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Kay H.

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...}

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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

4 8 9 7 1

5 6 3 2 0

1 2 3 4 9

+ 6 9 8 9 9

1 6 5 3 2 9

8 7 5 3

Scratch Method for Addition

This method involves doing a series of additon problems that involve any two single digits

Commutative Property of Addition

a+b=b+a

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Associative Property of Addition

(a+b)+c=a+(c+b)

Additive Identity

a+0=0+a

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

f(x)

F of X Leonard Euler used functions notation in 1734

Domain

the set of all X values specified for the functions

Range

the set of all Y values that result from the function

X

Independent variable

Y

dependent variable

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

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