Download 60 1-4PosNegRealNrs_W16

1-4 The Real Numbers
Review of Sets.
Set: a collection of objects.
Subset: a set contained completely within another set
Example: {2, 5, 8} is a subset of {1, 2, 3, 4, 5, 6, 7, 8}
Elements: the objects in the set.
Braces { }: used to indicate a set and enclose the elements.
Natural or Counting Numbers: {1, 2, 3, 4, 5, … }
Whole Numbers: {0, 1, 2, 3, 4, 5, … }
The set of Natural Numbers is a subset of the Whole Numbers.
----------------------------------------------------------------------------------------------------------Integers (p. 33)
The whole numbers do not allow us to describe certain everyday situations. If your
checking account is overdrawn, you need negative numbers. The set consisting of the
whole numbers and their opposites, is the set if integers.
{… , −4, −3, −2, −1, 0, 1, 2, 3, 4, … }
The Natural Numbers and Whole Numbers are subsets of the Integers.
The Natural Numbers can also be called the Positive Integers.
Negative integers can be used for situations such as:
A debt of $10 = –10
An elevation of 1,312 feet below sea level = –1,312
----------------------------------------------------------------------------------------------------------The number line is a good way to visualize the sets of numbers. (p. 32)
 Start with a line extending infinitely in opposite directions. (use arrowheads at both
ends)
 Pick a point on the line and mark it with a tick mark (not a dot).
This is 0 (the origin).
 Select a “unit distance” and mark off multiples of this distance to the left and right
of zero. Use tick marks not dots.
 Label the marks to the right of zero: 1, 2, 3, etc.
 Label the marks to the left of zero: –1, –2, –3, etc.
<------|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-------->
–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7
Numbers are graphed by placing a dot at the correct location on the number line.
Example: Graph –3, 4 and 0.
<------|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|--------->
–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7
----------------------------------------------------------------------------------------------------------Rational Numbers (p. 34)
Even with the integers, there are still situations that we cannot describe:
 any situation calling for a fraction
 a division where the quotient isn’t an integer
For these situations, we use a set of numbers called the rational numbers.
A rational number is any number that can be expressed as a quotient of two integers with
a nonzero divisor.
𝑎
{𝑏 |𝑎 and 𝑏 are integers and 𝑏 ≠ 0}
Since every integer can be written as a fraction with a denominator of 1, the integers are a
subset of the rationals.
Graphing rational numbers on a number line is more difficult when they are “between”
integer values.
 try to locate the number as accurately as possible
 label the dot with the number you are graphing
Example: Graph
7
2
=3
1
2
7
2
and –4.6
and is halfway between 3 and 4
–4.6 is a little more than halfway between –4 and –5
–4.6
7/2
<------|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|--------->
–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7
-----------------------------------------------------------------------------------------------------------
Expressing Rational Numbers as Decimals
Every rational number can be expressed as a fraction and as a decimal. The decimal will
either terminate (stop) or it will have a digit or block of digits that repeat. To change a
fraction to a decimal, divide the numerator by the denominator.
Example: Express
5
8
as a decimal.
.625
8 5.000
48
20
16
40
40
0
5
8
= 0.625
0.625 is called a terminating decimal because the division stops (zero remainder) at the
thousandths place.
Example: Express
7
11
as a decimal.
.6363...
11 7.0000 ...
66
40
33
70
66
40
33
70
.
.
.
7
11
̅̅̅̅
= 0.6363 … = 0. 63
̅̅̅̅ is called a repeating decimal because the division will never stop. It will keep
0. 63
repeating in the same pattern forever. The bar over the six and three is a quicker and more
concise way to show that the digits repeat.
Irrational Numbers (p. 35)
Any number that can be represented on the number line but is not a rational number is
called an irrational number.
Irrational numbers cannot be represented in fraction form. The decimal representation for
irrational numbers is nonrepeating and nonterminating.
Examples: √2 ≈ 1.41421356237 …
𝜋 ≈ 3.141592653589793238462643383279 …
Since the decimal for an irrational number never comes to an end, we
distinguish between the exact answer and an approximate answer.
Example: 8𝜋 is an exact number.
25.13272 is an approximate answer using 3.14159 for π.
NOTE: Not all roots are irrational. √25 = 5 which is a rational number.
----------------------------------------------------------------------------------------------------------The Set of Real Numbers (p. 36)
All numbers that can be represented by points on the number line are real numbers. The
set of real numbers is formed by combining the set of rational numbers and the set of
irrational numbers. Every real number is either rational or irrational.
For a summary of the sets that make up the real numbers, see the figure on page 36.
Classifying Real Numbers
Classifying means to identify every set to which a number belongs.
Example: –7 an integer, a rational and a real
3
4
a rational and a real
√5 an irrational and a real
0. 6̅ a rational and a real
3 a natural, a whole, an integer, a rational and a real
----------------------------------------------------------------------------------------------------------Ordering the Real Numbers (p. 37)
Ordering means to compare numbers by size: greater than, less than, equal to.
< means the number on the left is “less than” the number on the right
6 < 10
> means the number on the left is “greater than” the number on the right
–7 > –8
= means the two numbers are “equal to” each other
3=3
On the number line, numbers increase in size from left to right.
 If a number lies to the right of another number, it is larger.
 If a number lies to the left of another number, it is smaller.
<------|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|--------->
–6 –5 –4 –3 –2 –1 0 1 2 3 4
5 6 7
The < and > symbols can be combined with the equal sign.
≤ means “less than or equal to”
≥ means “greater than or equal to”
When a combined symbol is used, only one of the relationships has to be true for the
statement to be considered true.
Example: 5 ≤ 8 is true because 5 < 8.
6 ≤ 6 is true because 6 = 6.
-----------------------------------------------------------------------------------------------------------
Absolute Value (p. 38)
Absolute value describes how far a number is from zero without considering direction.
Two vertical segments, | |, are used to enclose the number or expression you want to
find the absolute value of. Absolute value is never negative.
Examples: |–3| = 3
–3 is three units from zero.
The absolute value of negative three is three.
|5| = 5
5 is five units from zero.
The absolute value of five is five.
|0| = 0
0 is zero units from zero.
The absolute value of zero is zero.