Transcript Ch 1

Chapter 1
Whole Numbers
1.1 Introduction to Numbers,
Notation and Rounding
• Definitions
Numbers : Amounts or quantities
Set: A group of elements
Natural Numbers : 1, 2, 3, 4, 5,….
Whole Numbers : 0, 1, 2, 3, ….
Place Values
Trillions period
Hundred
Trillions
Ten
Trillions
Billions period
Trillions
Hundred
Billions
Ten
Billions
Millions period
Billions
Hundred
Millions
Ten
Millions
Thousands period
Millions
Hundred
Thousands
Procedure
Write a number in expanded form:
1. Write each digit multiplied by its place value
2. Express it all as a sum
Example
68,456
=
Standard form
6x 10,000 + 8 x 1000 + 4 x 100 + 5 x 10 + 6 x 1
Expanded form
In words – 68 thousands, four hundred fifty six
Ten
Thousands
Thousands
Ones period
Hundreds
Tens
Ones
Use < , > , or = to make a true statement
Definition
Equation: A mathematical statement that contains an equal sign
Example
14 = 14 is an equation, which is true
14 = 5 is not an equation , it is false, they are not equal
Inequality: A mathematical statement that contains an inequality symbol.
Greater than >
Less than
<
14 is greater than 12
14 > 12
12 is less than 14
12 < 14
Round numbers to a specified place
 To round a number to a given place value, consider the digit to
the right of the desired place value.
 If this digit is 5 or greater, round up
If this digit is 4 or less, round down
Example
45,685,923
Millions
45,685,923
45,685,923
45,685,923
Hundred thousands
Ten thousands
Hundreds
45,685,923
Tens
1.2 Adding, subtracting, and solving equations
with whole numbers
Definition
Addition : Arithmetic operation that combines amounts
Commutative property of addition
a + b = b + a, where a and b are any numbers
Associative property of Addition
( a + b) + c = a + (b + c), where a, b, c are any numbers
To add whole numbers:
1.
Stack with corresponding place values aligned
2.
Add the digits
Example
1 1
1 1
3456
62345
+906
+ 9039
4362
71384
Estimate sum
Estimate the sum by rounding to the nearest
Thousand, then find the actual sum.
52,407
+31,596
Estimate
52,000
+32,000
84,000
Actual
52,407
+31,596
84,003
Perimeter: The total distance around a shape. To find the perimeter of a shape, add the
lengths of all the sides of the shape.
10 ft
8ft
8ft
10 ft
Variable: A symbol that can vary or change in value
Constant: Any symbol that does not vary in value.
P = 8 + 10 + 8 + 10= 36 ft
Subtraction
To subtract whole number:
1.
Stack the greater number on top of the smaller number, aligning the
place values.
2.
Subtract the digits in the bottom number from the digits directly above.
If a digit in the top number is less than the digit beneath it, then rename
the top digit.
14 9 9
3 4 10 10 13
45,00 3
- 8,36 5
Check by adding
36638
8365
Check by estimating
45,000
- 8,000
37,000
36 6 3 8
45,00 3
Keywords for subtraction
Subtract, minus, remove, decreased by, difference, take away, left, less than
Procedure: To find a missing addend, write a related
subtraction
sentence. Subtract the known addend from the sum.
Example 1
100 + x= 250
x = 250 – 100 = 150
Check
100+ 150 = 250
Example 2
65 + x= 93
x = 93 – 65
=28
Check
65 + 28 = 93
Definition
Solution: A number that can replace the variable(s) in an
equation and make the equation true
1.3 Multiplying whole numbers and
exponents
Commutative property of multiplication
a.b= b.a
Associative property of multiplication
Grouping three or more factors differently will not affect the product
(ab)c= a(bc), where a, b, c are any real numbers
Multiplicative property of 0
The product of 0 and a number is always 0
0.n = 0 and n.0= 0, where n is any number
Multiplicative property of 1
The product of 1 and a number is always the number
1.n = n and n.1 = n, where n is any number
Distributive Property
If a sum or difference is multiplied by a number, then each number inside
parenthesis may be multiplied by the number outside the paranthesis
a(b+c) = ab + ac
a(b – c) = ab – ac, where a, b and c are any numbers
Procedure To multiply two whole numbers, stack
them and then apply the distributive property
Multiply 503 x 62
1
503
X 62
1006 Multiply 2 times 503
+ 3018
Multiply 6(tens) times 503
31,186
Key words for multiplication
Multiply, times, product, each, of , by
Exponent or power
A symbol written to the upper right of a base number that
indicates how many times to use the base as a factor.
