Download The Real Number System

Integrated Algebra 1A
Real Numbers and the Number Line
Unit 1 – Foundations CW 1.1
Homework: HW 1.1
SET: Is a group of similar items
NUMBER SETS:
N = Natural Numbers or _______________________________________
Represent in set notation {
}
and on a number line
W = Whole Numbers is the set consisting of _________________________________ and ______________.
Represent in set notation {
}
and on a number line
Z = Integers is the set of _______________________ and __________________________________________________.
Represent in set notation {
}
and on a number line
We use dots(closed circles) to indicate each integer. What kind of numbers fall between each dot?
These are called ________________________________________________ and they fill in the entire number line.
R = Real Numbers is the set of all Rational and Irrational numbers.
Q = Rational Numbers are any numbers that can be expressed as:
a) a ratio or fraction
______________________________________
b) a terminating decimal (it has an end) ______________________________________
c) a repeating decimal
______________________________________
I = Irrational Numbers are any numbers that can be expressed as an
Infinite (never ending), non-repeating decimal ______________________________________
Determine if the following numbers are rational (Rat.) or irrational (Irr.):
22
________
7
3 __________
3.14 _________
3 __________
 __________
1
Fill in the diagram with the set names:
Natural, Whole, Integer, Rational, Irrational
The Real Number System
1. The smallest Whole Number is _________.
3. The largest Negative Integer is _________.
2. The smallest Positive Integer is _________.
4. Zero is what type of number?
Positive Negative Both
5. All Natural Numbers are also Whole Numbers.
True
False
6. Zero (0) is a Natural Number.
True
False
7. All Irrational numbers are Rational.
True
False
8. Every real number is a rational number.
True
False
9. Every rational number is a real number.
True
False
10. Some numbers are both rational and irrational.
True
False
11. All Whole numbers are Integers.
True
False
12. All rational numbers are integers.
True
False
13. Which fraction represents a repeating decimal?
a)
4
5
1
2
b)
14. Write the following numbers in order from smallest to largest:
3
4
d) 3.1,  ,
d)
5
5
6
d)
 , 8 , 3. 1 , 2 3 , 2
15. Which numbers are arranged from largest to smallest?
22
22
22
a) 10 ,
,  , 3.1
b) 3.1,
,  , 10
c)  ,
, 3.1, 10
7
7
7
16. Which is a rational number?
a) 2
b)
c)
4
2
c)
Neither
4
5
22
, 10
7
Integrated Algebra 1A
Real Numbers and the Number Line
Homework: HW 1.1
1. In Column I, sets of numbers are described in words. In Column II, the sets are listed using patterns
and dots. Match the patterns in Column II with their correct sets in Column I.
_____
_____
_____
_____
_____
_____
_____
_____
_____
_____
_____
_____
COLUMN I
1. Counting Numbers
2. Whole Numbers
3. Even Whole Numbers
4. Odd Whole Numbers
5. Even Counting Numbers
6. Odd Integers
7. Even Integers
8. One-digit Whole Numbers
9. One-digit Counting Numbers
10. Odd Whole Numbers less than 10
11. Even Whole Numbers less than 10
12. Integers greater than -3
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
l.
COLUMN II
0, 1, 2, …, 9
0, 1, 2, …
0, 2, 4, 6, …
0, 2, 4, 6, 8
0, 2, -2, 4, -4, 6, -6, …
1, 2, 3, 4, …
1, 2, 3, … , 9
1, 3, 5, 7, …
1, 3, 5, 7, 9
2, 4, 6, 8, …
-2, -1, 0, 1, 2, 3, 4, …
1, -1, 3, -3, 5, -5, …
2. Are there numbers that are both rational and irrational? If yes, name one. If not, why?
_________________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________________
3. Tell whether each number is rational (Q) or irrational (I).
a) .36 _________
b) .36363636… _________ c) .363663666… ________
d) .36 _______
4. Zero (0) is a positive integer.
True
False
5. All irrational numbers are real.
True
False
6. All Integers are Whole numbers.
True
False
7. All integers are integers.
True
False
e)
36 ______
22
. Do you agree with Carlos? Explain.
7
_________________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________________
8. Carlos said that 3.14 is a better approximation for  than
9. In which list are the numbers in order from least to greatest?
1
a)
3.2,  , 3 , 3
c)
3
1
b)
d)
3 , 3.2,  , 3
3
3 ,  , 3.2, 3
1
3.2, 3 ,
3
1
3
3, 
3
Integrated Algebra 1A
Operations of Integers
Unit 1 – Foundations CW 1.2
Homework: HW 1.2a(Odds)
HW 1.2b(Evens)
Warm-up: Using the words Rational, Integers, Whole, Irrational and Natural, fill in the Venn
diagram to illustrate the set of all Real Numbers.
