chapter 1 section 1 properties of real numbers. 1-1 algebra 2 lesson 1-1 simplify. 1.–(–7.2)2.1...
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Chapter 1
Section 1
Properties of Real Numbers
ALGEBRA 2 LESSON 1-1ALGEBRA 2 LESSON 1-1
Simplify.
1. –(–7.2) 2. 1 – (–3)
3. –9 + (–4.5) 4. (–3.4)(–2)
Properties of Real NumbersProperties of Real Numbers
5. –15 ÷ 3 6. – + (– )25
35
Solutions
1. –(–7.2) = 7.2
3. –9 + (–4.5) = –13.5
5. –15 ÷ 3 = –5
6. – + (– ) = = – = –125
35
–2 + (–3)5
55
2. 1 – (–3) = 1 + 3 = 4
4. (–3.4)(–2) = 6.8
Classifications of Numbers
Imaginary Numbers will be introduced later.
Real Numbers
The largest classification we will deal with Include any number that you can tell me
Ex:
Split into Rational and Irrational Numbers17
6 ,3 21,-54,
Real Numbers
Irrational Numbers Numbers that cannot be written as ratios Decimals that never terminate and never
repeat Square roots of positive non-perfect
squares Ex: √2, -√7, √(8/11),
1.011011101111011111…
Real Numbers
Rational Numbers
All the numbers that can be written as a ratio (fraction)
This includes terminating and repeating decimals.
Ex: 8, 10013, -54, 7/5, -3/25, 0, 0/6, -1.2, .09, .3333….
Real NumbersRational Numbers
Integers “Complete” numbers (no parts – fractions
or decimals) Negative, Zero, and Positive Each negative is the additive inverse (or
opposite) of the positive Ex: -543, 76, 9, 0, -34
Real NumbersRational NumbersIntegers
Whole Numbers Zero and positive integers
Ex: 0, 1, 2, 3, 4, …
Real NumbersRational NumbersIntegersWhole Numbers
Natural Numbers Also known as Counting Numbers Think of young children
Ex: 1, 2, 3, 4, 5, 6, …
Which set of numbers best describes the variable?a) The cost C in dollars for admission for n peopleb) The cost C of admission is a rational number and
the number n of people is a whole numberc) The maximum speed s in meters per second on a
roller coaster of height h in metersd) Since the speed s is calculated using a formula with a
square root, s is real (either rational or irrational). The height h is measured in rational numbers.
e) The park’s profit (or loss) P in dollars for each week w of the year
f) The profit P is a rational number and the week number w is natural.
Try This Problemp. 6 Check Understanding
The number r is the ratio of the number of adult tickets sold to the number of children’s tickets sold. Which set of numbers best describes the values of r? Which set of numbers best describes the average cost c per family for tickets?
r is a ratio, so it is a rational number; c is the cost which means a terminating decimal, so it is a rational number.
Graphing Numbers on a Number Line Make sure your number lines have zero Make them fairly accurate Label Important points
-3 -2 -1 0 21 3 4 5
0 15 20
0-120 -119
Ordering Real Numbers
Less Than <Or equal to ≤
Greater Than >Or equal to ≥
Properties of Real Numbers
Opposite or Additive Inverse – of any number a is –a The sum of opposites is 0.
Reciprocal or Multiplicative Inverse – of any number a is
Think of it as a flip Remember you may have to make a decimal into a fraction before
flipping it. Use the place (hundredths) to write a fraction. The product of reciprocals is 1.
a
1
Find the opposite and reciprocal of each number.a) .
b) -3.2
3125.16
5
32
10
2.3
1
2.3)2.3(
reciprocal
opposite
5
3
3
55
3
reciprocal
opposite
Try These Problemsp. 7 Check Understanding
State the opposite and reciprocal of each number.
a) 400
b) 4 1/5
c) -.002
d) -4/9
Opposite Reciprocal
-400 -1/400
-4 1/5 5/21
.002 -500
4/9 -9/4
Properties of Real Numbers
Property Addition MultiplicationClosure a+b is a real number ab is a real number
Commutative commute = to move
a + b = b + a Ab = ba
Associative associate = regroup
(a+b)+c = a+(b+c) (ab)c = a(bc)
Identity a+0=a,0+a=a a*1=a, 1*a=a
Inverse a+(-a)=0 a*(1/a)=1,a≠0
Distributive a(b+c) = ab + ac
Identify the propertyExample 5
Which Property is illustrated?a) 6 + (-6) = 0
a) Inverse Property of Addition
b) (-4 ∙ 1) – 2 = -4 – 2a) Identity Property of Multiplication
You MUST state Addition or Multiplication.Appropriate abbreviations:
Prop. Of Add. Or Prop. Of Mult.Comm. Assoc. Ident. Inv. Dist
Try these Problemsp. 7 Check Understanding
Which Property is illustrated?
a) (3 + 0) – 5 = 3 – 5a) Identity Property of Addition
b) -5 + [2 + (-3)] = (-5 + 2) + (-3)a)Associative Property of Addition
Absolute Value
The absolute value of a number is the distance from zero on the number line.
Its always positive. Be careful to watch for negatives outside
the absolute value bars (then the answer is negative).
Absolute Value Symbols │a│
Example 6
a) │-4 │ =
b) │0 │=
c) │-1 ∙ (-2) │=
d) │-10 │=
e) │1.5 │=
f) │0 - 3 │=
a) 4
b) 0
c) 2
d) 10
e) 1.5
f) 3