Numbers,
Inequalities
and Absolute
Values
, Numbers and Sets
A number is a mathematical object used to count, measure and label. Numbers
are classified into sets, called number systems, such as the natural numbers
and the real numbers.
We start with the natural numbers:
N = {0, 1, 2, . . .}.
To this set we add the negative whole numbers and get the integers:
Z = {. . . , −3, −2, −1, 0, 1, 2, 3, . . .}.
The rational numbers, denoted Q, are ratios of integers. Thus, any
𝑚
rational number r can be expressed as r =
𝑛
where m and n are integers and n ≠ 0.
3
Division by 0 is not allowed, so expressions like are undefined. Some
0
numbers, such as √ 2 and π, can’t be expressed as a ratio of integers and are
therefore called irrational numbers.
We say a is less than b and write a < b if b − a is positive. Equivalently, we say b
is greater than
a and write b > a. When we write a ≤ b (or b ≥ a) we mean that either a < b or a
= b and we
read it as “a is less than or equal to b”.
We use this order property of R to represent real numbers as points on a line,
which is called a real number line, or simply a real line.
Inequalities
and Absolute
Values
, Numbers and Sets
A number is a mathematical object used to count, measure and label. Numbers
are classified into sets, called number systems, such as the natural numbers
and the real numbers.
We start with the natural numbers:
N = {0, 1, 2, . . .}.
To this set we add the negative whole numbers and get the integers:
Z = {. . . , −3, −2, −1, 0, 1, 2, 3, . . .}.
The rational numbers, denoted Q, are ratios of integers. Thus, any
𝑚
rational number r can be expressed as r =
𝑛
where m and n are integers and n ≠ 0.
3
Division by 0 is not allowed, so expressions like are undefined. Some
0
numbers, such as √ 2 and π, can’t be expressed as a ratio of integers and are
therefore called irrational numbers.
We say a is less than b and write a < b if b − a is positive. Equivalently, we say b
is greater than
a and write b > a. When we write a ≤ b (or b ≥ a) we mean that either a < b or a
= b and we
read it as “a is less than or equal to b”.
We use this order property of R to represent real numbers as points on a line,
which is called a real number line, or simply a real line.
