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NATURAL NUMBERS
1, 2, 3, 4, 5, 15/3, √16

RATIONAL NUMBERS
RATIOS OF INTEGERS 1/2, -3/7, 46, .17, .6, .317
IRRATIONAL NUMBERS
REAL NUMBERS THAT CANNOT BE EXPRESSED BY RATIOS real numbers that cannot be represented as terminating or repeating decimals.

√3, √5, 3√2, PI, 3/PI SQAURED
PROPERTIES OF REAL NUMBERS:
COMMUTATIVE PROPERTIES
WHEN WE ADD OR MULTIPLY TWO NUMBERS, THE ORDER DOESN'T MATTER
PROPERTY                   EXAMPLE
a + b = b + a            7 + 3 =  3 + 7
ab = ba                    3 * 5   =  5 * 3

PROPERTIES OF REAL NUMBERS:
ASSOCIATIVE PROPERTIES
WHEN YOU ADD OR MULTIPLY THREE NUMBERS IT DOESN'T MATTER WHICH TWO YOU ADD OR MULTIPLY FIRST (A + B) + C = A + (B + C)
(AB)C = A(BC)

 

LIST THE PROPERTIES OF REAL NUMBERS:

  COMMUTATIVE 
ASSOCIATIVE
DISTRIBUTIVE PROPERTY
PROPERTIES OF REAL NUMBERS:
DISTRIBUTIVE
WHEN WE MULTIPLY A NUMBER BY A SUM OF TWO NUMBERS, WE GET THE SAME RESULT AS WE WOULD IF WE MULTIPLY THE NUMBER BY EACH OF THE TERMS AND THEN ADD THE RESULT
A(B + C) = AB + AC
(B+C)A = AB + AC
PROPERTIES OF NEGATIVES



1. (-1)A= -A
2. -(-A)= A
3. (-A)B= A(-B) = -(AB)
4. (-A)(-B) = AB
5. -(A+B) = -A - B 
6. -(A-B) = B -A

PROPERTIES OF FRACTIONS
WHEN MULTIPLYING FRACTIONS
MULTIPLY NUMERATORS AND DENOMINATORS
1. A/B * C/D =  AC / BD



PROPERTIES OF FRACTIONS
WHEN DIVIDING FRACTIONS
INVERT THE DIVISOR AND MULTIPLY
2. A/B  ÷  C/D   =  A/B  ×  D/C
PROPERTIES OF FRACTIONS
WHEN ADDING FRACTIONS WITH THE SAME DENOMINATOR
ADD THE NUMERATORS
3. 2/5 + 7/5 = 9/5
PROPERTIES OF FRACTIONS
WHEN ADDING FRACTIONS WITH DIFFERENT DENOMINATORS
 FIND A COMMON DENOMINATOR AND THEN ADD THE NUMERATORS
4. A/B + C/D = AD + BC  /   BD
PROPERTIES OF FRACTIONS
WHEN CAN YOU CANCEL NUMBERS?
WHEN THEY ARE COMMON FACTORS IN NUMBERATOR AND DENOMINATOR
AC / BC  = A/B
PROPERTIES OF FRACTIONS
IF A/B = C/D    THEN
AD =  BC   CROSS MULTIPLY
LEAST COMMON DENOMINATOR
It is the smallest positive integer that is a multiple of the denominators.
DEFINE A SET
A COLLECTION OF OBJECTS, AND THESE OBJECTS ARE CALLED THE ELEMENTS OF THE SET.
GIVE AN EXAMPLE OF SET BUILDER NOTATION
A=  {X| X IS AN INTEGER AND O < X , 7}
WHAT IS A UNION?
THE SET THAT CONSIST OF ALL ELEMENTS THAT ARE IS 'S' OR 'T' (OR BOTH)
WHAT IS AN INTERSECTION?
UPSIDE 'U'
IS THE SET THAT CONSISTS OF ALL ELEMENTS ARE ARE IN BOTH 'S' AND 'T'
WHAT IS AN OPEN INTERVAL?
CONSISTS OF ALL NUMBERS BETWEEN TWO POINTS. (A,B)
WHAT IS A CLOSED INTERVAL?
CONSISTS OF ALL NUMBERS BETWEEN AND INCLUDING THE END POINTS [A,B]
ABSOLUTE VALUE
|A| 
THE DISTANCE FROM A TO 0 ON THE REAL NUMBER LINE. DISTANCE IS ALWAYS POSITIVE OR ZERO
A to the nth power of 0=
1
A to the power of -n =
1/ a to the power of n
negative exponent means  
TO DIVIDE
x1 =
X  
61 = 6
x0 =

70 = 1
x-1 =
1/x
4-1 = 1/4
LAWS OF EXPONENTS
1. xmxn =
xm+n  x2x3 = x2+3 = x5
TO MULTIPLY TWO POWERS OF THE SAME NUMBER, ADD THE EXPONENTS
LAWS OF EXPONENTS
2. xm/xn =
 xm-n
x6/x2 = x6-2 = x4   TO DIVIDE TWO POWERS OF THE SAME NUMBER, SUBTRACT THE EXPONENTS

LAWS OF EXPONENTS   3.  (xm)n =
xmn (x2)3 = x2×3 = x6

TO RAISE A POWER TO A NEW POWER, RAISE EACH FACTOR TO THE POWER
LAWS OF EXPONENTS
4. (xy)n =
xnyn
(xy)3 = x3y3
TO RAISE A PRODUCT TO A POWER, RAISE EACH FACTOR TO THE POWER
5. (x/y)n =
xn/yn  
(x/y)2 = x2 / y2
x-n =
 1/xn
x-3 = 1/x3

LAWS OF EXPONENTS
6. (A/B)-n = 


(B/A)
(3/4)-2 =  (4/3)2
LAWS OF EXPONENTS
7. A-n / b-m    =   
3-2 / 4-5    =

A-n / b-m    =   bm / an
3-2 / 4-5    =     45 / 3-2  

A positive number x is said to be written in scientific notation if it is express as follows:
x= ax10n
where 1 ≤ a < 10 and n is an integer
4 x 1013 =
= 40,000,000,000,000  
move decimal from the '4' 13 places to the right
1.66 x 10-24= 
0.00000000000000000000000166
move decimal place 24 places to the left
the symbol √ means
the positive square root of 
thus √a = b    means  b2 = a   and b ≥ 0
√a = b    means
 b= a 
√9 = 3   because   32 = 9
definition of nth root if n is any positive integer, then the principal nth root of a is defined as follows:
n√a = b      means
 
bn = a   If n is even, we must have a ≥ 0 and b ≥ 0 

PROPERTIES OF nth Roots
law 2 n√a/b = 
PROPERTIES OF nth Roots
law 3 mn√a =  
√729 =
mn√a
6√729 = 3
PROPERTIES OF nth Roots
law 4.  n √an  =a   if n is odd
PROPERTIES OF nth Roots
law 1.n√ ab
PROPERTIES OF nth Roots
law 5. n√an =
|a| if is even
by the definition of nth root
a1/2 = 
n√a
definition of rational exponents
for any exponent m/n in lowest terms, where mand  are integers and  n> 0,  we definie
am/n =
am/n =(n√a)m or equivalently    am/n n√am
if n is even, then we require that a ≥ 0


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