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Week 1

Given problems with signed real numbers, perform basic arithmetic operations, using a hand help calculator, including conversions of

percentages, decimals, fractions, evaluations of mathematics expressions containing exponents

and roots, and processing complex orders of operation.

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Objectives Real Number System Sets and Venn diagrams Signed Numbers Powers and Roots Order of Operations

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Real Number System

Arithmetic uses only constants like -13, 0, 2/3 which have fixed value

Algebra uses not only constants, but also variables like a,b,x,y, which represent different numbers

The constants and variables we use in Algebra are called the Real numbers.

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Real Number System The origins of number systems date back to the

Egyptians, Babylonians, and Chinese. However, these earliest systems were much simpler than the real number system. For example, the number 0 was not widely accepted before the 13th century and the use of negative numbers (-1, -2, -3…) was not generally accepted before the 17th century.

In this first week, you will learn about natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers. If you would like to learn more about the historical development of number systems you can visit:

http://mathforum.org/alejandre/numerals.html

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The Natural Numbers Natural numbers or counting numbers

are the most fundamental set of numbers

{1,2,3,…} Braces, { }, are used to indicate a set of

numbers The … after 1,2, and 3, which are read

“and so on” mean that the pattern continues without end – there are infinitely many natural numbers

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The Whole Numbers The Natural number together with

Zero {0,1,2,3,…} Not adequate for indicating losses

or debts. Example: Bank statement Previous balance: $ 50 Checks paid: $ 70 New balance: -$ 20

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The Integers

The Whole numbers together with the negatives of the counting numbers form the set of integers

{…-3,-2,-1,0,1,2,3,…}

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The Rational Numbers

Any number that can be express as a ratio (or quotient) of two integers

{a/b, where a and b are integers, with b≠0}

Negative and positive fractions Example: 3/1; 5/4; -77/3

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The Irrational Numbers Real number that is not rational Can not be written as a ratio of two

integers Example: √2, √3,π

In Computation with irrational numbers we use rational approximation for them Example: √2≈1.4141; π≈3.14 : the ratio of the circumference and

diameter of every circle Note: not all square roots are irrational

Example: √9 = 3

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The Number Line Every real number corresponds to one and

only one point on the number line. Every point on the number line corresponds

to one and only one real number

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Set notation N = {…-3,-2,-1,0,1,2,3,…} This is infinite set A = {0,1,2,3} is finite set 0 is a member of the set; so 0 ϵ A Equal sets contain the same numbers B = {0,1,2,3}; hence A=B subsets: set C = {1,2,3} is a subset of

set A and set B Set D = {1,2,3,4} is not a subset of A or B Is C subset of D?

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Interval of Real numbers Interval of real numbers is the set of real numbers that is

between two real numbers Interval notation is representing intervals (4,6): all real numbers between 4 and 6, but not 4 and 6 [4,6]: all real numbers between 4 and 6, including 4 and 6 (-3, ∞): all real numbers between -3 and infinity, but not -3

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Applications of SetsVenn Diagrams

Venn Diagrams: useful way of visualizing the possible relationship that can exist between various sets.

The set is represented by A and the universal set is represented by U.

A

U

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Venn Diagrams

The set is represented by green color A’ is the complement of set A: includes all elements that does not belong to A.

A’

A

U

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Union of Sets

The union of sets A and B, written as A U B is the set that contains all the elements of A as well as all the elements of B.

A U B is represented by the shaded area. In terms of set notation, suppose A = {1, 2, 3} and that B = {d, f, g}. Then A U B = {d, f, g, 1, 2, 3}

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Union of Sets

BA A U B A={0,1, 2, 3} B={d, f, g} A U B={d, f, g, 0, 1, 2, 3}

U

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Union of Sets

BA A U B A={0,1, 2, 3} B={d, f, g,1,2,3} A U B={d, f, g,0,1, 2, 3}

U

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Intersection of Sets

BA

A ∩ B A={0,1, 2, 3} B={d, f, g,1,2,3} Members of A are members of B: 1,

2, 3 A ∩ B = {1, 2, 3}

x

U

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Counting formula

BA

Strictly speaking A U B=A + B – A ∩ B

A={0,1, 2, 3} B={d, f, g,1,2,3} A U B={d, f, g,0,1, 2, 3} A ∩ B={1, 2, 3} U

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Example Venn Diagrams A total of 1000 students at Concorde Career College were surveyed to determine their course-scheduling preferences.