INTEGERS (-2, +5, - 4, 1, 0) The “_______________” part tells you the distance away from zero,
the “___________” tells the direction. (+ _______________) (- ________________)
ABSOLUTE VALUE: The ______________________ away from zero on a number line (distance is ________)
The “number” part
Symbol: ______________
Ex.
|2|=
On the Calculator: MATH  NUM  1: abs( )
|-2|=
-|2|=
-|-2|=
INTEGER RULES:
Adding:
Integers with the SAME sign: _______________________________________________________________________
4+4=
-3 + - 7 =
Integers with DIFFERENT signs: __________________________________________________________________
-10 + 3 =
3 + -4 =
Subtracting:
CHANGE:
1. _________________________________________________________
2. _________________________________________________________
Then , use the rules for ______________________________
15 – 20 =
-3 – 18 =
5 – (-9) =
-7 – 2 =
4
Examples to try:
1. 4 - 7 =
7.
14 - -4 =
2.
-3 - 4 =
8.
2-6=
3.
5 - -8 =
9.
- 4 + -9 =
4.
-2 + -5 =
10. 10 + -10 =
5.
-3 + 8 =
11.
6.
-7 - -6 =
12. -17 + -11 =
-3-7=
Multiplying/Dividing:
SAME sign:
2(3) =
DIFFERENT signs:
-3(4) =
the answer is _______________________
-4(-5) =
12 ÷ 6 =
-7
-21 =
the answer is _______________________
4(-6) =
-24 ÷ 6 =
64 =
-8
Examples to try:
1.
(12)(-3) =
5.
8 ÷ -2 =
2.
2x4=
6.
-72 ÷ -9 =
3.
(-9)(-3) =
7.
81 ÷ 27 =
4.
-4(6) =
8.
-64 ÷ 8 =
Mixed Practice: Determine if they are the same(S) sign or different(D) signs, and solve.
1. 4  6
7.
4  6
2. 5 – 3
8.
9  3
3. 6 + 2
9.
13  5
4. 10 – 5
10.
16 : 4
5. 13 + 4
11.
27  3
6. -2 + 6 4  6
12.
48  16
5
Integrated Algebra 1A
Operations of Integers
Unit 1 – Foundations HW 1.2a(Evens)
HW 1.2b(Odds)
Perform the indicated operation:
1.
6  (3) 
2.
5  (3) 
3.
15  7 
4.
8  (11) 
5.
3  6 
6.
4  (3) 
7.
9  11 
8.
22  (13) 
9.
13  (4) 
10.
85
11.
16  7 
12.
30  (13) 
13.
(7)(3) 
14.
(2)(3)(5) 
15.
(3)(8) 
16.
(3)(8) 
17.
(7)(8)(2) 
18.
(9)(6) 
19.
(4)(6) 
20.
32  8 
21.
40  (5) 
22.
42  (14) 
23.
50  (10) 
24.
81  (9) 
In 25-34, find the value of the expression.
25.
7  (9)
26.
(3)(17)
27.
(75)  (5)
28.
28  (4)
29.
1.7  (2.3)
30.
(3)(10)(8)
31.
3
(14)( )
7
32.
0  (12)
33.
(5)2
34.
(7)2
35. The temperature at the start of the day was 10 degrees. At noon the temperature was
three times that temperature. At dinner time the temperature dropped 15 degrees. By
midnight the it had dropped another 20 degrees. What was the temperature at midnight?
6
Integrated Algebra 1A
PEMDAS – Order of Operations
Unit 1 – Foundations CW 1.3
Homework: HW 1.3
Warm-up: Determine if the number is rational(R) or irrational(I):
4 _________ 2.18 _________
17 __________ 32 __________  __________ .343434… _________
2
_________
9
ORDER OF OPERATIONS
Memory Tools:
PEMDAS (Please Excuse My Dear Aunt Sally)
P _______________________________________
(always start with the inner most and work outwards)
E _______________________________________
(Remember -22 = -4 and (-2)2 = +4)
M ______________________________________
D _____________________________________
(work from left to right)
A _______________________________________
S ______________________________________
(work from left to right)
When using a calculator, put in parentheses exactly where they are in the problem!
Evaluate each expression using the Order of Operations. Check with your calculator.
