕ= {1,2,3,4,...}
0 not included
integers
The set of integers consists of zero, the natural numbers, and all the negatives of the naturals.
Denoted by ℤ= {...,-2,-1,0,1,2,...}
the set of naturals is a subset of integers
rational numbers
A rational number is a number that can be expressed as a ratio of two integers. i.e. a rational number has the form A/B, where A&B∈ℤ, and B≠0.The set of all rational numbers is denoted by Q. ℕ⊂Q, ℤ⊂Q.
examples, 1, 2, 4, 47, 2/3, -4/3, 1/2
irrational number
An irrational number is a number that cant be expressed as a ratio of two integers. i.e., an irrational is a number that is not ratinal.
Examples: π, √2, √p where p=prime,
denoted by ℝ\Q
real numbers
the set of real numbers are all the numbers on a number line, denoted by ℝ
which types of numbers cant be listed
rational, real, irrational
what percent of ℝ do rational numbers take up
0
how to change remainder to a decimal
keep adding zeros to the end
fraction to decimal- 1/4
4⟌1
4⟌1.00= 0.25
steps to changing a repeating decimal to a fraction
Step 1. Let x=0.3'. 10x=10(0.3')= 3.3' the "10" is determined by amount of digits in your number
Step 2. 10x-x = 3.3-x => 9x=3
Step 3. x= 3/9 -> 1/3
repeating decimal to fraction that doesnt start with 0
2.56'
Step 1. let x=0.56'. 100x= 100(0.56')= 56.56
Step 2. subtract x from both sides. 100x-x= 99x = 56.56-x= 56. => 99x=56
Step 3. x= 56/99
Step 4. add back the whole number. 2 56/99= 254/99
cardinality
number of elements in a set. |{}|= x
|{Ø}|, |{}|
1, 0. becuase |{Ø}| has an empty set, but |{}| is an empty set.
density
Def. Let S⊆R. The set S is dense in R if for any a,b∈R such that a⊂b, there exists a number C∈S, such that a<c<b.
the set of rationals is dense in R. are the following sets dense in R? N, Z, Even integers, empty set, positive rational numbers
N- no. for example, I could choose 0 and 1/2, but there doesnt exist an element of N between 0 and 1/2. So, by the definition of density, N is not dense.
Z- no. for example, -4 , -3,5. There is no element of Z between those, so Z is not dense in R by definition
All even integers- no. same reason
Empty set- no.
The set of all positive rational numbers- no, ex -2 and -2.5
limit
a value that a sequence approaches as the number of terms approaches infinity.
ex) 8.1, 8.01, 8.001, 8.0001... has a limit of 8.
0,5,10,15,20... no limit bc it approaches infinity
fundamental theorem of arithmetic
any natural number n>1 can be written as a unique product of primes. ex. 6=2x3
124=2squaredx31
gcd
find the prime factorization, whatever they both share. denoted by gcd(m,n)
ex. 12=2x3, 30=2x3x5. GCD=2x3=6
ex. 36=2(2)x3(2) 48=2(4)x3 60=2(2)x3x5 GCD= 2(2) and 3. Therefore, 2(2)x3=12
lcm
denoted by lcm(m,n), is the smallest natural number that is divisible by both m and n
ex. lcm (12,30) = 60
ex. lcm(36,48,60) = 2(4) x 3(2) x 5 = 720
other way to find lcm
lcm(a,b)= axb/gcd(a,b)
ex. 24 and 35. 24= 2(3)x3, 35= 5x7. gcd(24,35)=1.
lcm(24,35)=24x35/1= 840
square root
let a∈R. The square root of a is a number x≥0, Whose square is equal to a.
i.e. √a=x, <-> x2=a
cant be negative bc √-4∉R, and two negatives make a positive
cube root
The cube root x of a is a number whose cube is equal to a.
i.e. 3√a=x <-> x3=a
ex. 3√8=2
can be negative bc 3√-8=-2
how to find square root
1296
Step 1. find the prime factorization
126= 2(4) x 3(4)
Step 2. rewrite the result from step 1 as a product of two equal factors.
(2x2x2x2) (3x3x3x3)
= 2x3x2x3x2x3x2x3
= (2x3x2x3)2
= 36(2)
So, √1296=36
finding the cube root
2744
Step 1. 2744= 2(3) x 7(3)
= (2x2x2) (7x7x7)
= (2x7) (2x7) (2x7)
= 14(3)
3√2744= 14