The survey results were as follows: 620 students liked morning classes 320 students liked afternoon classes 230 students liked both morning and afternoon classes Of those students surveyed, how many: Like morning classes? Like afternoon classes Like morning classes but do not like afternoon classes? Like afternoon classes but do not like morning classes? Like neither morning nor afternoon classes?

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Using numbers to show opposites

Can you give me an examples of opposite situations in life?

In mathematics, signed numbers are used to represent quantities that are opposites. Examples:

On a temperature scale, the opposite of 10 above zero is 10 below zero;

A profit of $100 is the opposite of losses of $100 etc.

On the real number line, +4 (read “positive 4”) is the opposite of – 4 (read “negative 4”). Both +4 and – 4 are the same distance from zero.

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Using numbers to show opposites We can say that for every positive number

there is a negative number, which is the same distance from zero on the opposite side of the zero. Opposites have the following properties

For any real number a, ( a) = a Example: -1 is the opposite of 1 and vice-

versa The sum of a real number and its

opposite is zero Example: -1 + 1 = 0

Graphical representation

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Absolute Value The absolute value of a number is the distance

between 0 and that number on a number line. Another way to put it is that the absolute value of a number is its numerical value, regardless of its sign.

In any pair of opposites, the positive number in the pair is the absolute value of each of the numbers. There is a symbol used to express the absolute value of the number. For any number a,a is how to write an absolute value of a. The absolute value of a (written as a) can be found from the rule:

If a 0, then, |a|=a If a 0, then, |a|= a |5 |= 5 |-5|= -(-5)=5

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Mathematical Operations The four basic operations used in arithmetic as

well as in algebra are: + addition, usually written as a + b. It means

the sum of a and b; a plus b; a added to b. - subtraction, usually written as a - b. It means

the difference of a and b; a minus b; b less than a;

* multiplication, usually written as a * b or ab. It means the product of a and b; a times b.

/ or , division, usually written as a/b or a:b. It means the quotient of a and b; a divided by b; a into b.

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Fractions

The terms of a fraction are numerator and denominator. Thus, in the fraction a/b (which we read as “a divided by b”), a is the numerator and b is the denominator.

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Type of Fractions

Proper fraction a/b: when a<b Example: 1/2, 3/4, 5/7, etc.

Improper fraction a/b: when a>b Example: 5/3, 6/5 etc.

Mixed number c a/b: where c is a whole number and a/b is proper fraction Example: 3 1/2, 5 2/3

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Equivalent fractions

Equivalent fractions are fractions that have the same value, but not the same numerator and denominator. There are two ways to find an equivalent fraction.

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Equivalent fractions Multiply the numerator and the

denominator by the same number. That number cannot be zero. Why? Example: 3/4=3*2/4*2=6/8;

3/4=3*3/4*3=9/12. The fractions: 3/4, 6/8, 9/12 are all equivalent. Converting a fractions to an equivalent fractions with a larger denominator is called building up the fraction.

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Equivalent fractions Divide the numerator and the denominator

by the same number- common factor. That number cannot be zero. Why? Example: 6/8=6÷2/8÷2=3/4. In this case

2 is the common factor of 6 and 8. We have to continue dividing until the numerator and denominator have no common factors greater then 1.

3/4 is in lowest terms or written in simplest form.

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Greatest Common Factor To find the simplest form of a

fraction is to look for the greatest common factor (GCF) of the numerator and denominator. Example: Let’s find the simplest form

of 6/24. The factors of 6 are: 1,2,3,6; the factors of 24 are: 1,2,3,4,6,8,12,24. The GCF of 6 and 24 is 6. So, 6÷6=24÷ 6=1/4; So ¼ is the simplest form of 6/24.

