Math 150 – Fall 2015
Chapter 1A
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Notation
Use 12 , x2x−7 ,
Don’t use
Is this
x
x2
− 7 or
x
x2 −7 ?
It’s confusing!
For multiplication use 2 · 4 = 8 instead of 2 × 4 = 8.
It’s easy to confuse multiplication × with the variable x.
Always use exact numbers for your√ answers
(unless asked otherwise)
√
√
3
4
7
Exact numbers: 31 , 5π, e3 , 7 13, 4−
3
Approximate answers: .33, 3.14, 2.718, 25.239, −0.01315 Multiplication
Chapter 1A - Properties of Real Numbers
Real numbers have two operations, addition (+) and multiplication (·), that are defined
for all real numbers. For any real numbers a, b, and c the following properties hold for
addition and multiplication:
Commutative:
Example:
a+b=b+a
5+3=3+5=8
ab = ba
2 · 7 = 7 · 2 = 14
Associative:
Example:
a + (b + c) = (a + b) + c
5 + (4 + 6) = 5 + 10 = 15
(5 + 4) + 6 = 9 + 6 = 15
a(bc) = (ab)c
4 · (3 · 5) = 4 · (15) = 60
(4 · 3) · 5 = 12 · 5 = 60
Distributive:
Example:
a(b + c) = ab + ac
2(6 + 8) = 2 · 6 + 2 · 8 = 12 + 16 = 28
(b + c)a = ba + ca
Identities: The numbers 0 and 1 are the unique additive and multiplicative identities
with the following properties:
Additive Identity
Example:
a+0=0+a=a
9+0=0+9=9
Multiplicative Identity
Example:
a·1=1·a=a
5·1=1·5=5
Additive Inverse: For each real number a, there is a real number, denoted by −a,
called the negative of a, for which
Additive Inverse:
a + (−a) = (−a) + a = 0
Example:
6+(-6)=(-6)+6=0
Math 150 – Fall 2015
Chapter 1A
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Subtraction: Subtraction, denoted by a − b, is defined as follows for any real numbers
a and b:
a − b = a + (−b)
Multiplicative Inverse: For each a 6= 0, there is a real number, denoted by
or a−1 , called the reciprocal of a, for which:
Inverse: a a1 = aa = 1
Example:
9·
1
9
=
9
9
1
a
or 1/a
=1
Division: Division, denoted by a ÷ b,
a
b,
or a/b, where b 6= 0, is defined as follows:
a
a
a ÷ b = = a/b = a
= ab−1
b
b
LC 3,4
Types of Real Numbers
The natural numbers, positive integers, or counting numbers, are
N = {1, 2, 3, . . .}
The whole numbers or nonnegative integers are
W = {0, 1, 2, 3, · · · }
The negative integers are
{. . . , −4, −3, −2, −1}
The integers are the positive integers, the negative integers, and zero:
Z = {. . . , −3, −2, −1, 0, 1, 2, 3, . . .}
The rational numbers are real numbers that can be written as a fraction where the
numerator and denominator are integers, this includes any number with repeating or
terminating decimal expansion (including the integers).
o
nm
Q=
m, n ∈ Z, n 6= 0
n
Examples:
The Irrational numbers, R − Q, are the numbers that have non-repeating and nonterminating decimal expansions. They cannot be written as fractions of integers.
Examples:
Math 150 – Fall 2015
Chapter 1A
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Summary: All of the above numbers are still real numbers. These numbers are also
related, all positive integers are whole numbers, all whole numbers are integers, all
integers are rational numbers. The rational and irrational numbers are disjoint and
together make up the real numbers.
Diagram:
LC 5,6
Number Lines and Absolute Value
A number line is an easy method of picturing the set of real numbers. Each point
on the number line corresponds to exactly one real number:
Definition. The absolute value of a real number x, denoted by |x|, refers to the distance
from that number to the origin of the number line, the point corresponding to 0.
x
if x ≥ 0
|x| =
−x if x < 0
Note. An absolute value is ALWAYS POSITIVE! (above −x is just using the − to
change x to positive)
Example. Find the absolute value
• |5| =
• | − 13| =
• |0| =
• |2 − e| =
Math 150 – Fall 2015
Chapter 1A
The distance between any two numbers a and b is |a − b|.
Example. Find the distance between the numbers −15 and 7
LC 7,8,9
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