Section 4.6 And 4.9: Rational Numbers and Scientific Notation

Section 4.6: Rational Numbers By Ms. Dewey-Hoffman October 20 th , 2008

Rational Numbers
Any number that can be written as a quotient of two integers. a/b = a/b = quotient
Rational Numbers Integers Whole Numbers

Writing Rational Numbers
There are three ways to write negative rational numbers.
For each rational number, there are an unlimited number of equivalent fractions.

Write Three Equivalent Fractions
1/3 =
-4/5 =
5/8 =
Possible answers: 2/6, -2/-6, and –1/-3 Possible answers: -8/10, -4/5, and 4/-5 Possible answers: 10/16, -10/-16, and -5/-8

You can graph Rational Numbers…
½
-8/10
1
-0.2
0 1 -1

Evaluating Fractions Containing Variables
Remember that a fraction bar is a grouping symbol!
First : Substitute for the variables.
Second : Simplify the expressions in the numerator and denominator.
Third : Write the fraction in simplest form.

Example:
1 + 9 + 2 / 2 – 5 = ???
First: No variables to replace. Next.
Second: Simplify the top and bottom.
1 + 9 + 2 = 12
2 – 5 = -3
Third: Write the fraction in simplest form. 12/-3 = -4

Acceleration…
The speed of a car changes from 37ft/s 2 to 102ft/s 2 in five seconds. What is its acceleration in feet/second 2 (ft/s 2 )? Use the formula a = f – i/t, where a = acceleration, f = is final speed, i = initial speed and t = time.
Solve!

Now try these!
a = 6 and b = -5, for all expressions.
a + b/ -3 =
7 – b/ 3a =
a + 9/b =

Switching Gears Section 4.9: Scientific Notation Its Still October 20 th =] Put your THINKING CAPS on.

Patterns in Scientific Notation
5 x 10 4 = 5 x 10,000 = 50,000
5 x __ = 5 x 1,000 = 5,000
5 x 10 2 = 5 x ___ = 500
5 x 10 1 = 5 x 10 = 50
5 x 10 0 = 5 x 1 = 5
5 x 10 __ = 5 x .1 = 0.5
5 x 10 -2 = 5 x ___ = 0.05
_ x 10 -3 = 5 x .001 = 0.005
5 x 10 -4 = 5 x .0001 = 0.0005

Did You See It?
5 x 10 4 = 5 x 10,000 = 50,000
5 x 10 3 = 5 x 1,000 = 5,000
5 x 10 2 = 5 x 100 = 500
5 x 10 1 = 5 x 10 = 50
5 x 10 0 = 5 x 1 = 5
5 x 10 -1 = 5 x .1 = 0.5
5 x 10 -2 = 5 x .01 = 0.05
5 x 10 -3 = 5 x .001 = 0.005
5 x 10 -4 = 5 x .0001 = 0.0005

Scientific Notation
Is a shorthand way of writing numbers using powers of 10. (Exponents!)
Scientific notation lets you know the size of a number without having to count digits.
You can write a number in scientific notation as the product of two factors.
7,500,000,000,000 = 7.5 x 10 12
The exponent is the number of times the decimal is moved so that it lies between the new ones and tenths place and the number to the left of the decimal is between 1 and 10.

Scientific Notation
10 to the 3 rd power means the numbers in the thousands.
10 to the 6 th power means the numbers is in the millions.
10 to the 9 th power means the number is in the billions.

Visitors to the Statue of Liberty
About 4,200,000 people visit the Statue of Liberty every year. Write this number in scientific notation.
Move the decimal point to get a decimal greater than 1 but less than 10.
4,200,000  4.200000
Drop the zeros after the 2
4.2
The decimal point removes 6 places to the left.
Use 6 as the exponent of 10.
There are 4.2 x 10 6 visitors every year.

Convert to scientific notation.
54,500,000
723,000
602,000,000,000
0.00021
0.00000005
0.0000000000803

From Scientific to Standard Notation
You can change expressions from scientific notation to standard notation by simplifying the product of the two factors.
8.9 x 10 5 =
Add zeros while moving the decimal point.
Rewrite in standard notation.
890,000

Write in Standard Notation
2.71 x 10 -6 =
3.21 x 10 7 =
5.9 x 10 -8 =
1.006 x 10 10 =
Hint: negative exponents make numbers very small, where as positive exponents make numbers very large.

Comparing and Ordering
You can compare and order numbers using scientific notation.
First, compare the powers of 10.
Then, compare the decimals.

Compare and Order These:
0.064 x 10 8 , 312 x 10 2 , and .58 x 10 7
Write each in scientific notation.
6.4 x 10 6 , 3.12 x 10 4 , and 5.8 x 10 6
Order the powers of 10. Arrange the decimals with the same power of 10 in order.
3.12 x 10 4 , 5.8 x 10 6 , and 6.4 x 10 6
Write the original numbers in order.
312 x 10 2 , .58 x 10 7 , and 0.064 x 10 8

Calculating with Scientific Notation
You can multiply numbers in scientific notation using the Rule for Multiplying Powers with the Same Base.
In this case, our SAME BASE, is 10.
Did you notice?
So get multiplying!

Multiplying Scientific Notation
3 x 10 -7 and 9 x 10 3 (Multiply and express result in scientific notation)
(3 x 10 -7 )(9 x 10 3 ) = 3 x 9 x 10 -7 x 10 3
27 x 10 -7 x 10 3
27 x 10 -4
2.7 x 10 1 x 10 -4
2.7 x 10 -3

Multiply:
(4 x 10 4 )(6 x 10 6 )
(7.1 x 10 -8 )(8 x 10 4 )

Assignment #26
Two Handouts.
Do the Odd Problems.
Start Now…
If there is time.

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Section 4.6 And 4.9: Rational Numbers and Scientific Notation

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Section 4.6 And 4.9: Rational Numbers and Scientific Notation — Presentation Transcript