Download Chapter 1 Review for Test on Integers and Operations 2016-2017

Chapter 1
Review for Test on Bases, Integers, Opposites, Absolute Value,
Integer Operations, Div. Rules
Integers
Natural Numbers (aka “Counting Numbers”) : {1, 2, 3, 4, 5, … }
Whole Numbers: {0, 1, 2, 3, 4, 5, … }
Integers: the set of whole numbers and their opposites { … -3, -2, -1, 0, 1, 2, 3, … }
Tell if the following belong to the Naturals (N), Whole Numbers (W), or Integers (Z).
1)
-40 _____
2)
12.5 _____
3)
0 _____
4)
18
_____
3
Comparing Integers
To compare integers (<, >, or =), you can use a number line. The number further to the right is
greater in value. Comparing negative numbers is a
matter of reversing your train of thought. 7 > 5,
however -5 > -7 (since -5 is further right on the
number line).
Compare using <, >, or =.
5) -4 __ 3
6) -12 __ -15
7) -5 __ 0
8) -1010 __ -1011
Absolute Value
The absolute value of a number is its distance from 0 on the number line. It is represented
symbolically as: │7│ means the absolute value of 7, which is 7.
Evaluate the following.
9) │-12│ = _____
10) │8│ = _____
11) │0│ = _____
12) │26│-│-14│ = _____
Opposites
The opposite of a number is a number with the same absolute value but a different sign
(pos/neg). 0 is its own opposite. Ex: - (-5) is read as the opposite of negative 5
Evaluate the following.
13) - (-15) = _____
14) -│-45│ = _____
15) Which is greater? -x or │x│ if x < -6
Divisibility Rules
These are the shortcuts for testing if a number is divisible by:
2:
3:
4:
5:
6:
7:
if the number ends in an even digit
if the sum of the digits is a multiple of 3
if the number formed by the last 2 digits is a multiple of 4
if the number ends in 0 or 5
if the number is divisible by 2 and 3
a) double units digit and subtract from the number formed before it
b) repeat until it is easy to see if the remaining number is a multiple of 7
8: if the number formed by the last 3 digits is a multiple of 8
9: if the sum of the digits is a multiple of 9
10: if the number ends in 0
11: a) Add every other digit = sum 1 b) Add remaining digits = sum 2
If the difference between sum1 and sum 2 is a multiple of 11
Find all possible digits that will make the following divisible by the # in parentheses.
16)
3_24 (4) _______
17)
_145 (9) _______ 18)
752_ (11) _________
Combining Integers
When combining:
Like signs: a) find sum of absolute values; b) result gets the sign of the addends
Unlike signs: a) find difference of abs. val.; b) result gets sign of the larger abs. val.
Before using these rules, eliminate double signs using these rules:
+- = Ex: 4 + (-7) becomes 4 - 7 (combine 4 and -7 = -3)
-- = +
Ex: 8 – (-12) becomes 8 + 12
(combine 8 and 12 = 20)
Evaluate the following by using combining rules and eliminating double signs.
19) -9 + 11 _____
20) -14 – 8 _____
21) -4 – (-4) ____ 22) -6 + (-7) + │-3│____
Multiplying and Dividing Integers
When multiplying or dividing:
Like signs: a) Mult/Div abs. val. ; b) result is positive
Unlike signs: a) Mult/Div abs. val. ; b) result is negative
PxP=P NxN=P
PxN=N NxP=N
**In an exponent expression with a negative base:
Even exponent = positive result
Odd exponent = negative result
Evaluate the following.
23)
-12 (8) ____
24)
 20
3  (14)
2
_____ 25) -7 – 9 (-2 – 4)2 _____ 26) -(-3)3 ____
Bases
Our decimal number system involves place value where each placement is a power of 10.
Non-decimal place-value systems powers of different numbers.
Ex: Binary uses powers of 2 ; Octal uses powers of 8
The base, n, utilizes n digits ranging from 0 to n-1.
Ex: Decimal – we use the digits 0-9 ; Binary uses the digits 0-1
Converting from Non-Decimal to Decimal
Start at your furthest right and identify the place value of each digit. Calculate the sum and
the result is your base 10 equivalent.
5418 = (5 x 82) + (4 x 81) + (1 x 80) = 320 + 32 + 1 = 353
Converting from Decimal to Non-Decimal
Divide the decimal number by the base you are converting to. Identify the remainder.
Repeat this process with the base into the quotient until you get a quotient that’s smaller
than the base. At this point the number formed by the last quotient and each remainder
before it (in order) is the converted number in the desired base.
Ex:
27)
157
= ________________5 = ________________2
28)
14236 = ________________10 = ________________4
Expressions
An expression is a mathematical phrase that can include numerals, operations, and
variables.
Ex: 8 ; -8x; 2+ 5 ; -32 + 5(2x)
A numerical expression is an expression that only includes numerals and operations.
Ex: 12 ; -9 – 7(3) ; 43 + 12
To evaluate (find the value of) a numerical expression, use the
Order of Operations (Parentheses, Exponents, Mult/Div, Add/Sub)
1) 35 – 3 (2 +5)
2)
9 (2)3 – 52 - │-6 │
An algebraic expression is an expression that includes at least one variable.
Ex: x ; 3x ; n – 9 ; n2 – 8n
To evaluate an algebraic expression, substitute values for all variables and evaluate
the resulting numerical expressions.
m = -3; n = -5
3)
7 + 4n – mn
4)
5 - 2m – n2 - m3
Translating Word Phrases
To translate word phrases to algebraic expressions, look for key words that indicate
operations. Add (more than, sum, increased by,…)
Subtract (difference, less than, decreased by, …)
Multiply (times, product, … )
Divide (quotient, ratio, … )
5)
seven more than the product of eleven and a number’s absolute value________
6)
the opposite of eight times the difference of a number and twelve ________
Set Notation and Rostering
A set is a well-defined collection of objects.
Set notation: { n │ n E , n P, n 10 } is read as “the set of all elements n such that n is an
element of the set of Evens and is not an element of the Primes and is less than 10”
Description – verbally describe set: The set of evens under 10 that are not prime
Rostering – list elements in the set: { 0, 4, 6, 8 }
For the following, give a written description and roster:
7)
{ n │ n W n  3 }
8)
{ n │ n Multiples of 3 , nO , n  20 }
Union and Intersection
The union of is all the elements of 2 or more sets joined.
If A = {0, 1, 2, 3} B = {3, 4, 5, 6}
A  B  {0,1,2,3,4,5,6}
(don’t repeat 3)
The intersection is a set composed of all common elements between 2 or more sets
If A = {0, 1, 2, 3} B = {3, 4, 5, 6}
A  B  {3}
The universal set is the set of all elements under consideration.
For the following, the universal set is N < 20.
If A = {primes}
B = {multiples of 5}
9)
A B
_____________ 10)
C = { odds}
B C
________________
Pascal’s Triangle
Pascal’s Triangle can be used to find the number of subsets – review in notes
Pascal’s Triangle can also be used to find the number of ways to win in a
“best of n” series – review POW “Winning The Series”
11)
How many subsets can be created for {1,2,3,4,5} _____
12)
In a best of 5 series, how many ways can a team win in 3, 4, or 5 games? _