Transcript Chapter 1

1.1 Numbers
•
Classifications of Numbers
Natural numbers
Whole numbers
Integers
Rational numbers – can be
p
expressed as q where p
and q are integers
Irrational numbers – not
rational
{1,2,3,…}
{0,1,2,3,…}
{…-2,-1,0,1,2,…}
-1.3, 2, 5.3147,
7
13
,
5 ,
23
5
47 , 
1.1 Numbers
• The real number line:
-3 -2 -1
0
1
2
3
• Real numbers:
{xx is a rational or an irrational number}
1.1 Numbers
•
•
Double negative rule:
-(-x) = x
Absolute Value of a number x: the distance
from 0 on the number line or alternatively
x 
x if x  0
 x if x  0
How is this possible if the absolute value of a
number is never negative?
1.1 Numbers
• 3 > -3 means 3 is to the right on the number
line
-4 -3 -2 -1
0
1
2
3
4
• 1 < 4 means 1 is to the left on the number line
1.2 Fundamental Operations of Algebra
• Adding numbers on the number line (-2 + -2):
-4 -3 -2 -1
-2
-2
0
1
2
3
4
1.2 Fundamental Operations of Algebra
• Adding numbers with the same sign:
Add the absolute values and use the sign of
both numbers
• Adding numbers with different signs:
Subtract the absolute values and use the
sign of the number with the larger absolute
value
1.2 Fundamental Operations of Algebra
• Subtraction:
x  y  x  ( y )
• To subtract signed numbers:
Change the subtraction to adding the
number with the opposite sign
5  (7)  5  (7)  12
1.2 Fundamental Operations of Algebra
• Multiplication by zero:
x0  0
For any number x,
• Multiplying numbers with different signs:
For any positive numbers x and y,
x( y )  ( x) y  ( xy)
• Multiplying two negative numbers:
For any positive numbers x and y,
(  x)(  y )  xy
1.2 Fundamental Operations of Algebra
• Reciprocal or multiplicative inverse:
If xy = 1, then x and y are reciprocals of
each other. (example: 2 and ½ )
• Division is the same as multiplying by the
reciprocal:
x
y
 x
1
y
1.2 Fundamental Operations of Algebra
• Division by zero:
x
For any number x,
0  undefined
• Dividing numbers with different signs:
For any positive numbers x and y,
x
y

x
y
 ( )
x
y
• Dividing two negative numbers:
For any positive numbers x and y,
x
y

x
y
1.2 Fundamental Operations of Algebra
• Commutative property
(addition/multiplication)
• Associative property
(addition/multiplication)
ab  ba
ab  ba
(a  b)  c  a  (b  c)
(ab)c  a (bc)
1.2 Fundamental Operations of Algebra
• Distributive property
a (b  c )  ab  ac
(b  c) a  ba  ca
1.2 Fundamental Operations of Algebra
•
PEMDAS (Please Excuse My Dear Aunt Sally)
1. Parenthesis
2. Exponentiation
3. Multiplication / Division
(evaluate left to right)
4. Addition / Subtraction
(evaluate left to right)
• Note: the fraction bar implies parenthesis
1.3 Calculators and Approximate Numbers
• Significant Digits – What’s the pattern?
Number
Significant Digits
4.537
0.000056
70506
40.500
4
2
5
5
1.3 Calculators and Approximate Numbers
• Precision:
Number
4.537
56
Precision
thousandths
units
56.00
40.500
hundredths
thousandths
• Meaning of the Last Digit:
56.5 V means the number of volts is
between 56.45 and 56.55
1.3 Calculators and Approximate Numbers
• Rounding to a number of significant digits
Original
Number
Significant Rounded
Digits
Number
4.5371
4.5371
4.5371
4.5371
1
2
3
4
5
4.5
4.54
4.537
1.3 Calculators and Approximate Numbers
• Adding approximate numbers – only as
accurate as the least precise. The following
sum will be precise to the tenths position.
12.123
 13.1
 10.1253
1.4 Exponents
• Power Rule (a) for exponents:
a 
m n
 a nm
• Power Rule (b) for exponents:
ab 
m
a b
m
• Power Rule (c) for exponents:
a
 
b
m
m
a
 m
b
m
1.4 Exponents
• Definition of a zero exponent:
a 0  1 (no matter wha t a is)
• Definition of a negative exponent:
a
n
1
1
 n  
a
a
n
1.4 Exponents
• Changing from negative to positive exponents:
a m
bn
 m
n
b
a
• This formula is not specifically in the book but is
used often:
p
p
m
n
a

