Download Section 1.7: Properties of Real Numbers

Algebra Concepts Test #1 Review
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Section 1.1: Fractions Refresher
Converting between improper and mixed fractions:
3 2 8  3
2 
3
25
4
8
8
 3 because 7 25 r.4
19
7
7

8
Reducing fractions: You can reduce the numerator and denominator if they are both divisible by the same
number. To do this, find the prime factorization of the numerator and denominator, then cancel identical
factors. Here is a list of the first twenty prime numbers:
2
3
5
7
11
13
17
19
23
29
31
37
41
43
47
53
59
61
67
71
24 2  2  2  3

60 2  2  3  5
2

5
Multiplying fractions: Convert all mixed numbers to improper fractions, then take numerator times numerator
and denominator times denominator.
1 2 3 11
1 3  
2 3 2 3
3 11

23
11

2
Dividing fractions: Convert all mixed numbers to improper fractions, then invert the second fraction and
multiply.
1
2 3 11
1 3  
2
3 2 3
3 3
 
2 11
9

22
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Adding (and subtracting) fractions: Find the Least Common Denominator (LCD), convert each fraction to
fractions using the LCD, then add (or subtract) numerators. To find the LCD, use every factor in both
denominators, and if a factor is repeated, then use the largest number of repeats.
13 5
13
5



 LCD  2  2  2  3  3  7  504
72 42 2  2  2  3  3 2  3  7
13 7 5 12
   
72 7 42 12
91 60


504 504
91  60

504
31

504
Section 1.2: Exponents, Order of Operations, and Inequality
The language of exponents:
Exponents represent repeated multiplication with a superscript number (the “exponent”) after the number that
gets repeatedly multiplied (the “base”):
3  3  3  3  3 = 35.
1. In many calculators and computer programming languages, the operation of repeated multiplication does
get a special symbol called the “caret” symbol ^ .
2. Special exponents:
a. Any number raised to the power of 1 is simply that number. Example: 271 = 27.
b. Any number raised to the power of 0 is 1. Example: 8,923,2760 = 1.
Order of Operations:
 Step 0: Do any calculations within grouping symbols (like parentheses ( ), square brackets [ ], curly
brackets { }, fraction bars, and absolute values ) first, from the innermost to the outermost.
 Step 1: Do any calculations with exponents.
 Step 2: Do any multiplications or divisions from left to right.
 Step 3: Do any additions or subtractions from left to right.
Algebra Concepts Test #1 Review
1  2   8  1
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2
 3  10
3  4  5(6  7)
  1  2  9 
2
 7

3  4  5(6  7)




1  2    9 
2
7

3  4  5(6  7)
1  2  81  7 
3  4  5(6  7)
 3 74 
3  4  5(13)
 3 74 
12  65
222

53
222

53
Relational Symbols: symbols for comparing the amounts of things.
1
 = (“is equal to”) Example:  0.5 , x = 7
2
1
  (“is not equal to”) Example:  0.3 , x + 2  x + 3
3
1
  (“is approximately equal to”) Example:  0.333
3
 > (“is greater than”) Example: 5 > 2
 < (“is less than”) Example: -1 < 1
  (“is greater than or equal to”) Example: 7  5, 7  7
  (“is less than or equal to”) Example: 3  21, 3  3
Section 1.3: Variables, Expressions, and Equations
A variable represents a number to stick into a calculation at a later time.
An algebraic expression is a multi-step calculation represented with one line using variables.
Sticking a number into an expression and calculating the answer is called evaluating the algebraic expression.
Section 1.4: Real Numbers and the Number Line
Natural (Counting) Numbers: {1, 2, 3, 4, … }
Whole Numbers: {0, 1, 2, 3, 4, … }
Integers: { …, -4, -3, -2, -1, 0, 1, 2, 3, 4, … }
Rational Numbers: fractions that can be made with integers; their decimal equivalent always repeats;
7
 0.07
99
Irrational Numbers: numbers whose decimal equivalent never repeats; they can not be written as a rational
number
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Real Numbers: the combined set of all rational and irrational numbers
Double Negatives: -(-3) = 3
Absolute Value: |-56| = 56; |128| = 128
Section 1.5: Adding and Subtracting Real Numbers
Big Idea: The number line can be used to understand how to add and subtract signed numbers, if you remember
that adding a positive number means move to the right and adding a negative number means move to the left.
Rule for Adding Numbers with the Same Sign: 7 + 17 = 24; -8 + (-12) = -20
Rule for Adding Numbers with the Different Signs:
New way to think of Subtracting Numbers: it is the same as adding the negative of the second number.
Section 1.6: Multiplying and Dividing Real Numbers
Big Idea: Multiplying numbers of the same sign results in a positive number, while multiplying numbers of
different signs results in a negative number.
Multiplying Numbers with Different Signs: x(-y) = (-x)y = -xy
Implication for odd powers of negative numbers, like (-2)5 = -32
Multiplying Numbers with the Same Signs: (-x)(-y) = xy
Implication for even powers of negative numbers, like (-2)6 = +64
Reciprocal or Multiplicative Inverse:
Pairs of numbers whose product is 1 are reciprocals, or multiplicative inverses of each other.
To find a reciprocal, just invert the fraction.
Zero has no multiplicative inverse.
1
1  2
3
The notation for a reciprocal using exponents is to raise the number to the power of -1: 5  ,   
5  3
2
1
Division:
Division can be written as multiplication by a fraction, which is the multiplicative inverse of the denominator:
x
1
 x
y
y
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Converting English Statements into Mathematical Statements:
English
Is
Is equal to
Equals
Sum of
Increased by
Added to
More than
Plus
Difference, difference between, or
difference of
Subtracted from
Less or less than
Decreased by
Minus
Product of
Times
Twice
Of (used with fractions and %)
As much as (?)
Quotient of
Divided by
Ratio of
Math
=
=
=
+
+
+
+
+







