Real Numbers

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Real Numbers

• Rational and Irrational

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Lets look at the relationships between number
sets. Notice rational and irrational numbers
make up the larger number set known as Real
Numbers
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A number represents the value or quantity of
something Like how much money you have.. Or how
many marbles you have Or how tall you are.As
you may remember from earlier grades there are
different types of numbers.
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Rational Numbers

• Fractions are rational numbers in disguise

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Rational number the set of all numbers that can
be written in the form of a/b, where a and b are
integers and b ? 0.

• Because a fraction can be written as a decimal,
  any rational number a/b can be written in decimal
  form.
• When you divide the denominator of a fraction
  into the numerator, the result is either a
  terminating or repeating decimal.

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A terminating decimal has a finite number of
nonzero digits to the right of the decimal point,
such as 0.25 or 0.10.

• A repeating decimal has a string of one or more
  digits that repeat infinitely, such as 1.555555,
  or 0.345634563456 These repeating decimals can
  be indicated by a bar over the repeating digits,
  1.5 or 0.3456

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Write 0.375 as a fraction in simplest
form.0.375 375/1000 3/8, so 0.375
3/8Write 1/3 as a decimal.Divide 1 by 3 and
you will see how the process will repeat
infinitely.0.333333333333
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A number line - is an infinitely long line whose
points match up with the real number system.
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Here are the rational numbers represented on a
number line.
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Integers

• The coldest temperature on record in the U.S. is
  -80 F, recorded in 1971 in Alaska

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Integers are used to represent real-world
quantities such as temperatures, miles per hour,
making withdrawals from your bank account, and
other quantities. When you know how to perform
operations with integers, you can solve equations
and problems involving integers.
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By using integers, you can express elevations
above, below, and at sea level. Sea level has an
elevation of 0 feet. Badwater Basin in Utah is
-282 below sea level, and Clingmans Dome in the
Great Smokey Mountains is 6,643 above sea level.
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If you remember, the whole numbers are the
counting numbers and zero0, 1, 2, 3,Integers
- the set of all whole numbers and their
opposites. This means all the positive integers
and all the negative integers together.
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Opposites two numbers that are equal distance
from zero on a number line also called additive
inverse.The additive inverse property states
that if you add two opposites together their sum
is 0-3 3 0
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Integers increase in value as you move to the
right along a number line. They decrease in
value as you move to the left. Remember to order
numbers we use the symbol lt means less than,
and the symbol gt means is greater than.
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A numbers absolute value - is its distance from
0 on a number line. Since distance can never be
negative, absolute values are always positive.
The symbol represents the absolute value of a
number. This symbol is read as the absolute
value of. For example -3 3.
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Finding absolute value using a number line is
very simple. You just need to know the distance
the number is from zero. 5 5, -6 6
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Lesson Quiz

• Compare, Use lt, gt, or .
• 1) -32 ? 32
• 2) 26 ? -26
• 3) -8 ? -12
• 4) Graph the numbers -2, 3, -4, 5. and -1 on a
  number line. Then list the numbers in order from
  least to greatest.

• 5) The coldest temperature ever recorded east of
  the Mississippi is fifty-four degrees below zero
  in Danbury, Wisconsin, on January 24, 1922.
  Write the temperature as an integer.

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Integer Operations

• Rules for Integer Operations

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Adding IntegersWhen we add numbers with the same
signs,1) add the absolute values, and2) write
the sum (the answer) with the sign of the
numbers.When you add numbers with different
signs,1) subtract the absolute values, and2)
write the difference (the answer) with the sign
of the number having the larger absolute value.
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Try the following problems

• 1) -9 (-7) -16
• 2) -20 15 -5
• 3) (3) (5) 8
• 4) -9 6 -3
• 5) (-21) 21 0
• 6) (-23) (-7) -30

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Subtracting IntegersYou subtract integers by
adding its opposite.9 (-3)9 (3) 12-7
(-5)-7 (5) -2
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Try the following problems

• 1) -5 4
• -5 (-4) -9
• 2) 3 (5)
• 3 (-5) -2
• 3) -25 (25)
• -25 (-25) -50
• 4) 9 3
• 9 (-3) 6
• 5) -10 (-15)
• -10 (15) 5

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Adding and Subtracting Real NumbersWhen there is
a negative sign in front of an expression in
parentheses, such as (3 4), there are two
methods that you can use to simplify the
expression.

• The first method is to simplify within the
  parentheses and then apply the rule for the
  addition of opposite signs as shown here.
• Simplify 10 (-5 3)
• 10 (-5 3) 10 (-2) Work within
  parentheses. Add -5 and 3 to get -2.
• 10 2 Write subtraction as
  addition.
• 12

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A second method that can be used to simplify (3
4) is based on the (Multiplicative Property of
-1)For all real numbers a, -1(a) -a, or the
opposite of a.This property can be extended to
more than one term with the parentheses.The
Opposite of a Sum -(a b) -a bThe
Opposite of a Difference -(a b) -a b b
aSimplify (- 3 5)-1(-3 5) 3 5
-2
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Try the following problems

• 1) -12 (4 9)
• 2) -(2x 3)
• 3) -(3 4) (5 6)
• 4) -(x 7)
• 5) -(3 4) (5 6) (2 5)
• 6) -(-2 8) (-5 11) (1 10) (-3-3)

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Multiplying and Dividing IntegersIf the signs
are the same, the answer is positive.If the
signs are different,the answer is negative.
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Try the following problems

• Think of multiplication as repeated addition.
• 3 2 2 2 2 6 and 3 (-2) (-2) (-2)
  (-2) -6
• 1) 3 (-3) Remember multiplication is fast
  adding
• 3 (-3) (-3) (-3) (-3) -9
• 2) -4 2 Remember multiplication is fast
  adding
• -4 2 (-4) (-4) -8

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Multiply the following
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Dividing Integers

• Multiplication and division are inverse
  operations. They undo each other. You can use
  this fact to discover the rules for division of
  integers.
• 4 (-2) -8 -4 (-2) 8
• -8 (-2) 4 8 (-2) -4
• same sign positive
  different signs negative
• The rule for division is like the rule for
  multiplication.

