Transcript Multiplying and Dividing Real Numbers

Multiplying and Dividing Real
Numbers
Section 1-6
Goals
Goal
• To Find products and
quotients of real numbers.
Rubric
Level 1 – Know the goals.
Level 2 – Fully understand the
goals.
Level 3 – Use the goals to
solve simple problems.
Level 4 – Use the goals to
solve more advanced problems.
Level 5 – Adapts and applies
the goals to different and more
complex problems.
Vocabulary
• Multiplicative Inverse
• Reciprocal
Multiplying Real Numbers
When you multiply two numbers, the signs of the
numbers you are multiplying determine whether
the product is positive or negative.
Factors
3(5)
Product
Both positive
15
Positive
3(–5) One negative
–15
Negative
15
Positive
–3(–5) Both negative
This is true for division also.
Rules for Multiplying and
Dividing
Example: Multiplying and
Dividing Real Numbers
Find the value of each expression.
A.
–5
The product of two numbers
with different signs is negative.
12
The quotient of two numbers
with the same sign is positive.
B.
Example: Multiplying and
Dividing Real Numbers
Find the value of each expression.
C.
Multiply.
The quotient of two numbers
with different signs is negative.
Your Turn:
Find the value of each expression.
a. 35  (–5)
–7
The quotient of two numbers
with different signs is negative.
b. –11(–4)
44
The product of two numbers
with the same sign is positive.
c. –6(7)
–42
The product of two numbers with different
signs is negative.
Reciprocals
• Two numbers are reciprocals if their product is 1.
• A number and its reciprocal are called
multiplicative inverses. To divide by a number,
you can multiply by its multiplicative inverse.
• Dividing by a nonzero number is the same as
Multiplying by the reciprocal of the number.
Reciprocals
Multiplicative inverses
10 ÷ 5 = 2
1 10
10 ∙ 5 = 5 = 2
Dividing by 5 is the same as multiplying by the
reciprocal of 5, .
Helpful Hint
You can write the reciprocal of a number by
switching the numerator and denominator. A whole
number has a denominator of 1.
Example: Dividing with
Example
2 Dividing by Fractions
Fractions
Divide.
To divide by
, multiply by
.
Multiply the numerators and
multiply the denominators.
and
have the same sign,
so the quotient is positive.
Example: Dividing with
Fractions
Divide.
Write
as an improper fraction.
To divide by
and
, multiply by
have different signs,
so the quotient is negative.
.
Your Turn:
Divide.
Write
as an improper fraction.
To divide by
, multiply by
.
and –9 have the same signs,
so the quotient is positive.
Your Turn:
Divide.
To divide by
, multiply by
.
Multiply the numerators and
multiply the denominators.
and
have different signs,
so the quotient is negative.
Check It Out! Example 2c
Divide.
Write
as an improper fraction.
To divide by
multiply by
The signs are different, so the
quotient is negative.
.
Zero
• No number can be multiplied by 0 to give a
product of 1, so 0 has no reciprocal.
• Because 0 has no reciprocal, division by 0 is
not possible. We say that division by zero is
undefined.
• The number 0 has special properties for
multiplication and division.
Example: Multiplying &
Dividing with Zero
Multiply or divide if possible.
0
A.
15
Zero is divided by a nonzero number.
0
The quotient of zero and any nonzero
number is 0.
B. –22  0
undefined
A number is divided by zero.
C. –8.45(0)
0
A number is multiplied by zero.
Division by zero is undefined.
The product of any number and 0 is 0.
Your Turn:
Multiply or divide.
a.
Zero is divided by a nonzero number.
0
b.
0÷0
The quotient of zero and any nonzero
number is 0.
A number is divided by zero.
undefined
c. (–12.350)(0)
0
A number divided by 0 is undefined.
A number is multiplied by zero.
The product of any number and 0 is
0.
Example: Application
The speed of a hot-air balloon is 3 34 mi/h. It
travels in a straight line for 1 1 hours before
3
landing. How many miles away from the liftoff
site will the balloon land?
Find the distance traveled at a rate of 3
To find distance, multiply rate by time.
rate
times
3
3
4

3
4
time
11
3
1
mi/h for 13 hour.
Example: Continued
3
1 = 15 4
3
• 1
•
4
3
4
3
15(4)
= 60
4(3)
12
=5
Write 3
3
and 1 1
4
3
as improper fractions.
Multiply the numerators and
multiply the denominators.
3
3
4
and 1 1 have the same sign, so
3
the quotient is positive.
The hot-air balloon lands 5 miles from the liftoff site.
Your Turn:
What if…? On another hot-air balloon trip, the
wind speed is 5.25 mi/h. The trip is planned for 1.5
hours. The balloon travels in a straight line parallel
to the ground. How many miles away from the
liftoff site will the balloon land?
5.25(1.5)
Rate times time equals distance.
= 7.875 mi
Distance traveled.
Joke Time
• What’s a fish with no eyes?
• A fsh.
• What do you call a nun that walks in her sleep?
• A roamin’ Catholic.
• Why did the cookie go to the doctor?
• Because he was feeling crummy.
Assignment
• 1.6 Exercises Pg. 49 – 51: #8 – 74 even.