Base : The number that is repeatedly multiplied
28
Exponent = number of times the base is used as a factor
Base
28 = 2.2.2.2.2.2.2.2 = 256
Exponential form
Factored form
Standard form
Powers of 10 and Period Names
Period names
103 =1,000 = One thousand
106 = 1,000,000 = One million
109= 1,000,000,000 = One billion
1012= 1,000,000,000,000 = One trillion
1015= (1 with 15 zeros) = One quadrillion
The names continue using the preceding pattern. Some names are
rather colorful.
10100 = (1 with 100 zeros) = googol
10googol = ( 1 with a googol of zeros)= googolplex
• Write in expanded form using powers of ten
Example 1
3,029,408
= 3x 106 + 0 x 105 +2x 104 + 9 x 103 + 4 x 102 + 8 x 1
Example 2
24,902
2 x 104 +4 x 103 + 9 x 102 + 2 x 1
1.4 Dividing, square roots, and solving equations with
whole numbers
Division Property
 When 1 is the divisor, the quotient is equal to the dividend
n divided by 1 = n/1 =, where n is any number
 When 0 is the divisor with any dividend other than 0, the quotient is undefined.
n divided by 0, or n/0 , is undefined, when n = 0
 When 0 is the dividend, the quotient is 0 as long as the divisor is not also 0
0 divided n = 0/n , when n = 0
 If both dividend and divisor are 0, the quotient is indeterminate
0 divided 0 or 0/0 is indeterminate
 When a number (other than 0) is divided by itself, the quotient is 1
n is divided by n = n/n = 1, when n = 0
Divisibility Rules
a)
2 is an exact divisor for all even numbers. Even numbers have 0, 2, 4, 6, or
8 in the ones place
Example- 2478 is divisible by 2 as even number 8 is in ones place
b)
To determine whether 3 is an exact divisor for a given number
Add the digits in the dividend
If the resulting sum is a number that is divisible by 3, then so is the dividend
Example- 35,616 is divisible by 3 as sum of the digits = 3 + 5 + 6 + 1 + 6= 21
divisible by 3
c) 5 is an exact divisor for numbers that have 0 or 5 in the ones place
Example – 71,315 is divisible by 5 because it has a 5 in the ones place
Divide
28
387
10836
84
243
224
196
196
0
Key words for division
Divide, distribute, each, split, quotient, into, per, over
42070
93 3912517
372
192
186
651
651
07
Definition
Square root : A base number that can be squared to equal a given number
The symbol for square root is the radical sign. The number we wish to find the
square root of is called the radicand
Radical sign
625
= 25
Radicand
Perfect Square : A number that has a whole – number square root
25 2 = 625
Roots and their square
Page 44
Order of Operations
1.
Grouping symbols. These include parentheses( ), 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
G
Grouping
E
Exponents
M
D A
S
Multiplications Division Addition Subtraction
or
P
Parenthesis
E
M
D A
S
Exponents Multiplications Division Addition Subtraction
1.6 Variables, Formulas, and Solving Equations
Formula for Perimeter of a rectangle
P = 2L + 2W
L
w
w
L
To Use a formula
1.
2.
Replace the variables with the corresponding given values
Solve for the missing value
Use the formula to find
Parallel lines : Straight lines that never intersect
Parallelogram : A four sided figure with two pair of parallel sides
The area of a parallelogram
A = bh
Right angle: An angle that measures 900
h
b
Volume of a Cube which is a measure of the
amount of space inside a three-dimensional object.
V= lwh
h
w
l