 bn





b

 am





1.4 Exponents
• Quotient rule for exponents:
m
a
mn
a
n
a
1.4 Exponents
• A few tricky ones:
 2   2  2  2  8
3
3
 2  2   2  2  2  8
4
 2   2  2  2  2  16
4
4
 2  2   2  2  2  2  16
3
1.4 Exponents
• Formulas and non-formulas:
 a  b n  a  n  b  n (distributive property)

a  b n  a n  b n
 a  b n  a n  b n ,
 a  b 2  a 2  b 2 ,

a 2  b2  a  b
( power rule b)
a  b n  a n  b n
a  b 2  a 2  b 2
1.4 Exponents
• Examples (true or false):
t t  t
4
3
12
( t 4 ) 3  t 12
s  t 
2
s  t 
3
 s t
3
3
 s2  t 2
1.4 Exponents
• Examples (true or false):
0
  10   1
1  2
0
2
3
2
x
x
 2
2
y
y
23
5

2
22
1.4 Exponents
• Putting it all together (example):
3xy 2 x y 
2
3
2
 3xy2 23 x 6 y 3
1 6
 3(8) x
 24 x y
7
5
y
23
1.4 Exponents
• Another example:
3
 2x y 
 3 xy 

   2 1 
2 
 3 xy 
 2x y 
3 3 6
3 x y
27 3 6 6  3
 3 6 3  8 x y
2 x y
2
1
9
27
y
27 3 9
8 x y 
3
8x
2
3
1.5 Scientific Notation
•
•
A number is in scientific notation if :
1. It is the product of a number and a 10 raised to a
power.
2. The absolute value of the first number is between 1
and 10
Which of the following are in scientific notation?
– 2.45 x 102
– 12,345 x 10-5
– 0.8 x 10-12
– -5.2 x 1012
1.5 Scientific Notation
•
Writing a number in scientific notation:
1. Move the decimal point to the right of the first nonzero digit.
2. Count the places you moved the decimal point.
3. The number of places that you counted in step 2 is the
exponent (without the sign)
4. If your original number (without the sign) was
smaller than 1, the exponent is negative. If it was
bigger than 1, the exponent is positive
1.5 Scientific Notation
• Converting to scientific notation (examples):
6200000  6.2 10?
.00012  1.2 10?
• Converting back – just undo the process:
6.203 1023  620,300,000,000,000,000,000,000
1.86 105  186,000
1.5 Scientific Notation
• Multiplication with scientific notation (answers
given without exponents):
4 10  5 10   4  5 10
5
8
5
10 8

 20 10 3  .02
• Division with scientific notation:
4 10   4  10
5 10  5 10
12
12
4
4
 80,000,000
 .8 1012 4  .8 108
1.6 Roots and Radicals
•
a is the positive square root of a, and  a
is the negative square root of a because
 a
2


2
 a and  a  a
• If a is a positive number that is not a perfect
square then the square root of a is irrational.
• If a is a negative number then square root of a is
not a real number.
• For any real number a: a 2  a
1.6 Roots and Radicals
• The nth root of a:
n
a is the nth root of a. It is a number whose nth
power equals a, so:
 a
n
n
a
• n is the index or order of the radical
• Example:
5
32  2 because 2  32
5
1.6 Roots and Radicals
• The
nth
root of
nth
powers:
– If n is even, then
n
– If n is odd, then
n
a
n
a
n
a
n
a
a
n
• The nth root of a negative number:
– If n is even, then the nth root is an imaginary number
– If n is odd, then the nth root is negative
1.7 Adding and Subtracting Algebraic
Expressions
• Degree of a term – sum of the exponents on
the variables
3 2
5a b degree  3  2  5
• Degree of a polynomial – highest degree of
any non-zero term
5x  3x  2 x  100 degree  3
3
2
1.7 Adding and Subtracting Algebraic
Expressions
• Monomial – polynomial with one term
5x
3
• Binomial - polynomial with two terms
5y  y
2
• Trinomial – polynomial with three terms
5 x  3x  100
3
2
• Polynomial in x – a term or sum of terms of
n
4
2
the form ax for example : x  3x  x
1.7 Adding and Subtracting Algebraic
Expressions
• An expression is split up into terms by the +/sign:
2
2
3x  4 x
 3 xy  35
• Similar terms – terms with exactly the same
variables with exactly the same exponents are like
terms:
• When adding/subtracting polynomials we will
need to combine similar terms:
3 2
3 2
3 2
5a b  3a b  2a b
1.7 Adding and Subtracting Algebraic
Expressions
• Example:
(4 x  3x  5)  (2 x  5 x  12)
2
2
 4 x 2  3x  5  2 x 2  5 x  12
 4 x 2  2 x 2  3x  5 x  5  12
 2 x  8 x  17
2
1.8 Multiplication of Algebraic
Expressions
• Multiplying a monomial and a polynomial:
use the distributive property to find each
product.
Example:
2
4 x 3 x  5




