Transcript Square roots

Chapter 2
Integers
2.1 Introduction to Integers
The whole numbers { 0, 1, 2, 3, 4,…..} with all negative natural
numbers { …., -3, -2, -1} form a set of numbers called the
integers
Graph integers on a number line
-4
-3
-2
-1
0
1
2
3
4
-4
-3
2
-2
-1
0
1
2
3
4
-3
Use < or > to make a true statement
8 > -7
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
-3
-2
-1
1
2
3
4
5
6
7
8
9
-5> -8
-9
-8
-7
-6
-5
-4
0
1
2
3
4
5
6
7
8
9
Absolute Value
Absolute Value : A given number’s distance from zero on a number line
-6
-5
-4
-3
-2
-1
0
1
2
Absolute value -6 = 6
3
4
5
6
Absolute value 6= 6
Absolute value of a number is always positive or zero
The absolute value of a negative number is always positive
The absolute value of a positive number is always positive
The absolute value of zero is zero
Example
-9 = 9
21 =
21
Additive inverse
6 and -6 are additive inverses and sum = 0
6 + (-6) = 0
2.2 Adding integers
Add integers with like sign
300 + 40
+ 40 means Moves 40 steps to the right
300
340
Starting position
Final Position
- 300 + (-40)= -340
+ (-40) means move 40 steps to the left
- 340
Starting position
-300
Final Position
Add integers with different signs
-300 + 80
+ 80 means move 80 steps to the right
-300
Starting position
- 220
Final Position
Examples
-
40+ (-23) = -63
24 + (-16) = 8
10 – 25 = -15
Add
-15 + 24 + ( -16) + 15 + (-30)
9 + (-16) + 15 + (-30)
-7 + 15 + (-30)
8 + (-30)
-22
Add -15 and 24 to get 9
Add -9 and 16 to get -7
Add -7 and 15 to get 8
Add 8 and – 30 to get -22
Other process
-15 + 24 + ( -16) + 15 + (-30)
24 + 15 + (-15) + (-30) ( Rearrange the addends so the positives and negatives
are grouped separately)
= 39 + (-45)
( Add positives and negatives separately )
= -6
( Reconcile the positive and negative sums)
No 49( Pg 102)
The following table lists the assets and debts for the Smith family. Calculate
their net worth.
+
Assets
Debts
Savings = $1498
Credit card balance = $ 1841
Checking = $2148
Mortgage =
$ 74,614
Furniture = $18,901
Automobile =
$5 488
Jewelry = $3845
Adding all assets and debts
1498 + 2148 + 18901 + 3845 + (-1841) + (-74614) + (-5488)
= 26392 + (-81943)
= -55551
The net worth = -$55,551
2.3
Subtraction statements as equivalent
addition statements
20 – 42= -22
20 + (-42) = -22
The operation changes from – to +
20 – 42
= 20 + (-42)
= -22
The subtrahend changes to
its additive inverse
-10 – 15= -25
-10 + (-15)= -25
The operation changes from – to +
-10 – 15= -25
= -10 + (-15)
-= -25
The subtrahend changes to
its additive inverse
No 48( Pg 111)
Gary bought a Fender Stratocaster guitar in 1960 for $ 150.
In 2004, he takes the guitar to a guitar show and sells it
for $12,000. What was the net ? Is it a profit or loss ?
Net = Revenue – cost
Revenue = 12000
Cost 150
Net = 12000- 150 = 11850 ; Profit
2.4 Multiplying, Dividing with integers
Rules
 When multiplying two numbers that have different signs, the product is
Negative
7.( -6) = -42
 When multiplying two numbers that have the same sign, the product is
positive
- 5 .(-9) = 45
 When multiplying signed numbers, count the total number of negative
factors:
If there are an even number of negative factors, then the product is positive
(-2).(-4).(-5).(-6) = 240
If there are an odd number of negative factors, then the product is negative
(-2).(-3).(-4)= -24
Rules
 When evaluating an exponential form that has a negative base:
If the exponent is even, the product is positive
(-6)2 = (-6).(-6) = 36
If the exponent is odd, the product is negative
(-4)3 = (-4).(-4).(-4) = -64
 When multiplying or dividing two integers:
If they have the same sign, the result is positive
45 divide by 5 = 9
If they have different signs, the result is negative
- 45 divide by 5 = -9
Rules
 To solve a missing factor, write a related division sentence,
dividing the product by the known factor
6x = -36
x=  36 = - 6
6
Square roots
Every positive number has two square roots, a positive root and a negative root
n
means to find the principal square root of n, which is its positive square root.
- n means to find the additive inverse of the principal square root of n which is
the negative square root of n.
The square root of a negative number is not an integer.
9
is not an integer
2.5 Order of Operations
Perform Operations in the following order:
1. Grouping symbols: parenthesis(), brackets[], braces{},
absolute value , radicals , and fraction bars
2. Exponents
3. Multiplication or division from left to right, in order as they
occur
4. Addition or subtraction from left to right, in order as they
occur
Applications and Problem Solving
1.
Solve problems involving net
N = R – C, where N represents the net amount, Represents revenue, and C represents cost
2.
Solve problems involving force
The formula for force
F = ma, F represents force, m represents mass, and a represents acceleration
In the American system of measurement
Force (F) measured in lbs, mass (m) in slugs, acceleration (a) in ft. sec2
The Earth accelerates all objects downward at the same rate, about -32ft/sec2
In metric system
Force (F) in Newtons , mass (m) in kilograms (kg), acceleration (a) in m/sec2
Metric system – 10 meters /second/second
3.
Solve problems involving voltage
V = ir
V = Voltage (volts), I = current (Ampere) , r = resistance (Ohms)
4. Solve problems involving average rate
Distance = rate . Time
d = rt
Average rate : A measure of the rate at which an object travels a
total distance in a total amount of time