2y





Section 1.7: Properties of Real Numbers
Commutative Properties: You can add or multiply two numbers in any order. Algebraic way of saying this:
ab  ba
ab  ba
Associative Properties: If you have to add three numbers, you can add the first two first and then add the third
one, or you can add the last two first and then add the first one. The same applies for multiplication. Algebraic
way of saying this:
 a  b   c  a  b  c 
 ab  c  a  bc 
Identity Properties: There are special numbers that when added to or multiplied by any real number leaves the
original number unchanged. These special numbers are called the identity elements and are 0 (for addition) and
1 (for multiplication).
a0  a
a 1  a
Inverse Properties: Every number has a special number called its inverse such that when the inverse is added
to or multiplied by that number, the answer is the identity element. The additive inverse is the negative of a
number, while the multiplicative inverse is the reciprocal of a number.
Algebra Concepts Test #1 Review
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a   a   0
a
1
1
a
Distributive Property: If you have to multiply a sum of numbers by a factor, you can instead sum up the
products of each addend with the factor. Algebraic way of saying this:
a  b  c   ab  ac
 b  c  a  ba  ca
Section 1.8: Simplifying Expressions
Big Idea: Since an algebraic expression represents a multi-step calculation, we look for ways to simplify them
before we use them for calculating to save work later on. This section focuses on the simplifying strategy of
using the distributive property and combining like terms.
The distributive property for simplifying expressions:
4(3x + 7y) = 4(3x) + 4(7y) = 12x + 28y
Combining like terms for simplifying expressions:
 A term is a number, a variable, or a product or quotient of numbers and variables raised to powers.
 Like terms are terms with exactly the same variables and exponents (but possibly different coefficients),
and can be combined using addition or subtraction into a single term.
 Number trick example:
English statement
Mathematical translation
Pick a number.
n
Double it.
2n
Add 5.
2n + 5
Now double that answer.
2(2n + 5)
Subtract four times your original number.
2(2n + 5) – 4n
The answer you get is 10, no matter what number you 2(2n + 5) – 4n = 4n + 10 – 4n
picked.
= 10

Find the perimeter and area of the following shape in terms of the variable x:
Picture
Perimeter
Perimeter = sum of edge lengths
Area
Area = Area of rectangle + Area of square
 2x  x  x  x  x  2x
  2 x  x    x  x 
 8x
 2x2  x2
 3x 2
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Quiz example:
English statement
Multiply the number of guests by 4
Then add 8
Divide by 24
Multiply by 6.50
Multiply by 1.055
Simplify:
Mathematical translation
4n
4n + 8
4n  8
24
4n  8
 6.50 
24
4n  8
 6.50 1.055
24
1
 4n  8    6.50 1.055   4n  8 0.285729 
 24 
 1.142917n  2.285833