Transcript 3 - NEHSMath

Section 1-6 Multiply and Divide Real Numbers
SPI 12B: Identify the reciprocal of a real number
Objectives:
• Apply properties of real numbers by multiplying
and dividing
Identity Property of Multiplication
For every real number n, 1 ∙ n = n
Multiplication Property of Zero
For every real number n, 0 ∙ n = 0
Multiplication Property of (- 1)
For every real number n, -1 ∙ n = - n
Example
1 ∙ 5 = 5 and 1 ∙ (-5) = -5
Example
35 ∙ 0 = 0 and (-35) ∙ 0 = 0
Example
-1 ∙ 5 = -5 and -1 ∙ (-5) = 5
Rules for Multiplying derived from the Properties
Numbers with the same sign
The product of 2 positive numbers or 2 negative numbers
is positive.
Example
2 ∙ 5 = 10 and (-2)(-5) = 10
Numbers with different signs
The product of a positive number and a negative numbers
is negative.
Example
(-2) ∙ 5 = -10 and 6 ∙ (-5) = -30
Simplify each expression.
a. –3(–11)
–3(–11) = 33
The product of two negative
numbers is positive.
b. –6 (3)
4
–6 (3 ) = – 18 The product of a positive number and
4
4 a negative number is negative.
= –4 1 Write – 18 as a mixed number.
4
2
Real- World Example
Temperature.
You can use the expression
a
5.5(1000 ) to calculate the changes in the air
temperature in degrees Fahrenheit for an
increase in altitude a, measured in feet. A hot
7200
air balloon starts on the ground and then
rises 7200 feet. Find the change in
temperature at the altitude of the balloon.
Use the expression –5.5( a ) to calculate the change in
1000
temperature for an increase in altitude a of 7200 ft.
a
–5.5(
) = –5.5 (7200)
1000
1000
Substitute 7200 for a.
= –5.5(7.2)
Divide within parentheses.
= –39.6°F
Multiply.
The change in temperature is –39.6°F.
Evaluate the Expression
Evaluate 5rs for r = –18 and s = –5.
5rs = 5(–18)(–5)
Substitute –18 for r and –5 for s.
= –90(–5)
5(–18) results in a negative number, –90.
= 450
–90(–5) results in a positive number, 450.
Exponents and Multiplication using Negative Numbers
Use the order of operations to simplify each expression.
Do you think the answers to a and b will be the same?
a. –0.24 = –(0.2 • 0.2 • 0.2 • 0.2) Write as repeated multiplication.
= –0.0016
Simplify.
b. (–0.2)4 = (–0.2)(–0.2)(–0.2)(–0.2)
= 0.0016
Write as repeated multiplication.
Simplify.
Rules for Dividing Real Numbers
Dividing numbers with the same sign
The quotient of 2 positive numbers or 2 negative numbers is
positive.
Example: 6 ÷ 3 = 2 and (-6) ÷ (-3) = 2
Dividing numbers with different signs
The quotient of a positive number and a negative numbers is
negative.
Example: -6 ÷ 3 = -2 and 6 ÷ (-3) = -2
Simplify each expression.
a. 70 ÷ (–5) = –14
The quotient of a positive
number and a negative
number is negative.
b. –54 ÷ (–9) = 6
The quotient of a
negative number and a
negative number is
positive.
Division using Reciprocal (Multiplicative Inverse)
For every real number a, there is a multiplicative inverse 1 such that
a
a ∙ 1 = 1.
a
Example: -5 ∙ 1 = 1
-5
Divide real numbers by using the reciprocal of a number.
KEEP the 1st term…
CHANGE the sign to multiply…
FLIP the 2d term ….
Evaluate p for p = 3 and r = – 3 .
r
2
p
=p÷r
r
3
3
= 2 ÷ (– 4
3
4
= 2 (– 3 )
= –2
4
Rewrite the equation.
)
3
3
Substitute 2 for p and – 4 for r.
4
3
Multiply by – 3 , the reciprocal of – 4 .
Simplify.