 4 x 2 3 x   4 x 2 5
 12 x 3  20 x 2
1.8 Multiplication of Algebraic
Expressions
• Multiplying two polynomials:
x2  x  2
x 3
 3x  3x  6
2
x  x  2x
3
2
x  4 x  5x  6
3
2
1.8 Multiplication of Algebraic
Expressions
•
Multiplying binomials using FOIL (First –
Inner – Outer - Last):
1.
2.
3.
4.
5.
F – multiply the first 2 terms
O – multiply the outer 2 terms
I – multiply the inner 2 terms
L – multiply the last 2 terms
Combine like terms
1.8 Multiplication of Algebraic
Expressions
• Squaring binomials:
x  y   x  2 xy  y
2
x  y   x 2  2 xy  y 2
2
2
2
• Examples:
2
2
2
2
m  3  m  23m   3  m  6m  9
5 z  1
2
 5 z   25 z   12  25 z 2  10 z  1
2
1.8 Multiplication of Algebraic
Expressions
• Product of the sum and difference of 2
terms:
x  y   x  y   x 2  y 2
• Example:
3  w 3  w  3
2
w  9w
2
2
1.9 Division of Algebraic
Expressions
• Dividing a polynomial by a monomial:
divide each term by the monomial
4x y  6x y 4x y
6x y

 2  2 xy  3
2
2
2x y
2x y 2x y
3
2
2
3
2
2
1.9 Division of Algebraic
Expressions
• Dividing a polynomial by a polynomial:
2x2  x  2
2 x  1 4 x3  4 x 2  5x  8
4 x3  2 x 2
 2 x 2  5x
 2x2  x
4x  8
4x  2
6
1.9 Division of Algebraic
Expressions
• Synthetic division:
2
1
1
5
2
3
x3  5x 2  7 x  3
x2
3
2
1
7
6
1
answer is: x 2  3 x  1
1
remainder is: -1  x  3 x  1 
x2
2
1.10 Solving Equations
• 1 – Multiply on both sides to get rid of
fractions/decimals
• 2 – Use the distributive property
• 3 – Combine like terms
• 4 – Put variables on one side, numbers on the
other by adding/subtracting on both sides
• 5 – Get “x” by itself on one side by multiplying or
dividing on both sides
• 6 – Check your answers (if you have time)
1.10 Solving Equations
• Fractions - Multiply each term on both sides by the Least
Common Denominator (in this case the LCD = 4):
1
 x  5  1 x  3
4
2
Reduce Fractions: 4  x  5   4 x  4  3
4
2
 x  5  2 x  12
Subtract x:
5  x  12
Subtract 5:
x  17
Multiply by 4:
1.10 Solving Equations
• Decimals - Multiply each term on both sides by the
smallest power of 10 that gets rid of all the decimals
Multiply by 100:
Cancel:
Distribute:
Subtract 5x:
Subtract 50:
Divide by 5:
.1 x  5  .05 x  .3
100  .1 x  5  100  .05x  100  .3
10 x  5  5 x  30
10 x  50  5 x  30
5 x  50  30
5 x  80  x  16
1.10 Solving Equations
• Example:
Clear fractions:
2
3
x  12 x  16 x  3
Combine like terms:
4x  3x  x  18
Get variables on one side:
Solve for x:
7 x  x  18
6 x  18
x3
1.11 Formulas and Literal Equations
• Example: d = rt; (d = 252, r = 45)
then 252 = 45t
divide both sides by 45:
27
3
t 5
5
45
5
1.11 Formulas and Literal Equations
• Example: Solve the formula for B
A  12 h(b  B)
multiply both sides by 2:
2 A  h(b  B )
divide both sides by h: A
2
h
subtract b from both sides:
bB
A
B  2 b
h
1.12 Applied Word Problems
• 1 – Decide what you are asked to find
• 2 – Write down any other pertinent information
(use other variables, draw figures or diagrams )
• 3 – Translate the problem into an equation.
• 4 – Solve the equation.
• 5 – Answer the question posed.
• 6 – Check the solution.
1.12 Applied Word Problems
• Example: The sum of 3 consecutive integers is
126. What are the integers?
x = first integer, x + 1 = second integer,
x + 2 = third integer
x  ( x  1)  ( x  2)  126
3 x  3  126
3 x  123
x  41
41, 42, 43
1.12 Applied Word Problems
• Example: Renting a car for one day costs $20 plus $.25
per mile. How much would it cost to rent the car for
one day if 68 miles are driven?
$20 = fixed cost,
$.25  68 = variable cost
$20  68  $.25 
$20  $17 
$37