1.
(5  3)  2  2
2.
3 6  9  3  6
3.
(3  6)  32
4.
(8  3)(12  4)  5
5.
20  4  5  2  10
6.
5 x 7 - 6 ÷ 2 + 32
7.
(6  8)(8  3)  (9  3  2)2
8.
8[62 - 3(2 + 5)]÷4 + 3
9.
1
50  (17  5)
2
10.
(8  5)  3 2  4  32
11.
(6  2)2
 3 9
16
12.
[6 2  (2  4)2 ]3
Use parentheses around all numerators and demominators.
13.
12  12
11  5
14.
32  2 2
12  4
15.
6 [
27
 (2  3  5)]
3
7
Integrated Algebra 1A
PEMDAS – Order of Operations
Homework: HW 1.3
Evaluate each expression using order of operations.
1.
7  5  3 4
2.
53  8  4
3.
6  (12  7.5)  7
6.
3(2 3  4 2 )
9.
62  4 2
 23
2(3  2)
4.
2.1  (0.5  0.2)
5. 125  [5(2  3)]
7.
6(0.2 + 0.3) – 0.25
8.
8 2  6(4)
4
2(5)
19  2[3  2(6)]
10.
*13
8
12[10 
(5  6)3
]
6
2
1
3 5 5
5[  (  )  ]
2 5 6 8
11.
*14.
11  3[2  6(3)]
3 2 1 1
[   (  )] 12
4 3 2 3
17[3  2(6)]
17[3  12]
17[15]
255
15. Jon calculated the expression
above incorrectly. Describe where
he made a mistake and provide the
correct answer.
INSERT MAKE IT CLOSE WS HERE
9
Integrated Algebra 1A
Properties of Real Numbers/Closure
Unit 1 – Foundations CW 1.4
Homework: HW 1.4
VARIABLE: ______________________________________________________________________
ADDITIVE IDENTITY ELEMENT. ____
(lets a number keep its identity when added)
a + ____ = a
ADDITIVE INVERSE
_________________________________
_________________________________
a + _____ = 0
-a + _____ = 0
MULTIPLICATIVE IDENTITY ELEMENT. ____
(lets a number keep its identity when multiplied)
b x ____ = b
MULTIPLICATIVE INVERSE
_________________________________
_________________________________
b x _____  1
1
x ____  1
b
a
x ____  1
b
when b  0
0
 ____
x
x
 ________
0
1.
List the additive inverses of the following numbers:
5:
2.
7:
1/6:
-41:
-19:
1/7:
List the multiplicative inverse of the following numbers:
2/3:
10
-3:
-3/5:
7:
3.
Additive Inverse
COMMUTATIVE PROPERTY (“Commute”)
Addition:
a + b = ___________
ASSOCIATIVE PROPERTY (“Associate”)
Addition:
a + (b + c) = ____________
Multiplicative Inverse
__________ order
Multiplication:
a x b = _____________
__________ order _________ parentheses
Multiplication:
a x (b x c) = ______________
DISTRIBUTIVE PROPERTY (“Distribute”)
multiple all numbers inside the parentheses by number outside
a (b + c) = ______________
What property is illustrated?
1. 4(3  2)  4(3)  4(2)
2. 4  0  4
3. 2  3  3 2
4. 2x(3x4)  (2x3)x4
1
1
 (.6  3)  (3  .6)
5.
2
2
6. 1x8  8
7. 8  (8)  0
8. 4  (5  2)  (5  2)  4
9. 4  (5  6)  (4  5)  6
1
1
10. 3 
3
a(b – c) = _______________
______________________________________________________
______________________________________________________
______________________________________________________
______________________________________________________
______________________________________________________
______________________________________________________
______________________________________________________
______________________________________________________
______________________________________________________
______________________________________________________
11
CLOSURE: A set is closed under an operation when any two elements of the set produce an answer
that is also in the set.
** Open Party vs. Closed Party **
Natural
Whole
Integer
Closed under Addition?
Subtraction?
Multiplication?
Division?
