Download Lesson Notes 1.1doc.jnt

Lesson Notes 1.1
Math 10H
Classifying & Estimating Real Numbers
Recall from last year that the all numbers can be classified as either “Real Numbers” or
“Imaginary Numbers”.
For most human tasks, real numbers offer an adequate description of data.
2
1
and
are meaningless to a person counting stones, but essential to
Fractions such as
3
2
a person comparing the sizes of different collections of stones.
Negative numbers such as −3 and −5 are meaningless when measuring the mass of an
object, but essential when keeping track of monetary debits and credits.
Similarly, imaginary numbers have essential concrete applications in a variety of
sciences and related areas such as signal processing, control theory, electromagnetism,
fluid dynamics, quantum mechanics, cartography, and vibration analysis.
In high school mathematics, we will work only with the Real Number Set, which can be
split into two subsets: “Rational Numbers” and “Irrational Numbers”.
The Real Numbers:
All numbers that can be placed somewhere on the infinitely long number line.
( )
Real Number Subset II: Irrational Numbers Q :
p
, where p and q are integers,
q
( q ≠ 0 ) . In other words, an irrational number cannot be represented as a simple fraction.
You can also remember that irrational numbers, in their decimal form, cannot be
represented as terminating or repeating decimals.
π = 3.1415926536897932384626433833795……
Some examples:
http://www.popsci.com/node/47646/?cmpid=enews081210
7 = 2.645751311…..
The real numbers that cannot be expressed as a fraction
Real Number Subset I: Rational Numbers ( Q ) :
p
, where p and q are integers,
q
( q ≠ 0 ) . In other words, a rational number can be represented as a simple fraction. You
can also remember that rational numbers, in their decimal form, can be represented as
terminating or repeating decimals.
Some examples:
4
10.8
3
5
2.6
4.93859271
Special sets within the rational number set are: natural numbers, whole numbers and
integers.
The real numbers that can be expressed as a fraction
Natural Numbers ( N )
The most exclusive set of Rational numbers, the Natural Numbers (or counting
numbers) include only the following numbers:
1, 2, 3, 4, 5, 6, ……….
Whole Numbers ( W )
The second most exclusive set of Rational numbers, the Whole Numbers simply add
zero to the natural number set:
0, 1, 2, 3, 4, 5, 6, ……
Integers ( Z )
This set of numbers just adds all Negative Integers to the Whole Number set:
……. -6, -5, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, …….
Imaginary Numbers
Okay, it’s time for us to start putting numbers into their categories……
Name all of the sets to which each of the following belong.
1.
11
2.
4.
6
5.
7.
2.35
8.
10.
6.225225225....
13. −
9
16
2
5
(Use: Q , Q , Z , W , N )
−5
3.
6.
−2
4.389685....
9.
9.03754927
11.
6.252252225....
12.
3
14.
3.14
15.
9
8
−0.12
Use your calculator to change each of these Real Numbers into decimals (round to nearest
thousandth) and label as Q (Rational - decimal stops or repeats) or Q (Irrational decimal neither stops nor repeats).
1.
4.
1
3
3
2.
12
5.
4
9
3.
1.96
6.
−
10
π
All Real Numbers can be placed onto the number line. The ability to estimate an
irrational number without a calculator allows us to understand where it “fits” on the
number line.
1.
7
2.
Between ___ and ___
4.
3
− 20
Between ___ and ___
5.
Between ___ and ___
3.
3
15
Between ___ and ___
75
6.
Between ___ and ___
3
75
Between ___ and ___
Now place the corresponding letter of these Real Numbers on the number line below.
A. −2
1
3
B.
8
C. − 16.5
D.
4
9
E. 1 + 5
Absolute Value
The absolute value of a real number is its numerical value ignoring its sign.
The absolute value is defined as the distance from zero on the number line.
Straight brackets around an expression indicate the absolute value function.
Eg:
5 reads as “the absolute value of 5”
Distance cannot be a negative number. Both 5 and -5 are five units from zero.
0
-5
5
So, 5 = 5
−5 = 5
and
Simplify these:
a)
− 14 =
b)
10 − 6 =
c)
11 − ( −6 ) =
d)
11 − ( −6 ) =
e)
− −4 =
f)
2 − 8 + −3 =
g)
7 + −3 +
3−7
=
h) − 4 × −2 − 3
− 5× 2
=
Solve for x:
a)
x = 5
b)
x = − 19
c)
2x = 12
d)
3 − x = 10
e)
2x -1 = 13
f)
2x
+5 = 9
3