Download 806.2.1 Order and Compare Rational and Irrational numbers and

806.2.1 Order and Compare Rational and Irrational numbers and
Locate on the number line
Rational Number ~ any number that can be made by dividing
one integer by another. The word comes from the word "ratio".
Examples:
1/2 is a rational number (1 divided by 2, or the ratio of 1 to 2)
0.75 is a rational number (3/4)
1 is a rational number (1/1)
2 is a rational number (2/1)
2.12 is a rational number (212/100)
-6.6 is a rational number (-66/10)
But Pi is not a rational number, it is an "Irrational Number".
Famous Irrational Numbers
Pi is a famous irrational number. People have calculated Pi to
over a quadrillion decimal places and still there is no pattern.
The first few digits look like this:
3.1415926535897932384626433832795 (and more ...)
The number e (Euler's Number) is another famous irrational
number. People have also calculated e to lots of decimal
places without any pattern showing. The first few digits look
like this:
2.7182818284590452353602874713527 (and more ...)
The Golden Ratio is an irrational number. The first few digits
look like this:
1.61803398874989484820... (and more ...)
Comparison Property ~ If you are given any two numbers a and
b, then there are three possible relationships between them.
Either:
a = b
a > b
a < b
•
We are comparing the two numbers, or putting them in order.
•
To read < and > remember, we read from left to right.
•
In < the left side is smaller than the right side of the symbol,
therefore the symbol is representing less than
•
In > the left side is bigger than the right side of the symbol,
therefore the symbol is representing greater than
•
An inequality is a mathematical sentence using < or > to
compare two expressions.
•
We can use any of the following inequality symbols to compare
numbers:
Symbol
<
>
≤
≥
≠
•
Read as
less than
greater than
less than or equal to
greater than or equal
to
not equal to
Along with =, we can make any sentence true by using the
appropriate symbol.
Step 1: To Compare Rational numbers, you must first make them all
into decimals.
The simplest method is to use a calculator.
Example: What is 5/8 as a decimal?
Type in 5 / 8 and hit the F
The answer should be 0.625
D button
If you can’t use a calculator, then you can do one of a couple
things:
1. Make an equivalent fraction over 10 or 100 if possible.
75
=
100
¾
2. Divide the numerator by the denominator to make a decimal.
Example: 1/3
Step 2: Line up your numbers, making sure you line up the decimals.
Example: 5/8, .125, .4
5/8 = .625
.125
3.4
Step 3: Bring all your decimal numbers out to the same place value
by adding zeros. Once you do this, it’s easy to see the order.
0.625
0.125
0.400
Step 4: List them in order using the correct inequality symbols.
0.125
< .4 < 5/8
For irrational numbers, you must estimate the number in
relationship to the other numbers using the largest place value
necessary.
Practice:
1. Write the following rational numbers in order from greatest to
least:
19
/25, 0.33, 0.68,
1 , 0.5
2. Write the following rational numbers in descending order:
9 , 64/16, 3.63, 25/8, 2.125
3. Write the following rational numbers in order from greatest to
least:
31
/6, 4.121, 38/9, 47/12, 16
Now, let’s do some practice ☺
806.2.1 ~ Order rational and irrational numbers
Directions: Write the following sets of rational and irrational numbers in order
from least to greatest.
1) 16 , 4 2 , 3.4, 3
5
2) 6.5, 6 3 , 16 ,
8
3) 1.25,
2
,
10
20
,
3
6)
π
6.3333…
2 , 1 3 , 1.875, 1 3
8
5
20 , -2 3 ,
5
10
,
3
7) -4.5, -4 1 , 7 ,
5
, -3.875,
9) 1.75,
5) 4 5 , 4.375, 4.3, 4 3 , 4.161616…
10) 5.838383…, 5 3 ,
8
9
4
4.182182…
29
,
8
4) 8 1 , 8.22, 8 1 , 8.3, 8.35235246…
5
14
,
5
8
23
,
5
5
8) 3 2 ,
3.9, 3 7
3.6
5 , 1.9, 1 2
3
8
25 ,
Locate rational/irrational numbers on a number line:
Let's think about where 4.5, 1.838383... and π
should be placed on a number line.
1.838383... is placed closer to the 2 because as a rounded
number it would be rounded to 2.
π is placed closer to the 3 because π is approximately 3.1416.
4.5 is halfway between 4 and 5.
Let's place rational and irrational numbers on a number line.
Draw a number line for each practice problem and place the
number given on the number line.
1. Place -1.4,
placement.
2 , and
on a number line and justify their
-1.4 is almost halfway between -1 and -2
(closer to -1)
2
is approximately 1.4 so is almost halfway
between 1 and 2
is closer to the 2 than the 3 but not near
the halfway point
2. Place 3.9, 9/3, and -0.3
on a number line and justify
their placement.
3.9 is approximately 4
9/3 is equal to 3
-0.3 is close to the halfway point between 0
and -1, closer to the 0
3. Place –16/8, -0.5, and
0.4444... on a number line and
justify their placement.
-16/8 is equal to -2
-0.5 is halfway between 0 and -1
0.444... is close to 0.5
Now, let’s practice! ☺
806.2.1 ∼ Locate Rational and Irrational numbers on a Number Line Practice
Put the following sets of numbers in order on the number line below each set.
1.) 2.3
2
2.)
3.)
7
5
2
1
4
16
-2.3
-2
5
2
4
4
2.09
17
8
24
2.5
-3
7
8
1.9
3.4
1
4
4
2
-
4
2
9
8
3
806.2.1 Quiz
1. Write the following rational numbers in ascending order:
3
6.5, 6 8 ,
20
16 ,
3 , 6.3333…
2. Write the following rational numbers in order from smallest
to largest:
1.25,
3.
2
3
3
, 1 8 , 1.875, 1 5
Compare the following rational numbers using the symbols < or >:
3
π, 5 , 0.827, .075,
1
4. Which of the following rational or irrational numbers belongs
between the 5 and the 6 on the number line below?
35
9 ,
5
23
16 , , 6.8,
4
9+ π
6