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Addition and subtraction of fractions with same denominator

When the fractions have same denominators Example: 2/5+1/5=3/5; Add the

numerators and keep the same denominator

Example: 2/5-1/5=1/5; Subtract the numerators and keep the same denominator

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Addition of fractions with different denominator 5/9 + 3/2=? Find least common denominator: LCD

Make a list of all multiples of both denominators: For 9: 9,18; For 2: 2,4,6,8,10,12,14,16,18 LCD is the lowest number that both of these

denominators would be able to divide into evenly ; For 9 and 2 this number is 18; so the common denominator of the resulted fraction should be 18.

Multiply the numerator and denominator of the 5/9 with 2 and numerator and denominator of 3/2 with 9:

5*2/9*2 + 3*9/2*9 = 10/18+27/18= 37/18; this is improper fraction and should be converted to mixed number ( explanation of how to do this later in the lecture).

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Subtraction of fractions with different denominator

1/3-1/12=? Find least common denominator: LCD For 3: 3,6,9,12; For 12: 12,

LCD is the lowest number that both of these denominators would be able to divide into evenly ; For 3 and 12 this number is 12; so the common denominator of the resulted fraction should be 12.

Multiply the numerator and denominator of the 1/3 with 4 and numerator and there is no need to multiply second fraction because it already has denominator is 12.

1*4/3*4 – 1/12= 4/12 – 1/12 = 3/12 Reduce to lowest term:1/4 is in its simplest form

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Multiplication of fractions

Multiplication of fraction is the easiest of all fraction operations. All you have to do is to multiply straight across- multiply the numerators and the denominators. Example: 4/5 * 2/3 = 4*2/5*3=8/15

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Division of fractions

Let’s consider following example: 2/5÷1/3=? Division is a opposite of

multiplication; so we change ÷ to * sign and 1/3 with its reciprocal 3/1;

So 2/5÷1/3=2/5*3/1 = 6/5

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How to convert mixed number into improper fraction 3 1/2=?

Multiply denominator 2 by the whole number 3, so 2*3=6;

Add the numerator 1 to the product of the whole number and the denominator-this will be your new numerator: 1+6=7

Keep the same denominator; So, the new improper fraction is 7/2

3 1/2=7/2

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How to convert improper fraction into mixed number Convert 45/4 to a mixed number. First, do the long division to find

the "regular" number part:11 and the remainder:1

Since the remainder is 1 and you're dividing by 4, the fraction part will be 1/4.

so 45/4 = 11 1/4

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How to multiply/divide mixed numbers (4 1/2) * (3 1/3)=? Convert mixed numbers to

improper fractions 4 1/2=9/2 3 1/3=10/3 Perform multiplication/division of the

two improper fractions following the rules

9/2*10/3=90/6=15

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How to subtract/add mixed numbers

16 1/8 + 5 5/8 = 21 6/8 = 21 3/4 129/8 + 45/8 = 174/8 = 21 3/4 8 5/8 – 3 1/12

LCD = 24 8 15/24 – 3 2/24 = 5 13/24 207/24 – 74/24 = 133/24

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Carrying when adding mixed numbers

9 5/8 + 13 7/8 = 22 12/8 12/8 can be written in its lowest

terms as 3/2 = 1 ½ So, 22 12/8 = 22 3/2 = 22 + 3/2 22 + 1 ½ = 23 1/2

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Borrowing when subtracting mixed numbers 8 1/3 – 4 3/5

LCD = 15 8 5/15 – 4 9/15 You can not subtract 9/15 from 5/15 We borrow from the whole number 8 8 5/15 = 7+1+5/15 = 7+15/15+5/15 7 + 20/15 = 7 20/15 7 20/15 - 4 9/15= 3 11/15

