Download Integers, Rational, and Real Numbers

Integers, Rational, and
Real Numbers
A set is a collection of objects.
Typically in math we will refer to
collections of numbers as sets.
The objects in a set are called
elements.
In math, we represents our sets as
numbers in squiggly brackets.
{2, 6, 9} is a set of numbers.
The integers include the zero, the
counting numbers, and the counting
numbers “signed” negative.
{. . . , -3, -2, -1, 0, 1, 2, 3, . . .} is one
way to write the integers in set
notation.
Goal – to introduce 3 sets of
numbers
The counting numbers is a set of
numbers. We could represent the
counting numbers is set notation as
{1, 2, 3, . . .}.
Math need lots more numbers that the
counting numbers.
Zero is one of the most important
numbers, but is not a counting
number!
A number line is often labeled with
integers. The size of the integer
indicates how far from zero the
number is; the sign of the integer
indicates whether the integer is to the
left (negative) or to the right (positive)
of zero.
-4 -3 -2 -1 0 1
2 3 4
The size of an integer, or the distance
from zero of that integer along a
number line is called the absolute
value of that integer.
| 1 | = 1 because 1 is one unit away
from zero on a number line, but
| -1 | = 1 also, because -1 is also one
unit away from zero on a number line!
In fact, 1 and -1 are the only numbers
with absolute value of 1.
All rational numbers can be written as
decimals.
Some of the decimal representations
of rational numbers are very familiar.
Examples:
1
1
= 0.5, = 0.25
2
4
The decimal representation of all
rational numbers are terminating
or repeating decimals.
The set of rational numbers is the set
of all number that can be formed as
quotients, or fractions, using integers.
This set can be denoted
a
{ b such that a and b and integers,
b ≠ 0} or
a
{
| a and b and integers,
b
b≠0}
Where | stand for “such that”.
If you don’t know the decimal
representation of a rational number
you can find it by division by hand or
by calculator.
Example:
.11
9 1.00
1
9
9
10
1
= .111..., or 0.1
9
Example: Label the rational
numbers on the number line:
3.5, -2.7, 1
3
-4 -3 -2 -1 0 1
2 3 4
Some numbers have decimal
representations that are nonrepeating
and nonterminating. These numbers
are called irrational.
Real
Rational
Irrational
Integers
All of the sets of numbers discussed
so far fall into a larger set of numbers
called the real numbers.
Counting
numbers
All numbers on the number line are
real numbers.
The number line places the real
numbers in order.
To describe the order of the real
numbers, we use the symbols:
Finally absolute value works on any
real number, and describes the
distance from that real number to
zero on the number line.
> meaning greater than, and
Examples:
< meaning less than.
Example:
10 > 8
-3.7 < -3
| 0.1 | = 0.1
| -3.7 | = 3.7
−
10 10
=
3
3