Practice D
Real Numbers
Unit 1
Getting
Ready
Most likely, all the numbers you’ve encountered in math classes so far have
been real numbers. Real numbers are the ones you use in real life, from a career
in accounting to reading the date off a calendar. So, it’s important that you
get to know their characteristics. The following chart shows some important
sets of numbers that are subsets of the real numbers.
Subsets of Real Numbers
Natural Numbers
Whole Numbers
Integers
Rational Numbers
Irrational Numbers
The numbers you use to count.
1, 2, 3, 4, . . .
The natural numbers and 0.
0, 1, 2, 3, 4, . . .
The natural numbers, their opposites, and 0.
. . . , −4, −3, −2, −1, 0, 1, 2, 3, 4, . . .
Numbers that can be written as quotients of integers. In decimal
form, they can be written as terminating or repeating decimals.
8 ​ , 4
1  ​, 0.07, 5.​__
Ex. ​ __
67​ , −2​ __
2
9
Numbers that cannot be written as quotients of integers. In
decimal
__ form,
___ they do not terminate or repeat.
3
​  70 ​
Ex. ​√2 ​
, π, √
The diagram below can help you understand how the subsets of real numbers
are related. As you can see, all real numbers are either rational or irrational, and
the natural numbers are the smallest category in the diagram.
Real Numbers
Rational Numbers
© 2011 College Board. All rights reserved.
Integers
Whole Numbers
Natural Numbers
Irrational Numbers
EXAMPLE 1
Classify each of the following numbers as rational or irrational:
____
, 0.73625…, 0
9.​1234​
If the number is rational, list all other classifications that apply.
____
Rational numbers include repeating and terminating decimals, so 9.​1234​ is
rational. 0.73625… is neither repeating nor terminating, so it is irrational. 0 is
rational and it is also included in the set of integers and whole numbers.
____
Solution: 9.​1234​ is rational. 0.73625… is irrational. 0 is rational, as well as an
integer and a whole number.
Algebra 2, Unit 1 • Linear Systems and Matrices D-1
Unit 1
Practice D
Getting
Ready
Real Numbers continued
You’ve seen that there are a number of categories of real numbers. Although
each category has its own unique set of properties, there are a handful of
properties that they all share.
Properties of Real Numbers
Addition
Multiplication
a+b=b+a
ab = ba
a + (b + c) = (a + b) + c
a(bc) = (ab)c
Identity
a+0=a
a∙1=a
Inverse
a + (− a) = 0
1
a ∙ ​ __
a  ​= 1, (a ≠ 0)
Commutative
Associative
Distributive
a(b + c) = ab + ac
EXAMPLE 2
Name the property that justifies the equation −9 + 2 = 2 − 9.
This follows the pattern a + b = b + a, with a = −9 and b = 2.
Solution: commutative property of addition
This follows the pattern a ∙ 1 = a, with a = 5.
Solution: multiplicative identity property
The numbers a and −a are called additive inverses or opposites. Similarly, the
numbers a and __
​ a1  ​, where a ≠ 0, are called multiplicative inverses or reciprocals.
Additive inverses have a special feature in common. They share the same
absolute value. The absolute value of a given number can be thought of as the
distance between the number and 0 on a number line. The following notation
can be used for the absolute value of a number, x: ∙x∙.
D-2 Getting Ready Practice D Real Numbers
© 2011 College Board. All rights reserved.
EXAMPLE 3
Name the property that justifies the equation 5 ∙ 1 = 5.
Unit 1
Practice D
Getting
Ready
Real Numbers continued
EXAMPLE 4
Evaluate ∙−12∙.
There are 12 units between 0 and −12 on the number line, so the absolute value
of −12 is 12.
Solution: ∙−12∙ = 12
When evaluating an expression that involves an absolute value, make sure you
simplify inside the absolute value bars first. Take the absolute value only after
you have simplified the expression inside the bars to a single number.
EXAMPLE 5
Evaluate 2∙−2 ∙ 4 − 3∙ + 8.
Step 1:
Use the order of operations to simplify inside the
absolute value bars first.
Step 2:
Step 3:
Take the absolute value.
Use the order of operations to simplify.
2∙−2 ∙ 4 − 3∙ + 8
2∙−8 − 3∙ + 8
2∙−11∙ + 8
2 ∙ 11 + 8
22 + 8
30
© 2011 College Board. All rights reserved.
Solution: 2∙−2 ∙ 4 − 3∙ + 8 = 30
TRY THESE
Name all the ways to classify each number from among the following terms:
real, irrational, rational, whole, natural, integer.
__
3. √
​ 6 ​
1. −3
2. __
​ 1 ​
3
Name the additive and multiplicative inverses for each.
1  ​
4. −​ __
5. −2
4
6. __
​ 8 ​
3
Evaluate.
7. ∙−4∙
8. ∙−3(−7) − 8∙
9. −2 + ∙−3(5) + 9∙
Algebra 2, Unit 1 • Linear Systems and Matrices D-3
Unit 1
Practice D
Getting
Ready
Real Numbers continued
10. Ida is biking on a trail. The trail can be represented by the number line
below, with 0 representing a rest stop and one unit representing one mile.
Rest
Stop
–10 –9 –8 –7 –6 –5 –4 –3 –2 –1
0
1
2
3
4
5
6
7
8
9 10
© 2011 College Board. All rights reserved.
She is at position x when she sees a sign saying she is 4 miles from the
rest stop. Which point or points on the number line correspond to x?
Use mathematics including expressions or equations to explain your
­reasoning.
D-4 Getting Ready Practice D Real Numbers