Name: ___________________________________________Date: ___
Real Numbers
These are all REAL Numbers… Some are Rational and some are Irrational.
 The set of Whole numbers is the same as the set of Natural numbers, except that it includes
0. To help remember this, think “o” is in “whole.”
 The set of Integers is the same as the set of whole numbers and the negatives of the whole numbers.
 We can think of Rational numbers as fractions. To remind us, notice that the word “ratio” is embedded in the word “rational.” A ratio is a fraction.
 The set of Rational numbers includes all decimals that have either a finite number of decimal places or that repeat in the same pattern of digits. For example, 0.333333… = 1/3 and .245245245…. = 245/999.
 The set of Natural numbers is a subset of the set of Whole numbers, which is contained in the set of Integers, which is inside of the set of Rational numbers
Real Number Member of these sets of Real Numbers
0 Whole, Integer, Rational (can be written as ), Real
4 Natural, Whole, Integer, Rational (can be written as
Real
-9 Integer, Rational, Real
),
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Rational, Real
Natural ( ), Whole, Integer, Rational, Real
Integer ( ), Rational, Real
Irrational (
≈ 3.31662479036… This is not a terminating
decimal and it does not repeat), Real
Whole ( ), Integer, Rational, Real
Rational (
Irrational, Real
= ) , Real
Rational, Real
0.23 Rational (terminating decimal equal to ), Real
Irrational (It is a fraction, but not a quotient of two integers), Real http://www.jamesbrennan.org/algebra/numbers/real_number_system.htm
Fractions, Decimals, and Percents Conversions
F D To convert a fraction to a decimal, divide the numerator by the
denominator.
D P To convert a decimal to a percent, move the decimal two places to the
right. This is the same as multiplying by 100. Don’t forget to place the
percent (%) sign at the end of the number!
F P To convert a fraction to a percent, it must first be converted to a decimal. Then move the decimal point two places to the right. This is the
same as multiplying by 100. Don’t forget to place the percent sign (%) at
the end of the number!
D F To convert a decimal to a fraction, write the decimal number as a fraction just as you would pronounce it. Once it is in fraction form,
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reduce the fraction to its simplest form.
P D To convert a percent to a decimal, move the decimal point two places to the left. This is the same as dividng by 100. Don’t forget to leave off
the percent sign!
P F To convert a percent to a fraction, change the percent to a decimal using the procedure listed above. Then follow the procedure for
converting decimals to fractions. Be sure to reduce the fractions to their
lowest terms.


.



.

TO CONVERT A NUMBER INTO SCIENTIFIC NOTATION:
Create a number between 1 and 10 by moving the decimal to the left.
Count the number of spaces the decimal moved to determine the exponent of 1o.
EXAMPLE: 3,346,000,000. = 3.346 x 10 9
The decimal moved 9 places # between 1 & 10 power of ten. Add zeros as needed for place holders.
TO CONVERT A NUMBER INTO STANDARD FORM:
Move the decimal (right or left)* the number of spaces indicated by the exponent.
If the power of 10 is positive, then move the decimal point to the right.
If the power of 10 is negative, then move the decimal point to the left.
EXAMPLE: 1.312 x 10 6 = 1,312,000
The decimal moved 6 places to the RIGHT. Add zeros as needed for place holders.
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√
78 is between ___8___ and ___9____.
Now, approximate the answer to the nearest hundredth.
How many spaces are there between 64 and 81? 17 (Since 78 is in the
14th spot, you would divide 14/17 and get 0.82, rounded to the nearest hundredth)
√
78 ≈
8.82
http://www.khanacademy.org/math/arithmetic/exponents-radicals/radicalradicals/v/approximating-square-roots?topic=exponents-radicals http://www.mathsisfun.com/square-root.html
All numbers below must be real ~ RULES FOR EXPONENTS
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Name: ___________________________________________Date: ___ http://regentsprep.org/Regents/math/ALGEBRA/AO5/rule_pmu.htm
http://regentsprep.org/Regents/math/ALGEBRA/AO5/rule_pow.htm
http://regentsprep.org/Regents/math/ALGEBRA/AO5/rule_prd.htm
PROPERTIES OF REAL NUMBERS
The following table lists the defining properties of the real numbers
(technically called the field axioms).
Addition
Commutative
For all real a , b a + b = b + a
Associative
For all real a , b , c a + ( b + c ) = ( a + b ) + c
Identity
There exists a real number 0 such that for every real a a + 0 = a
Additive Inverse
(Opposite)
For every real number a there exist a real number, denoted
(
 a ), such that a + (– a ) = 0
Multiplication
Commutative
For all real a , b ab = ba
Associative
For all real a , b , c
( ab ) c = a ( bc )
Identity
There exists a real number 1 such that for every real a a
× 1 = a
Multiplicative Inverse
(Reciprocal)
For every real number a except 0 there exist a real number, denoted , such that a × = 1
Distributive Law
For all real a , b , c a ( b + c ) = ab + ac , and ( a + b ) c = ac + bc
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The commutative and associative laws do not hold for subtraction or division: a – b is not equal to b – a a ÷ b is not equal to b ÷ a a – (b – c) is not equal to (a – b) – c a ÷ ( b ÷ c) is not equal to (a ÷ b ) ÷ c
What these laws mean is that order and grouping don't matter for addition and multiplication, but they certainly do matter for subtraction and division. In this way, addition and multiplication are “cleaner” than subtraction and division. This will become important when we start talking about algebraic expressions. Often what we will want to do with an algebraic expression will involve rearranging it somehow. If the operations are all addition and multiplication, we don't have to worry so much that we might be changing the value of an expression by rearranging its terms or factors. Fortunately, we can always think of subtraction as an addition problem (adding the opposite), and we can always think of division as a multiplication (multiplying by the reciprocal).
You may have noticed that the commutative and associative laws read exactly the same way for addition and multiplication, as if there was no difference between them other than notation. The law that makes them behave differently is the distributive law, because multiplication distributes over addition, not vice-versa.. The distributive law is extremely important, and it is impossible to understand algebra without being thoroughly familiar with this law.
Example: 2(3 + 4)
According to the order of operations rules, we should evaluate this expression by first doing the addition inside the parentheses, giving us
2(3 + 4) = 2(7) = 14
But we can also look at this problem with the distributive law, and of course still get the same answer. The distributive law says that
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