College Algebra.
Lecture Notes – Part 1
Day 1.
Should know material
1. Sets of numbers – List each of the following sets
( Do not forget the set symbols → {1, 3, 9 } )
a) natural numbers(N) or counting numbers: ______
Whole Numbers ( W):______Integers ( I, Z ) : ______
b) rational numbers:
if we can write a decimal number as a fraction, then it must be a rational number. List some rational numbers that include
almost all types.
- 2, ______, ______, 3/11, ______, 0.13, ______
c) irrational numbers:
if we can not write a decimal number as a fraction, then it will be an irrational number. List some common examples that
include almost all types
2. Other sets of numbers - List them.
prime numbers: ______
even, odd counting numbers: ______
Nonnegative integers: ______
Positive real numbers: ______
3. True or False.
If a line has a slope, then that slope will be positive when the line leans to the right. ______
If a line has a slope, then that slope will be negative when it leans to the left. ______
If a line is vertical, then it has a slope of zero. ______
All lines have some slope ( a number) . ______
4. = ______ = ______ = ______
5. Simplify by combining similar terms
2 + x + x + 5 = ______
6. Factor
2x + 4y = ______
x2 - 4x = ______
7. Find
x + 2 • 3 + x = ______
8. Closure of sets
We say set A is closed under addition
if for any two numbers in the set (including the same number twice), the sum is also in the set.
A = { 0, 1 } is closed under multiplication but is not under addition because
even though 0 +1 = 1 is in the set, 1 +1 = 2 which not part of the set.
The set of natural numbers, integers, whole numbers are closed under addition. True or False.
The set of natural numbers, integers, whole numbers are closed under multiplication. True or False
{ 0, -1, 1 } is closed under addition. True or False
{ 0 , 1, -1 } is closed under multiplication. True or False
What about under division and subtraction ? Use the sets above – to determine which is commutative and/or associative
______
(subtraction) (division)
9. Properties of Real numbers.
a) a + b = b + a is called the commutative law (property) of addition. ______(true or false)
a(bc) = (ab)c because of the associative law of multiplication. ______( true or false)
Assume that x ≠ 0
b) x ÷ x = ______c. x • x = ______
c) Write down the distributive law: ______
d) The additive identity: _____ the multiplicative identity: ______the additive inverse off x : ______(opposite)
the multiplicative inverse of x : _____ (reciprocal)
10. Rules of Exponents.
a) x4 • x5 = ______b) 2 + 3 • 4 + 5 = ______
c) x4 ÷ x8 = ______d) x + 3 • x + 5 = ______
e) ( 2x3 ) 0 = ______f) ( 2x3 ) ( 5x4 ) = ______
11. Solving Equations.
a) 2 + 4x = -3 → x = ______b) → x = ______
c) 2 + + = ______d) = ______
12. More Exponents.
a) 24 = ______b) 0 3 = ______c) 80 = ______
d) 4-1 = ______e) - 42 = ______
13. 2 + 3x + 8xy has three ( factors or terms ?) ______(which one)
3xy has three (factors or terms ? ) ______(which one)
14. Find the greatest common factor of 12 and 18. ______
15. The least common multiple of 8 and 12 is ______
16. What is the graph (shape) of the equation 2x – y = 4 ? ______
18. If x = - 2, then find y if y = x2 - 2x – 2 . ______
19. Construct a rectangular coordinate system and label each of the four quadrants as well as the axes.
20. Sketch the graph of 2x + 3y = 12.
21. Find
½ + ¼ = ______½ • ¼ = ______
½ ÷ ¼ = ______ = ______
22. Which is larger
the smallest whole number the smallest counting number the first prime number
None , they are all equal not enough information
______
23. Which is even
the product of the 1st 2 prime numbers the sum of the 1st 2 odd numbers
Not enough information they both are even ______
24. As n gets larger and larger ( as )
what happens to the values of ( 1 + 1/n ) n ? ______
Math 1302 College Algebra
Sets Of Numbers
____ = { 1, 2, 3, 4, ..... } is called the set of ______.
Let x be one of these numbers. Find all natural number solutions of the following equations.
2x + 4 = 6 è x = ______x - 2 = 2 è x = ______
2x + 4 = 4 → ______
We can use the set of natural numbers to describe the
1) The number of times a student will have to enroll in a required class before the class is passed with a grade of C or above.
2) The number of jobs a currently employed individual has held during his/her lifetime.
NOTE:
a) the set of prime numbers: a natural number that is greater than one and is divisible (evenly) only by one and itself
{ 2, 3, 5, 7, 11, 13, 17, 19, 23, ... }
b) the smallest natural number is one, there is no largest
_____ = { 0, 1, 2, 3, ... } is called the set of ______
Let x be one of these numbers . Find all whole number solutions of the following equations.
x – 4 = 6 è x = ______x + 5 = 5 → x = ______
x + 6 = 2 , è x = ______
We can use whole numbers to describe the
1) number of days without an accident
2) the number of friends that you can wake up at 2:00 AM just to say hello
3) The number of problems you will answer correctly on the first exam.
Note:
1) Zero (0) is the smallest whole number and is the only whole number that is not a natural number
______= { ..., -3, -2, -1, 0, 1, 2, 3, ... } is called the set of ______
Let x be one of these numbers. Find all integer solutions of the following equations.
x – 4 = - 6 è x = _____ 2x + 1 = 1 è x = ______
3x = 1 → x = ______
1) A game costs $1 to play. If you lose, you lose your dollar and if you win, you win $5. Record the amount won.