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Examples

7 – 2 5/6

14 6/7+15 ½

19 2/3 – 11 ¾

12 8/15 + 18 3/5

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From Fractions to Decimals Fraction with a denominator of 10,

100, 1000 etc. can be written as decimals numbers: Examples: 3/10=0.3; 25/100=0.25;

5/1000=0.005 Fractions with a denominator of

100 are often written as percents Examples: 25/100=25%; 3/100=3%;

300/100=300%

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From Decimals to Fractions Count the positions to the right of decimals, i.e., tenth’s,

hundredth’s, thousandth’s, ten thousandth’s, etc. Whatever that number ends up being, that number

becomes your denominator. The number in the numerator now loses its decimal, any

zeros that remain to the left of any natural number are eliminated.

Example: 0.56=? Fraction; the position to the right of decimal is hundredth’s; hence the denominator of the fraction will be 100

So, 0.56 = 56/100 Example: 1.56=?; the denominator is 100, the

numerator is 56 and the whole number is 1; the whole number of original decimal now is the whole number of the mixed number

Hence: 1.56 = 1 56/100=1 14/25

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From Decimals to Percents

Fractions Decimals Percents

Example: 5.45=?% Move decimal point two places to the

right 5.45=545%

Move decimal point two placesFrom left to right

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From Percents to Decimals Fractions Decimals Percents

Examples: 3.67% = 0.0367 The process converting decimals to percents and

vice-versa calls for a two space movement in either direction. When the spaces are not there, add Zeros (0’s), as many as necessary, to make that 2 position movement and proceed.

Move the decimal point two placesFrom right to left

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Ratio Previously, we defined a rational number as

the ratio of two integers. More general definition of a ratio: If a and b are any two real numbers, with

b≠0, then the expression a/b is called the ratio of a and b, or the ratio of a to b. Example: During soccer game, 240 tickets were

sold. There were 500 tickets available. Find the ratio of tickets sold to the total number of tickets.

240/500 = 12/25; after reducing the initial fraction

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Proportion A proportion is any statement of equality

of two ratios. The statement a/b=c/d , (where b, d ≠0) is a proportion;

When two pairs of numbers have the same ratio we say the they are proportional Example: 4/8=1/2

Cross-multiplying: the cross products of a proportion are equal

If a/b=c/d then a*d=b*c Example: 4/8=1/2 , so 4*2=8*1

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Proportion

Cross products can be used to find missing numbers in a proportion

2/3 = n/12 2*12 = 3*n 24 = 3n n = 8

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Operations with signed numbers-Addition Adding numbers with like signs: add their

absolute values; the sum has the same sign as the given numbers Example: 23+56=79; (-23)+(-56)=-79

Adding numbers with unlike signs: subtract the absolute value of the smaller number from the absolute value of the larger number; the sum has the sign of the larger number Example: -23+56=33; -7 + 5 = -2

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Operations with signed numbers-Subtraction

For any real numbers a and b: a – b = a + (-b) To do any subtraction, we can

change to the addition of the opposite Example: -5 -3 = -5 +(-3)= -8 Example: -5 - (-3) = -5+ (3) = -2 Example: 5 - (-3) = 5+ (3) = 8

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Operations with signed numbers-Multiplication

To find the product of two real numbers, multiply their absolute values The product is positive if the numbers

have like signs, i.e. -*-=+; +*+=+ Example: (-2)*(-3)=2*3=6 The product is negative if the numbers

have unlike signs, i.e.-*+=- Example: (-2)*(3)=2*3=- 6

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Operations with signed numbers-Division

To find the quotient of two nonzero real numbers, divide their absolute values The quotient is positive if the numbers

have like signs, i.e. -÷-=+; +÷+=+ Example: (-8)÷(- 4)=8÷4=2 The quotient is negative if the numbers

have unlike signs, i.e.-÷+=- Example: (-8)÷(4)=8÷4=- 2

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Exponents

For any counting number a, an =a*a*a*a…*a

a is called the base; n is exponent Example: 24=2*2*2*2=16 Common mistake: 24≠2*4=8

n factors

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Exponents Example: - 42=-(4)2 = -(4)*(4)=-16 The base is 4 and than form the

opposite of the product Example: (- 4)2 = (- 4)*(- 4) = 16 Example: - (- 4)2 = -(-4)*(- 4) = -16 Example: (-1/5) 3 =(-1) 3 /(5) 3 =-1/125

