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REAL NUMBER SYSTEM
I. Sets of Real Numbers
A. Real numbers






Natural numbers N={1,2,3,…}
Whole numbers W={0,1,2,3,…}
Integers I={…,-2,-1,0,1,2,…}
Rational numbers Q={a/b | a,b integers}; fractions, terminating and
repeating decimals
Irrational numbers H={x | x is not rational}; Nonterminating,
nonrepeating decimals
Real numbers
B. Comparisons



Greater than
Less than
Equal to
C. Absolute value


Geometric definition: A measure of distance on the real number line.
a, when a  0
a 
 a, when a  0
Algebraic definition:
II. Operations with Real Numbers
Signed numbers
Addition
 Same signs: Add the absolute values and keep the sign.
 Opposite signs: Subtract the absolute values and keep the sign of the
larger.
Subtraction
 Add the opposite.
Multiplication
 Same signs: Product is positive.
 Opposite signs: Product is negative.
Division
 Same signs: Quotient is positive.
 Opposite signs: Quotient is negative.
Fractions
Addition
 If we have the same denominator, we simply add the numerators.
 If we have different denominators, we must find the LCD. Convert each
fraction to an equivalent fraction with the LCD.
 Add the numerators once common LCD has been established.
Subtraction
 Same process as addition except subtract the numerators.
Multiplication
a c ac
 

b d bd
Division

Invert the second fraction and multiply:

Division by zero is not defined.
a c a d ad
   
b d b c bc
III. Exponents
A. Definitions
1. Exponents are shorthand notation for repeated multiplication
a n  a 
a
 a
 
a.
n factors
2. Zero exponent b 0  1
3. Negative exponent b  n 
1
bn
To see why the zero exponent and negative exponents are defined this
way consider the pattern in the example below. In the left column the
exponents decrease by 1 and in the right column the numbers increase by
a factor of 10.
10 3  1000
10 2  100
101  10
10 0  1
1
1
 1
10 10
1
1

 2
100 10
1
1

 3
1000 10
10 1 
10  2
10 3
B. Rules of Exponents
1. Product rule a ma n  a m n
am
2. Quotient rule n  a mn
a
3. Power rule a m   a mn
n
4. Product to a power ( ab) m  a mb m
m
am
 a
5. Quotient to a power    m
 b
b
IV. Order of Operations
A. PEMDAS
Highest priority to lowest priority. Work from left to right.
 Parentheses
 Exponents
 Multiplication/Division
 Addition/Subtraction
B. Remove parentheses
Use the distributive law a(b  c)  ab  ac
C. Combining like terms
Like terms have the same variables with the same exponents.
Combine like terms by adding or subtracting their numerical coefficients.
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