Download The Product of Two Real Numbers

1-7 Multiplication and Division of Real Numbers
Multiplication
The answer to a multiplication problem is called the product.
The numbers that are multiplied are called factors.
Example: 4 ∙ 3 = 12
4 and 3 are the factors. 12 is the product.
--------------------------------------------------------------------------------------------There are three types of problems using signed numbers:
 the product of two positive numbers
 the product of a negative number and a positive number
 the product of two negative numbers
--------------------------------------------------------------------------------------------Product of Two Positive Numbers
One positive is interpreted as “how much” and the other is interpreted as
“how many.”
Example: 5 ∙ 3
“How much?” 5 “How many 5s?” 3
5+5+5
OR
“How much?” 3 “How many 3s?” 5
3+3+3+3+3
Both sums are 15.
Adding a positive number over and over always gives a positive sum.
The product of two positive numbers is always positive.
--------------------------------------------------------------------------------------------Product of a Negative Number and a Positive Number (p. 56)
The negative is “how much” and the positive is “how many.”
Example: 5(–3)
“How much?” –3 “How many –3s?” 5 –3 + (–3) + (–3) + (–3) + (–3)
The sum is –15
Adding a negative number over and over always gives a negative sum.
The product of a positive number and a negative number is always negative.
---------------------------------------------------------------------------------------------
Product of Two Negative Numbers (p. 57)
First, we state the property of Multiplication by –1
Multiplying a number by –1 equals the opposite of the number.
3 ∙ −1 = −3
7 ∙ −1 = −7
By this property, −8 ∙ −1 = 8
--------------------------------------------------------------------------------------------Any negative number can be written as the product of its opposite and –1.
−4 = 4 ∙ −1
− 11 = 11 ∙ −1
− 9 = 9 ∙ −1
So, the product of two negative numbers can always be written as the product
of a negative times a positive times –1.
−5 ∙ −6 = −5 ∙ (6 ∙ −1)
− 9 ∙ −3 = −9 ∙ (3 ∙ −1)
By the Associative Property,
−5 ∙ (6 ∙ −1) = (−5 ∙ 6) ∙ −1 = −30 ∙ −1 = 30
−9 ∙ (3 ∙ −1) = (−9 ∙ 3) ∙ −1 = −27 ∙ −1 = 27
The product of two negative numbers is always positive.
--------------------------------------------------------------------------------------------The Product of Two Real Numbers
The product of two real numbers with different signs is always negative.
The product of two real numbers with the same sign is always positive.
--------------------------------------------------------------------------------------------Multiplying More Than Two Numbers
 Problem contains an even number of negatives:
Each pair of negative numbers gives a positive product.
(–5)(–2)(–2)(3)(–4) = [(–5)(–2)] [(–2)(–4)] (3)
= (10)(8)(3) = (80)(3) = 240
A product with an even number of negatives is positive.
 Problem contains an odd number of negatives:
With an odd number of negatives, there will always be one extra negative
after pairing. The extra negative makes the product negative.
(–4)(–3)(–2) = [(–4)(–3)] (–2)
= (12)(–2) = –24
(–5)(–2)(–2)(–4)(–3) = [(–5)(–2)] [(–2)(–4)] (–3)
= (10)(12)(–3) = (120)(–3) = –360
A product with an odd number of negatives is negative.
--------------------------------------------------------------------------------------------Multiplicative Property of Zero (p. 56)
For any real number 𝑎, 0 ∙ 𝑎 = 𝑎 ∙ 0 = 0
NOTE: When multiplying, if any factor is zero, the product is zero.
(–37)(423)(0)(–55)(–3.7) = 0
--------------------------------------------------------------------------------------------Division
The result of dividing the real number a by a nonzero real number b is
called the quotient of a and b.
𝑎÷𝑏
𝑎
or
𝑏
The number you’re dividing by, b, is called the divisor.
The number you’re dividing into, a, is called the dividend.
----------------------------------------------------------------------------------Reciprocals
The reciprocal or multiplicative inverse of a number is formed by
interchanging the numerator and denominator. It has the same sign as the
original number. The product of a number and its reciprocal is 1.
Example: The reciprocal of
2
5
is
The reciprocal of 5 is
5
2
1
5
.
The reciprocal of −
.
The reciprocal of −
3
4
1
3
4
is − .
3
is − 3.
---------------------------------------------------------------------------------------------
When dividing by a fraction, you may change division to multiplication and
invert the divisor. This results in an expression equivalent to the original
expression. Thus, division can be defined as multiplying by the reciprocal of
the divisor.
𝑎÷𝑏 =𝑎∙
1
𝑏
--------------------------------------------------------------------------------------------Sign Rules for Division
Since division of two real numbers can be defined in terms of multiplying
two real numbers, the rules for dividing signed numbers are the same as for
multiplying.
The quotient of two real numbers with different signs is always negative.
The quotient of two real numbers with the same sign is always positive.
8
−2
= −4
−54
6
= −9
−15
−3
=5
18
6
=3
--------------------------------------------------------------------------------------------Division Involving Zero (p. 60)
Division of any number by zero is undefined.
𝑎
= undefined
0
Zero divided by any nonzero number is zero.
0
𝑎
=0
𝑎≠0
--------------------------------------------------------------------------------------------Multiplication and Algebraic Expressions
Negative Signs and Parentheses
If a negative sign precedes parentheses, you may remove the parentheses if
you change the sign of every term inside the parentheses.
Examples:
−(11𝑥 + 5) = −11𝑥 − 5
− (−11𝑥 − 5) = 11𝑥 + 5
You may also think of the negative sign as representing a factor of –1. The
parentheses are removed by distributing the –1 to each term.
−1(11𝑥 + 5) = −1 ∙ 11𝑥 + −1 ∙ 5 = −11𝑥 − 5
−1(−11𝑥 − 5) = −1 ∙ −11𝑥 + −1 ∙ −5 = 11𝑥 + 5
--------------------------------------------------------------------------------------------Simplifying Algebraic Expressions
Examples: Simplify.
−2(3𝑥 ) = (−2 ∙ 3)𝑥 = −6𝑥
Examples:
−3(2𝑥 − 5) = −3(2𝑥 ) + −3 ∙ −5 = −6𝑥 + 15
−(3𝑦 − 8) = −3𝑦 + 8
Example: Simplify 5(2𝑦 − 9) − (9𝑦 − 8)
Remove parentheses: Distributive Property 10𝑦 − 45 − (9𝑦 − 8)
Remove parentheses: negative precedes parentheses 10𝑦 − 45 − 9𝑦 + 8
Combine like terms (10𝑦 − 9𝑦) + (−45 + 8) = 1𝑦 + (−37) = 𝑦 − 37
--------------------------------------------------------------------------------------------Determining If a Number is a Solution of an Equation
We have expanded the possible values from the natural numbers to the real
numbers. The technique is the same (evaluate both expressions to see if they
have the same value) but you must be more careful now in your calculations
because of negative numbers, fractions and decimals.
Example: Determine if –3 is a solution of 6𝑥 + 14 = −7 − 𝑥
Substitute –3 for x:
6(−3) + 14 = −7 − (−3)
−18 + 14 = −7 + 3
−4 = −4
Both expressions have the same value, –3 is a solution of the equation.