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Odd and even powers Another aspect of exponents is the determination

of the sign of the number we develop from a number raised to a particular power. Example: (-1)2 states that (-1) * (-1) which

results in a positive The questions becomes then what sign results if

we have (-1)175 ? The rule states When a negative number is raised to an odd

power, the resulting answer will be negative and when a negative number is raised to an even power, the answer will be positive. Example: (-1)175 = -1 Example: (-1)174 = 1

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Square RootsFinding the square root of a number is

the inverse operation of squaring that number. So, 52 = 25 and (-5)2 =25

The principle root of √25 = 5√11+25 = √36 = 6It is not as: √11+√25 = √11+5

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Arithmetic expression The result of writing numbers in a

meaningful combination with the ordinary operations is called arithmetic expression or simply an expression. Example: (3+2)*5

The parentheses are used as grouping symbols and indicate which operation to perform first. Because of the parentheses, the expressions below have different values: Example: (3+2)*5=25 Example: 3+(2*5)=13

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Order of Operations Often there will be more than one

operation or set of grouping symbols in a problem.

TIP: Your may find it helpful in remembering the order of operations to memorize the expression: P. E. M. D. A. S, or Please Excuse My Dear Aunt Sally, which stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.

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Order of Operations 1. Evaluate each exponential expression in

order from left to right 2. Perform multiplication and division in order

from left to right 3. Perform addition and subtraction in order

from left to right Note: Multiplication and Division have equal

priority in order of operations. If both appear in an expression, they are performed in order from left to right.

The same holds for Addition and Subtraction.

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Order of Operations Example: 8÷4*3=2*3=6 Example: 9-3+5=6+5=11 Example: 2*(6-1) - 3*4 +42

Evaluate the parentheses first (6-1)=5 2*5 – 3*4 + 42

do the exponent second 42=16 2*5 – 3*4 + 16 Multiplication from left to right: 10 – 12+ 16 Addition and subtraction from left to right: -2 + 16 = 14

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Examples:

Consider the following example. Simplify:

4 + 6 2 + 3 * 5 – 4 Consider the following example.

Simplify: 6 + 7 + 8 2(3 + 2) * 32 - 6 + 3

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Algebraic expression

Evaluate the following expressions for x=3, y=-1, z=5 x-y+z; we substitute x, y, and z with

their values 3-(-1)+5=3+1+5=9 x*(y+z); substitute x, y, and z 3*(-1+5)=3*4=12

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Distributive Property For any real numbers a, b, and c, a*(b + c)=a*b + a*c

Example: 4*(5 +2) = 4*5 + 4*2 Example: 4(x+2) = 4x+4*2

a*(b - c) = a*b - a*c Example: 4*(5 -2) = 4*5 - 4*2 Example: 4(x-2) = 4x-4*2

Remember: to distribute the negative sign across the parentheses.

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Commutative Property

For any real numbers a and b a + b = b + a

Example: 5 + 4= 4 + 5 a*b= b*a

Example: 5*4=4*5 Valid only for addition and

multiplication

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Associative property

For any real numbers a, b, and c (a + b) + c = a +(b + c)

Example: (3+8) + 5 = 3 + (8+5) (a*b)*c=a*(b*c)

Example: (6*7)*8=6*(7*8) Valid only for addition and

multiplication

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Identity Property

The numbers 0 and 1 have special properties

For any real number a, a*1=1*a=a: Multiplication Identity a+0=0+a=a: Additive Identity

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Inverse Property

For any real number a, there is a number –a, such that:

a + (-a) = 0 For any nonzero real number a

there is a number 1/a such that: a*1/a=1