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Math 171 Flash Cards
Mathematics 171 with Furucci at Keene State College
About this deck
By: Kay Hedstrom
Created: 2010-11-15
Size: 105 flashcards
Views: 111
Created: 2010-11-15
Size: 105 flashcards
Views: 111
About StudyBlue
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Venn Diagrams
Geometric figures used to represent sets and set relations
Used by: Leonard Euler
Named after: John Venn
Venn was a mathematician and in the 19th century came up with the idea using rectangle to represent the universe.
Whole Numbers
the set composed of natural numbers and zero.
Ex: {0,1,2,3...}
Binary Operation
means that two whole numbers that are manipulated and form a third whole number.
Ex: Addition and Subtraction
Addends and Sum
Ex: a+b+c
a and b are addends
c is the sum
Addition of Whole Numbers
If a set A contains a element and set B contains b element and A^B does not equal 0 then a+b is the number of elements in AUB (Let A and B be two disjoint finite sets
If n(A)=a and n(B)=b then a+b=n(AUB)
Less than
for any whole number a and b
a is less than b written a<b if an only if, there exists a natural number K such that a+K=b
Ex: 3<7
Set Model of Addition
Number Line Model for Addition
Lattice Algorithm for Addition
Closure Property for Addition
If a and b are whole numbers than a+b exists and is a unique whole number.
Ex: {2,4,6} not closed
4+6=10 and 10 is not in the set
Subtraction of whole numbers
for any whole numbers a b and c a-b=c for some whole number a
subtraction is the inverse of addition
Termonlogy for Subtraction
a-b=c
a is the minuend
b is the subtrahend
c is the difference
Take Away Method for Addition
15-10=? 10+?=5
Missing Addend Model for Subtraction
If Juan had $6 and Susie has $2 how much more money does Juan have?
Comparison Model
Drawing picure to match up numbers and what does not match up is the answer
Number Line Model for Addition
93-27=93+3-27+3=96-30=66
Equal Addend Method for Subtraction
making it eaiser by adding same number to each a + b
Multiplication of Whole Numbers
for any whole number a and b of n(A)=a and n)B)=b then axb=n(AxB)
for all whole numbers a and b
axb=a+a+a+a...+a (b times) Repeated addition
Termonology for Multiplication
axb=c
a is a factor
b is a factor
c is a product
Commutative Property for Multiplication
axb=bxa
Associative Property for Multiplication
(axb)xc=ax(bxc)
Distributive Property of Multiplication with Respect to Addition
a(b+c)=ab+ac
Multiplicative Identity
ax1=1xa
Multiplicative Property of Zero
ax0=0xa=0
Closure Property of Multiplication
If a and b are whole numbers then axb exists and is a unique whole number
Ex:{1,2,3} not closed because 2x3=6
3x5=3+3+3+3+3=15
Repeated Addtion Model for Multiplicaiton
Array Model Method 1
Add the dotsArray Model Method 2
1 2 3 4
1X X X X
2X X X X
3X X X X
count the x's
Cartesian Products Method of Multiplication
{vanilla, choc, coff}
{sprink, syrup}
{(van, sprink), (van, sy), (choc, sprink), (choc, syr), (coff, sprink), (coff, syr)}
3x2=6
Lattice Multiplication Method
Russian Peasant Algorithm
27x68
Halves and Doubles
circle odd #'s in halves column and then add the double column that corrasponds to double side
Division of whole numbers
for all whole numbers a, b and c with b not equal to 0 a/b=c iff c is a unique while number such that bxc=a
division is repeated subtraction
divistion by zero is undefined
Termonology for Division
a/b=c
a is the dividend
b is the divisor
c is the quotient
Partition or Subset Model for Division
10/5=? 5x?=10
Missing Factor Method
10/5=? 10-5=5-5=0
Repeated Subtraction Method
how many times did you subtract?
2 to get to 0
6/726 add 100 20 1 and the answer is 121(do it out)
Scaffolding Method
Egyptian Numeration System
- 3000 B.C.
- used hieroglyphics (picture symbols)
- used papyrus and ink filled reeds
- did not have place value
had no symbols for zero
was very cumbersome
used additive system
Babylonian System
- 3000 B.C.
- used wedge-shaped marks on wet clay
- used base 60
- no symbol for zero
Ex: <<<^^ 32
Mayan System
- wrote vertically
- base 20
- had 3 symbols
- had a symbol for zero
Japanse and Chinese System
- wrote vertically
- wrote number 1st then place value
ex: 6 100 2 10 3=623
Roman Numerals
- used additive system with subtraction and multiplication
- used 7 symbols
- no symbol for zero
- no more than 3 identical symbols used in succession
- if a symbol has a smaller value than the symbol to its right, it must be subtracted
- bars written above symbols mean to multiply by 1000 or 1,000,000
Hindu Arabic System
- modern day system
- invented in India by Hindus and brought to Europe by the Arabs
- uses place value
- has 10 symbols and has a symbol for zero
Function
from set A to set B exists when every element of A is paired (by a rule) to one and only one element of B
Formula
Play an important role in mathematics
Domain
the set of all X values specified for the functions
Range
the set of all Y values that result from the function
Constant Function
any function of the form f(x)=c were c is the constant
Domain of a function
is the set of real numbers for which the function is defind. If the function has a denominator eliminate any values which will result in a denominator zero
Cartesian Graph
the domain is plotted on the x-axis while the range is plotted on the y-axis
Representing Functions
- Arrows
- Tables
- Equations
- Ordered Pairs
- Graphs
- Function Machines
Arrows Notation
f : x ----> 2x+1
A 2----->5B
(circle and draw arrow)
Tables
(X and Y table)
Equations
y=2x+1
f(x)=2x+1
Ordered Pairs
values and their squares
(0,0) (1,2) etc.