1. Which one of the following sets is NOT closed under addition?
(a) odd integers
(b) even integers
(c) natural numbers
(d) whole numbers
2. Under what operation is the set of positive integers NOT closed?
(a) addition
(b) subtraction
(c) multiplication
(d) division
3. Under what operation is the set of odd integers closed?
(a) addition
(b) subtraction
(c) multiplication
(d) division
4. Which one of the following sets has the property of closure under multiplication?
1 1
(a) { , 0, }
(b) {-1, 0, 1}
(c) {1, 2, 3, 6}
(d) {0, 2, 4}
2 2
5. Under which operation is the set {1, 2, 4, 8, 16,…} closed?
(a) addition
12
(b) subtraction
(c) multiplication
(d) division
Integrated Algebra 1A
Properties
Homework: HW 1.4
Place the letter of the appropriate property next to each example. (you may use them more than once)
A. Commutative for Addition
1. ________
B. Commutative for Multiplication
2. ________
C. Associative for Addition
3. ________
D. Associative for Multiplication
4. ________
E. Distributive
5. ________
F. Identity for Addition
6. ________
G. Identity for Multiplication
7. ________
H. Additive Inverse
8. ________
I. Multiplicative Inverse
9. ________
10. ________
(6  2)  5  6  (2  5)
1
7 1
7
13  18  18  13
5 5
6  6
6 6
909
1
(8  2)  .5(8)  .5(2)
2
(8  7)  6  8  (7  6)
3 3
 0
7 7
5(3  4)  5(3)  5(4)
3
3
1 
4
4
11. Which on of the following sets is closed under the operation of division?
(a) positive integers
(b) natural numbers
(c) real numbers
(d) integers
12. Which one of the following sets is not closed under addition?
(a) [1, 2, 3, 4,…}
(b) {3, 6, 9, 12,…}
(c) {2, 4, 6, 8,…}
(d) {1, 3, 5, 7,…}
13. Which on of the following sets is closed under the operation of subtraction?
(a) prime numbers (b) odd numbers
(c) counting numbers
(d) integers
14. For what operation is the set of integers not closed?
(a) addition
(b) subtraction
(c) multiplication
(d) division
15. Erika knows that the sum of two rational numbers is always a rational number. Therefore, she
concludes that the sum of two irrational numbers is always an irrational number. Give one
example that will convince Erika that she is wrong.
13
Integrated Algebra 1A
Evaluating Expressions
Unit 1 – Foundations CW 1.5
Homework: Finish CW 1.5
WARM-UP (Review): Name the property illustrated:
1.
A+B=B+A
______________________________________
2.
A + (B + C) = (A + B) + C
______________________________________
3.
A(B + C) = AB + AC
______________________________________
4.
What is the multiplicative inverse of
2
?
3
5.
What is the additive inverse of -4?
_________
___________
EVALUATING EXPRESSIONS
1. Rewrite the expression.
2. Substitute in the value for each variable using parentheses.
3. Evaluate using the Order of Operations (PEMDAS).
Evaluate if x = -2, y = 3, z = -1
1.
2x2 – z
2. 3y – z2
3.
5. 2xy + y2
6.
6x 2
3xz
Evaluate if x = -2 and y = -3
4.
14
x2y – y
3x
4 xy
Integrated Algebra 1A
Evaluating Expressions
Homework: Finish Packet
Evaluate each using the values given.
1)
y  2  x when x = -1 and y = 2
2)
a  5  b when a = 10 and b = -4
3)
p2  m when m = 1 and p = 5
4)
y  9  x when x = -1 and y = 3
5)
m  p  5 when m = 1 and p = 5
6)
y 2  x when x = 7 and y = -7
7)
z(x  y) when x = 6, y = -8 and z = 6
8)
x  y  y when x = 9 and y = 10
9)
p 3  10  m when m = -9 and p = 3
10)
6q  m  m when m = 8 and q = -3
11)
p2 m  4 when m = 4 and p = 7
12)
y  (z  z 2 ) when y = 10 and z = -2
13)
z  (y  3  1) when y = 3 and z = -7
14)
(y  x)  2  x when x = 1 and y = -1
15
15)
p  (9  (m  q)) when m = 4, p = 5 and q = 3
16)
(a2  b)  6 when a = -5 and b = 1
17)
2( p  4)  (m  n) when m = 4, n = 2 and p = 5
18)
y  (4  x  y  2) when x = 3 and y = -2
19)
x 3  3  y when x = 3 and y = -1
20)
pn  (n  m)2 when m = 1, n = 4 and p = 6
21)
12k  h2 when h = 2 and k = 3
22)
p  m  n  m 2 when m = 4, n = 5 and p = 5
23)
2  r  (5  q)  p when p = 2, q = -2 and r = 5
24)
y  z  xz  6 when x = -3, y = 4 and z = 4
25)
y
 x  4  z  y when x = 7, y = 2 and z = 4
2
26)
c
16
bc
 (7  a) when a = 4, b = 8 and c = 5
4