31/08/2022
Real Numbers
What You Should Learn
• Represent and classify real numbers.
• Order real numbers and use inequalities.
• Find the absolute values of real numbers and
the distance between two real numbers.
• Evaluate algebraic expressions and use the
basic rules and properties of algebra.
2
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, 31/08/2022
Real Numbers
3
Real Numbers
Real numbers are used in everyday life to describe
quantities such as age, miles per gallon, and population.
Real numbers are represented by symbols such as
–5, 9, 0, , 0.666 . . ., 28.21, , , and .
Here are some important subsets (each member of subset
B is also a member of set A) of the set of real numbers.
{1, 2, 3, 4, . . .} Set of natural numbers
{0, 1, 2, 3, 4, . . .} Set of whole numbers
{. . . , –3, –2, –1, 0, 1, 2, 3, . . .} Set of integers
4
2
, 31/08/2022
Real Numbers
A real number is rational when it can be written as the ratio
p/q of two integers, where q ≠ 0. For instance, the numbers
= 0.3333. . . = 0.3, = 0.125, and
= 1.126126 . . . = 1.126 are rational.
The decimal representation of a rational number either
repeats (as in = 3.145) or terminates (as in = 0.5).
A real number that cannot be written as the ratio of two
integers is called irrational. Irrational numbers have infinite
nonrepeating decimal representations.
5
Real Numbers
For instance, the numbers
= 1.4142135 . . . 1.41
and
= 3.1415926 . . . 3.14
are irrational. (The symbol means “is approximately equal
to.”) Figure P.1 shows subsets of real numbers and their
relationships to each other.
Subsets of Real Numbers
Figure P.1 6
3
Real Numbers
What You Should Learn
• Represent and classify real numbers.
• Order real numbers and use inequalities.
• Find the absolute values of real numbers and
the distance between two real numbers.
• Evaluate algebraic expressions and use the
basic rules and properties of algebra.
2
1
, 31/08/2022
Real Numbers
3
Real Numbers
Real numbers are used in everyday life to describe
quantities such as age, miles per gallon, and population.
Real numbers are represented by symbols such as
–5, 9, 0, , 0.666 . . ., 28.21, , , and .
Here are some important subsets (each member of subset
B is also a member of set A) of the set of real numbers.
{1, 2, 3, 4, . . .} Set of natural numbers
{0, 1, 2, 3, 4, . . .} Set of whole numbers
{. . . , –3, –2, –1, 0, 1, 2, 3, . . .} Set of integers
4
2
, 31/08/2022
Real Numbers
A real number is rational when it can be written as the ratio
p/q of two integers, where q ≠ 0. For instance, the numbers
= 0.3333. . . = 0.3, = 0.125, and
= 1.126126 . . . = 1.126 are rational.
The decimal representation of a rational number either
repeats (as in = 3.145) or terminates (as in = 0.5).
A real number that cannot be written as the ratio of two
integers is called irrational. Irrational numbers have infinite
nonrepeating decimal representations.
5
Real Numbers
For instance, the numbers
= 1.4142135 . . . 1.41
and
= 3.1415926 . . . 3.14
are irrational. (The symbol means “is approximately equal
to.”) Figure P.1 shows subsets of real numbers and their
relationships to each other.
Subsets of Real Numbers
Figure P.1 6
3