Graph
f(x)=5x+4 (linear equation)
f(x)=x^2 (parabola)
GRAPH IT!
Function Machine
input-operation-output
Vertical Line Test
If a vertical line drawn on a graph intersects in only one place then the graph is a funtion
Composite Function
The range of the first function becomes the domain of the second function
notation: f * g (f of g) (f composed of g)
If the first function
f is followed by the second function g, write it as g * f
When the domain
is also the range for every element in the composite function
Relation
Is any ordered pairing of the members from one set to the member of another set.
However it may have out puts associated with a single input
A Function
is a relation, specifically, a function is a relation in which each element in the domain is paired with exactly one element in the range.
not all relations are functions
Relations may be classified as being
Reflexive
Symmetric
Transitive
Reflexive Relation
A relation R on a set A is reflexive is and only if (a,a) E R for every a E A.
That is, every relation is equal to itself a=a
Symmetric Relation
A relation R is symmetric if and only if (a,b) E R implies that (b,a) A R
That is, any two relations can be reversed, Id a is related to be then b is related to a. If a=b then b=a
Transitive Relation
A relation R is transitive is relation is related to the second relation and the second is related to the third, then the first relation is related to the third.
If a=b and b=c
Equivalence Relation
A relation that is reflexive, symmetric, and transitive is an equivalence relation.
Factor
a divisor of a number
Ex: factors of 10 are: 1,2,5,10 (finite)
Multiple
a number that is formed by the multiplication of factors
Ex: multiples of 10 are: 10, 20, 30...
Divides
If a and b are any integers (with a not equal to 0) then a is said to divide b iff there exists an integer c such that b=ac
a/b is read as "a divides b"
Divisibility Tests 2
units digit must be an even number
Divisibility Tests 5
units digit must be a 0 or a 5
Divisibility Tests 10
units digit must be a 0
Divisibility Tests 4
last two digits must be divisible by 4
Divisibility Tests 8
last three digits must be divisible by 8
Divisibility Tests 3
sum of its digits must be divisible by 3
Divisibility Tests 9
sum of its digits must be divisible 9
Divisibility Tests 6
If the integer is divisible by both 2 and 3 That means it is an even number and the sum of its digits is divisible by 3
Divisibility Tests 7
If the integer without its units digits minus twice the units digits is divisible by 7
Ex: 462 divisible by 7
7/(46-2x2)
Divisibility Tests 11
Sum of alternating digits minus the sum of other alternating digits is divisible by 11?
Ex:8471980
0+9+7+8=24
8+1+4+13
24-13=11 so yes!
Proper Divisor
all the number that can divide evenly into a value except for the number itself.
Ex: 10- 1,2,5,
Perfect Number
the sum of its proper divisors equals the number
Ex: 1+2+3=6
smallest perfect number is 6 the next is 28
Deficient Number
the sum of its proper divisor is less than the number
Ex:10-1+2+5=8
Abundant Number
the sum of its proper divisors is greater than the number
Ex:12-1+2+3+4+6=16
Amicable Number
two positive numbers are amicable if each is the sum of the proper divisors of the other
Ex: 220 has a proper divisors of 1+2+4+5+10+11+20+22+44+55+110=284
284 has proper divisors of 1+2+4+71+142=220
Prime Number
smallest prime number is 2
a natural number with exactly two distinct factors (itself and one)
2,3,5,11,19,29
Composite Number
any natural number with more than two factors
4,6,8,12,16,24
Neither Composite or Prime
0 and 1
Factor Trees
Fundamental Theorem of Arithmetic (Gauss)
Each composite number can be written as a product of primes in one and only one way, regardless of order.
Sieve of Eratosthenes
Conjecture
a guess based on inconclusive evidence
Goldbach's Conjecture
every known even integer greater than 2 is the sum of two primes
Greatest Common Divisor
the greatest common divisor (gcd) and the greatest common factor (gcf) are the same thing.
Three methods
- List Methods
- Prime Factorization
- Euclidean Algorithm
List Method
list all the factors and circle ones that have in common and find the greater one
Prime Factorization
Factor trees find what is in common with the both of them and multiply it out
Euclidean Algorithm
divide the small number into the bigger number and keep dividing until you get to zero and the last divisor is your answer
Least Common Multiple
List Method
Prime Factorization
About this deck
By: Kay Hedstrom
Created: 2010-11-15
Size: 105 flashcards
Views: 111
Created: 2010-11-15
Size: 105 flashcards
Views: 111
About StudyBlue
STUDYBLUE makes things that make you better at school.
Things like online flashcards with photos and audio.
Things like personalized quizzes and friendly reminders about when (and what) to study next.
Think of it as a digital backpack™: access to all of your study materials online and on your phone.
STUDYBLUE exists to make studying efficient and effective for every student, for free. Join us.
“I have been getting MUCH better grades on all my tests for school. Flash cards, notes, and quizzes are great on here. Thanks!”
Kathy
Kathy