This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr.
The basic operations with real numbers are presented in this chapter. The concept of absolute value is discussed both geometrically and symbolically. The geometric presentation offers a visual understanding of the meaning of |x|. The symbolic presentation includes a literal explanation of how to use the definition. Negative exponents are developed, using reciprocals and the rules of exponents the student has already learned. Scientific notation is also included, using unique and real-life examples.Objectives of this module: be able to multiply and divide signed numbers.
Overview
- Multiplication of Signed Numbers
- Division of Signed Numbers
Multiplication of signed numbers
Let us consider first the product of two positive numbers.
Multiply:
.
means
.
This suggests that
.
More briefly,
.
Now consider the product of a positive number and a negative number.
Multiply:
.
means
.
This suggests that
More briefly,
.
By the commutative property of multiplication, we get
More briefly,
.
The sign of the product of two negative numbers can be determined using the following illustration: Multiply
by, respectively,
. Notice that when the multiplier decreases by 1, the product increases by 2.
We have the following rules for multiplying signed numbers.
Rules for multiplying signed numbers
To multiply two real numbers that have
- the
same sign , multiply their absolute values. The product is positive.
-
opposite signs , multiply their absolute values. The product is negative.
Sample set a
Find the following products.
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Practice set a
Find the following products.
Division of signed numbers
We can determine the sign pattern for division by relating division to multiplication. Division is defined in terms of multiplication in the following way.
If
, then
.
For example, since
, it follows that
.
Notice the pattern:
Since
, it follows that
The sign pattern for division follows from the sign pattern for multiplication.
- Since
, it follows that
, that is,
- Since
, it follows that
, that is,
- Since
, it follows that
,
that is,
- Since
, it follows that
, that is
We have the following rules for dividing signed numbers.
Rules for dividing signed numbers
To divide two real numbers that have
- the
same sign , divide their absolute values. The quotient is positive.
-
opposite signs , divide their absolute values. The quotient is negative.
Sample set b
Find the following quotients.
Practice set b
Find the following quotients.
Sample set c
Find the value of
.
Using the order of operations and what we know about signed numbers, we get
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Practice set c
Exercises
Find the value of each of the following expressions.
Exercises for review