2) The number of yards gained by a football player
Note: Now we can introduce the idea of positive and negative numbers
1) the set of positive integers { 1, 2, 3, ... }, 2) the set of negative integers: { -1, -2, -3, ... }
3) zero is neither positive nor negative
4) the set of nonnegative integers: { 0, 1, 2, 3, 4, .... }
Other Sets
The set that contains all real numbers that can be written as fractions ( ratio of integers )
is called the set of ______
We describe this set in what is called set-builder notation: Q = { a / b | a is an integer and b is a nonzero integer }
For example: -2/3, 7/11, 2/9, - 6/2=3, 4/1=4, ...
Note:
Rational numbers include simple fractions like 2/3 , - 7/11, ¼, 11/5, ... that can not be reduced
but also numbers that can be labeled as integers → 3, -2, -7, 0, 2, ... they can all be written as fraction by writing
them with a denominator of one (1).
Note:
Rational numbers consists of simple fractions and integers
integers consist of negative integers and whole numbers
whole numbers consist of natural numbers and zero
All rational numbers can be written as fractions and also as decimals that are either
terminating: 0.12, 3.1114, 5.0000000007,
or repeating block: 3.12121212... , 4.111111..., 6.123412341234...
Real numbers that are not rational numbers
( decimal numbers that can not be written as fractions)
are called ______
( this means the real numbers are made up entirely of rational and irrational numbers)
This implies that irrational numbers can not be written in a “nice” decimal pattern ( not terminating)
They do include the following numbers →
2.10100100010000100000... 423.122333112223333111222233333...
More commonly we also include the following as irrational numbers
π = 3. 14... = _____ = ______e = ______
as n gets larger and larger what happens to ?
1 / 2 / 5 / 10 / 20 / 50 / 100 / 200 / 500 / 1000 / 2000 / 50002 / 2.25 / 2.48832 / 2.593742 / 2.653298 / 2.691588 / 2.704814 / 2.711517 / 2.715569 / 2.716924 / 2.717603 / 2.71801
500 / 1000 / 2000 / 5000 / 10000 / 20000 / 50000 / 100000 / 200000 / 500000 / 1000000 / 2000000
2.715569 / 2.71692393 / 2.717603 / 2.71801 / 2.718146 / 2.718214 / 2.718255 / 2.718268 / 2.718275 / 2.718279 / 2.71828 / 2.718281
Closure:
We say a set is closed under a given operation if when you perform the operation on any two numbers of that set, the result is a third
number that is also a member of that set.
Example:
Time on a clock: Add any amount of time to a clock – do you still get a time on the clock.
1) natural numbers are closed under addition and multiplication: 3 + 7 = 10 all natural numbers, 4 * 5 = 20 all natural numbers
2) What other sets are closed under addition and multiplication:
3) What sets are closed under subtraction ? Under division ?
This concludes a brief review of the sets that make up the set of real numbers. If we wish to solve equations of the form
x2 + 4 = 0, we have to extend the idea of numbers to a larger set – The Set of Complex Numbers.
For now we will only look at the set of real numbers. Later on, we will look at the set of complex numbers.
Practice Problems #1
1. Identify each set that each of the following numbers can be classified as being a part
Number / natural # / whole # / Integer / Nonnegative / Rational / Irrational / Real #-2
0
7
0.23
1/7
0.232323...
0.121122111222...
2. Find the sum of the first four prime numbers. ______
3. Find the product of the first three prime numbers. ______
4. How many integer solutions does the equation x2 – 9 = 0 have ? ______
5. How many real number solutions does the equation 2x – 4 = 0 have ? ______
6. How many natural number solutions does the equation 3x + 9 = 0 have ? ______
7. True or False.
______a) the set of integers are closed under division.
______b) the set of irrational numbers are closed under multiplication.
______c) the set of rational numbers are closed under division.
8. We can write a natural number that is not prime and greater than one (composite # ) as a product of prime numbers.
For example: 6 = 2 * 3. Write 42 as a product of prime numbers. This is called the prime factorization of 42.
42 = ______
9. What is the smallest positive real number ? ______
What is the smallest nonnegative integer ? ______
10. All rational numbers can be written as a fraction. Write each of these rational numbers as fractions.
a) 0.66666... b) 2.1111... c) 0.232323...
Properties of Real Numbers
commutative law of addition commutative law of multiplication
a + b = ______ab = ______
3 + 2 = 2 + 3 5 +2 = ______12 • 7 = 7 • 12 6 • 3 = ______
( 91 + 17 ) + 9 = ______( 8 • 7 ) • 5 = ______
associative law of addition associative law of multiplication
a + ( b + c ) = ______( a b ) c = ______
(5 + 15 ) + 17 = 5 + ( 15 + 17 ) (8 • 5 ) • 13 = 8 • (5 • 13 )
(93 + 16 ) + 4 = 9 3 + 16 + 4 (17 • 4 ) • 5 = 17 • 4 • 5
Distributive law of multiplication over addition : this property combines both the addition and the multiplication properties
a(b + c ) = ______- 2 ( x – 3 ) = ______
- ( a – b ) = ______(a + b) ( x + y ) = ______
Commutative and the associative laws do not hold true with subtraction and division – do they ?
ex. 2 – 4 = 4 - 2 ? ______12 ¸ 6 = 6 ¸ 12 ? ______
ex. ( 12 – 4 ) - 2 = 12 - ( 4 – 2 ) ? ______(16 ¸ 8 ) ¸ 2 = 16 ¸ ( 8 ¸ 2 ) ? ______