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Try the following problems

• 1) 72 (-9)
• 72 (-9) Think 72 9 8
• -8 The signs are different, so the
  quotient is negative.
• 2) -144 12
• -144 12 Think 144 /12 12
• -12 The signs are different, so the
  quotient is negative.
• 3) -100 (-5) Think 100 5 20
• -100 (-5) The signs are the same, so the
  quotient is positive.

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Divide the following
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Lesson Quiz

• Find the sum or difference
• 1) -7 (-6)
• 2) -15 24 (-9)
• Evaluate x y for x -2 and y -15
• 3) 3 9
• 4) -3 (-5)
• Evaluate x y z for
• x -4, y 5, and z -10

• Find the product or quotient
• 1) -8 12
• 2) -3 5 (-2)
• 3) -75 5
• 4) -110 (-2)
• 5) The temperature in Bar Harbor, Maine, was -3
  F. During the night, it dropped to be four times
  as cold. What was the temperature then?

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Dividing Real NumbersBecause you can multiply
any two real numbers, and every nonzero real
number has a multiplicative inverse, you can
define division using multiplication and
multiplicative inverse.
Example 1
2
1
Try This
5
1
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Using Multiplication and division with other
operationsYou have learned the order of
operations and the rules for adding subtracting,
multiplying, and dividing real numbers. You can
now simplify many expressions that involve
positive and negative numbers.
Simplify
Another
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Simplify
1
3
-5
4
Try This
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The Distributive Property and Combining Like
TermsA monomial - is a real number or the
product of a real number and a variable raised to
a whole-number power.For example, 6, -4c, and
3x2 are monomials.When there is both a number
and a variable in the product, the number is
called the coefficient. In the monomials -4c and
3x2, -4 and 3 are coefficients.

• Recall that a monomial with no visible
  coefficient, such as x, actually has a
  coefficient of 1. Similarly, -x has a
  coefficient of -1.

Using the Distributive Property
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Simplify
Example 1
Example 2
Example 3
Try This
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Simplify
More Distributive Property
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When you have unlike terms in a sum or difference
within parentheses, you may have to use the
opposite of a sum or a difference when
simplifying.
Simplify
Example 1
Try This
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Using the Distributive Property to multiply and
divideYou can use the Distributive Property to
simplify a product. The Distributive Property
applies to both positive and negative numbers.
Simplify
Example 1
Example 2
More
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Example 3
Next Example
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Example 4
Try This
Try This
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Rational Numbers

• Fractions and Decimals

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Rational numbers, numbers that can be written in
the form a/b (fractions), with integers for
numerators and denominators.Integers and certain
decimals are rational numbers because they can be
written as fractions.
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Hint When given a rational number in decimal
form (such as 2.3456) and asked to write it as a
fraction, it is often helpful to say the
decimal out loud using the place values to help
form the fraction.
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Write each rational number as a fraction
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Hint When checking to see which fraction is
larger, change the fractions to decimals by
dividing and comparing their decimal values.
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Examples of rational numbers are

• 6 or 6/1 can also be written as 6.0
• -2 or -2/1 can also be written as -2.0
• ½ can also be written as 0.5
• -5/4 can also be written as -1.25
• 2/3 can also be written as .66
• 2/3 can also be written as 0.666666
• 21/55 can also be written as 0.38181818
• 53/83 can also be written as 0.62855421687
• the decimals will repeat after 41 digits

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Examples Write each rational number as a
fraction

• 0.3
• 0.007
• -5.9
• 0.45

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Since Real Numbers are both rational and
irrational , ordering them on a number line can
be difficult if you dont pay attention to the
details.As you can see from the example at the
left, there are rational and irrational numbers
placed at the appropriate location on the number
line.This is called ordering real numbers.
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Irrational numbers

• v2 1.414213562
• no perfect squares here

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Irrational number a number that cannot be
expressed as a ratio of two integers or as a
repeating or terminating decimal.

• An irrational number cannot be expressed as a
  fraction.
• Irrational numbers cannot be represented as
  terminating or repeating decimals.
• Irrational numbers are non-terminating,
  non-repeating decimals.

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Examples of irrational numbers are

• 3.141592654
• v2 1.414213562
• 0.12122122212
• v7, v5, v3, v11,
• 343v
• Non-perfect squares are irrational numbers

• Note
• The v of perfect squares are rational numbers.
• v25 5
• v16 4
• v81 9
• Remember Rational numbers when divided will
  produce terminating or repeating decimals.

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NOTE

• Many students think that is a
  terminating decimal, 3.14, but it is not. Yes,
  certain math problems ask you to use as
  3.14, but that problem is rounding the value of
  to make your calculations easier.
  It is actually an infinite decimal and is an
  irrational number.

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There are many numbers on a real number line that
are not rational. The number is not a
rational number, and it can be located on a real
number line by using geometry. The number
is not equal to 22/7, which is only an
approximation of the value. The number is
exactly equal to the ratio of the circumference
of a circle to its diameter.
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ENJOY YOUR PI p