# Chapter 2 Working with Real numbers (visual)

## 90 terms · Algebra Structure and Method Book 1 / The Classic Richard G. Brown Mary P. Dolciani Robert H. Sorgenfrey William L. Cole

### Chapter *2*

Working with Real Numbers

### Lesson *2-1*

Basic Assumptions

### *Objective* 2-1

To use number properties
to simplify expressions.

### *closure properties*

For all real numbers a and b,
a + b is a unique real number.
ab is a unique real number.

The sum and product
of any two real numbers
are also real numbers.
Moreover, they are unique.
(This means there is one and only one possible answer
when you add or multiply two real numbers.)

### *commutative properties*

For all real numbers a and b,
a + b = b + a
ab = ba

The order
in which you add or multiply
two numbers
does not affect the result.

### *associative properties*

For all real numbers a, b, and c,
(a + b) + c = a + (b + c)
(ab)c = a(bc)

any three real numbers,
the grouping (or association)
of the numbers
does not affect the result.

### *terms*

In the sum a + b,
a and b are called terms.

### *factors*

Numbers
that are multiplied together
to produce a product.

### *Properties of Equality*

For all real numbers a, b, and c:

### *reflexive property* *of equality*

If a is a real number,
then a = a.

### *symmetric property* *of equality*

If a and b are real numbers,
and a = b,
then b = a.

### *transitive property* *of equality*

For all real numbers a, b, and c,
if a = b and b = c,
then a = c.

### *Objective* 2-2

a number line or

Zero (0).

There is
a unique real number 0
such that
for every real number a,
a + 0 = a and 0 + a = a.

### *property of opposites*

For every real number a,
there is
a unique real number −a
such that
a + (−a) = 0 and (−a) + a = 0.

A number and its opposite.

### *property of the opposite of a sum*

For all real numbers a and b,
−(a + b) = (−a) + (−b)

The opposite
of a sum
of real numbers
is equal to
the sum
of the opposites
of the numbers.

### From the properties of addition, prove the property of the opposite of a sum: (a + b) + [ −(a + b) ] = 0 (a + b) + [ (−a) + (−b) ] = 0

−(a + b) = (−a) + (−b)

### *Rules for Addition* 1. If a and b are both positive,

then a + b = |a| + |b|.

### *Rules for Addition* 2. If a and b are both negative,

then a + b = −(|a| + |b|).

### *Rules for Addition* 3. If a is positive and b is negative and a has the greater absolute value,

then a + b = |a| − |b|.

### *Rules for Addition* 4. If a is positive and b is negative and b has the greater absolute value,

then a + b = −(|b| − |a|).

then a + b = 0.

### Lesson *2-4*

Subtracting Real Numbers

### *Objective* 2-4

To subtract real numbers and
to simplify expressions involving differences.

### *Definition of Subtraction*

For all real numbers a and b,
the difference a − b
is defined by

a − b = a + (−b).

To subtract b,

a sum.

### When you find the opposite of a sum or a difference,

you change the sign of
each term of
the sum or difference.

No.

No.

a

Multiplication

### Lesson *2-5*

The Distributive Property

### *Objective* 2-5

To use the distributive property
to simplify expressions.

### *distributive property* *(of multiplication* *with respect to addition)*

For all real numbers a, b, and c,

a(b + c) = ab + ac
and
(b + c)a = ba + ca.

### *distributive property* *(of multiplication* *with respect to subtraction)*

For all real numbers a, b, and c,

a(b − c) = ab − ac
and
(b − c)a = ba − ca.

### By applying the symmetric property of equality,

the distributive properties of multiplication
can also be written in the following forms:

### *ab + ac = a(b + c)*

ba + ca = (b + c)a

### *ab − ac = a(b − c)*

ba − ca = (b − c)a

### *equivalent expressions*

Expressions that represent
the same number
for all values
of the variable
that they contain.

### *simplifying an expression*

Replacing an expression
containing variables
by an equivalent expression
with as few terms as possible.

### Lesson *2-6*

Rules for Multiplication

### *Objective* 2-6

To multiply real numbers.

One (1).

### *identity property* *of multiplication*

There is
a unique real number 1
such that
for every real number a,
a⋅1 = a and 1⋅a = a.

### When one (or at least one) of the factors of a produce is zero,

the product itself is zero.

### *multiplicative property of zero*

For every real number a,
a⋅0 = 0 and 0⋅a = 0.

### Multiplying any real number by −1 produces

the opposite of the number.

### *multiplicative property of −1*

For every real number a,
a(−1) = −a and (−1)a = −a.

### *property of opposites in products*

For all real numbers a and b,
(−a)(b) = −ab,
a(−b) = −ab,
and (−a)(−b) = ab.

### *Rules for Multiplication* 1.

If two numbers have the same sign,
their product is positive.

If two numbers have opposite signs,
their product is negative.

### *Rules for Multiplication* 2.

The product
of an even number
of negative numbers is positive.

The product
of an odd number
of negative numbers is negative.

### Lesson *2-7*

Problem Solving: Consecutive Integers

### *Objective* 2-7

To write equations
to represent relationships
among integers.

### *consecutive integers*

Numbers
obtained by
counting by ones
from any number
in the set of integers.

### *natural order* *of integers*

Order from least to greatest.

### *even integer*

An integer
that is the product of
2 and any integer.

### *odd integer*

An integer
that is not even.

### *consecutive even integers*

Integers obtained by
counting by twos,
beginning with
any even integer.

### *consecutive odd integers*

Integers obtained by
counting by twos,
beginning with
any odd integer.

### *multiple*

The product of
any real number
and an integer
is a multiple
of the real number.

### Consecutive multiples of a are

multiples of a
by consecutive integers.

Division

### Lesson *2-8*

The Reciprocal of a Real Number

### *Objective* 2-8

To simplify expressions
involving reciprocals.

### *reciprocals*

Two numbers
whose product is 1;
also called multiplicative inverses.

5 ·¹/₅ = 1

−1.25·(−0.8) = 1

### 0 has no reciprocal because

0 times any number is 0, not 1.

### *property of reciprocals*

For every nonzero real number a,
there is
a unique real number 1/a
such that
a⋅1/a = 1 and 1/a⋅a = 1.

### *property of* *the reciprocal of* *the opposite of a number*

For every nonzero number a,
1/−a = −1/a.

Read, "The reciprocal of -a is -1/a."

### *property of* *the reciprocal of a product*

For all nonzero numbers a and b,
1/ab = 1/a⋅1/b.

The reciprocal
of the product
of two nonzero numbers
is the product of
their reciprocals.

the number

### Lesson *2-9*

Dividing Real Numbers

### *Objective* 2-9

To divide real numbers and
to simplify expressions
involving quotients.

### *Definition of Division*

For every real number a
and every nonzero real number b,
the quotient a ÷ b, or a/b,
is defined by

a ÷ b = a⋅1/b.

To divide by a nonzero number,
multiply by its reciprocal.

a product.

### *Rules for Division*

If two numbers have the same sign,
their quotient is positive.
If two numbers have opposite signs,
their quotient is negative.

### Why can you never divide by zero?

Dividing by 0
would mean
multiplying by the reciprocal of 0.
But 0 has no reciprocal
(because 0 times any number is 0,
not 1).
Therefore, division by zero
has no meaning
in the set of real numbers.

Yes.

No.

No.

### Properties of division

For all real numbers, a, b, and c
such that c ≠ 0,

### *(a + b) / c = (a / c) + (b / c)*

and (a − b) / c = (a / c) − (b / c)

### *average*

The average of a set of numbers
is the sum of the numbers
divided by
the number of numbers.