Algebra I
2.1 (Real Numbers) and 2.5 (Algebraic Property) notes
Date:____________
I. Real Numbers
A. Real Number System
1) Integers = _______ = A number that can be written without a decimal or
fraction {…-2, -1, 0, 1, 2, 3…}. Integer values can be less than, equal to, or
greater than zero. Also, the set of whole numbers and their opposites:
{…, -3, -2, -1, 0, 1, 2, 3, …}.
2) Whole Numbers = _______ = Any positive integer, including 0.
{0, 1, 2, 3, 4, 5, …1345…}
3) Natural Numbers = ______ = Any counting #. The number 1 and any other
number obtained by adding 1 to it repeatedly. The set of
whole numbers without zero: {1, 2, 3, 4, 5, …}.
4) Rational Numbers = ______ = Any number that can be written in the form
a/b, where a and b are integers and b ≠ 0
5) Irrational Numbers = _____ = A # whose decimal part does not terminate or
repeat
6) Real Numbers = ______ = Any Rational or Irrational Number
7) Classify each number in as many ways as possible.
(a) 3/5 ________________
(e) 17π _________________
(b) √
(f) 0/5 ___________________
______________
(c) .1925… _____________
(g) - 4/5 ____________________
(d) .̅̅̅̅̅̅ ________________
(h) 5/0 ____________________
B. Ordering Real numbers by using a number line
1) A Number line is ordered, meaning that _____________________________________
_____________________________________. Always include at least three numbers on
a number line.
2) A coordinate is __________________________________________________________
Place a symbol on the line to make a true mathematical statement. Draw a number line
below and plot the following values:
(a) 3/5 ________________ - 4/5
(b) √
______________17π
(c) 0/5_________________5/0
(d) .̅̅̅̅̅̅ _______________.1925…
3) Definitions of Absolute Value
(a) The distance from _______ on a number line. It’s always positive!
(b) Formal Definition
(c)
| |
| |
4) Opposites: In math a ___________________ is the same as saying the word
“opposite”.
5) Use the given number line to find the indicated coordinates.
M
P
K
<----------o----------o----------o----------o----------o----------o----------o----------o---------->
-2
-1
0
1
2
3
4
5
What is the coordinate of …
(a) M? __________________
(b) The point halfwaybetween M and P? ________
(c) The point three-fourths of the way between M and K? _______
(d) The point one-third of the distance from K to P? ___________
(e) The point 2.7 units left of P? _____________
Property
Addition
Multiplication
Closure
a + b is a real number
ab is a real number
Commutative
a+b=b+a
ab = ba
Associative
(a + b) + c = a + (b + c)
Identity
There is a number 0 such that
There is a number 1 such that
a+0=a
a(1) = a
For every nonzero real number
For every real number a, there
a there is a real number
Inverse
is a real number –a such that
a + (-a) = 0
such that
( )
2) Identify the algebraic properties that justify the equivalence of the following pairs of algebraic
expressions
a) (
)
b) (
)
c) (
)
(
(
)
)
(
)
(
d)
)
(
)
C. The Distributive Property
For all real numbers a, b, and c:
(
)
( )
( )
(
)
D. Order of Operations (identify the correct order in which to work components of a problem)
1) ____________________________________________________________________
2) ____________________________________________________________________
3) ____________________________________________________________________
4) ____________________________________________________________________
5) Follow the order of operations to evaluate each expression.
(a)
[
(
) ]
(c)
– [ –(
) ]
(b) (
)
(d) (
– )
The TI 83/84 Plus and the Order of Operations.
Your calculator is only as smart as the operator, so you need to make sure that if you are
going to enter values, they will be represented the way you want them to be.
a) (
) (
b) (
)
)
c)
6) Numeral - A written symbol referring to a number.
7) Value - a numerical quantity measured or assigned or computed; "the value assigned to the car
was 5,200 dollars"
8) Equation – A statement of equality between two expressions that may be true or false
9) Inequality Symbol - <, >, ≤, ≥, or ≠
__________________________________________________________________________
10) Inequality – A mathematical sentence that contains <, >, ≤, ≥, ≠
11) Sum - The result of adding two or more numbers.
12) Terms – Parts of an algebraic expression separated by addition or
subtraction signs.
13) Difference - The result of subtracting one number from another.
14) Product – The result of multiplying two or more numbers.
15) Factors – Numbers that are multiplied together to form a product.
16) Quotient –The result of dividing two or more numbers (Example: the
quotient of 42 divided by 7 is 6!
17) Base – In an exponential expression of the form b^x. b is the base.
18) Power – an expression of the form a^n
19) Exponent - a mathematical notation indicating the number of times a quantity is multiplied by
itself
20) Grouping Symbols - symbols such as parentheses, ( ), and brackets, that are used to
group expressions.
21) Simplify - make simpler or easier or reduce in complexity or extent