Download Topics - Semester 2

SLEs
Investigate some of the approximations to π which have been used.
11 Mathematics C
Topics - Semester 1
•
Real and Complex Number Systems I
•
Vectors and Applications I
•
Matrices and Applications I
•
Introduction to Groups
•
Structures and Patterns I
•
Real and Complex Number Systems II
Topics - Semester 2
•
Vectors and Applications II
•
Structures and Patterns II
•
Matrices and Applications II
•
Dynamics I
•
Periodic and Exponential Functions I
TOPIC 1
Real and Complex Number
Systems I
(4 weeks)
Subject Matter
Structure of the Real Number System including:
• rational numbers
• irrational numbers
Simple manipulation of surds
Unit 1
Real and Complex
Number Systems I
Real
Rational (a/b)
Integers
(…-2,-1,0,1,2,…)
Non-integers
(¼,-3.453,43/17,…)
Irrational (a/b)
Surds
(5,5.17,-17…)
Transcendental
(π ,e)
Real numbers are any numbers which can be placed
on the real number line.
Rational numbers are any numbers which can be
expressed as a ratio p where p and q are integers
q
All integers, terminating decimals or recurring
decimals are rational
-5
-4
-3
-2
-1
0
1
2
3
4
5
Get real!

i
Get rational!
Get real!

i
Model
Express each of the following as fractions:
(a) 0.454545….
(b) 0.14676767…..
(a) 0.454545….
Let x = 0.454545….
 100x = 45.454545….
100x – x = 45
99x = 45
x = 45/99
= 5/11
(b) 0.14676767…..
Let
x = 0.14676767…..
 10000x = 1467.676767…..
100x = 14.676767…..
9900x = 1453
x = 1453/9900
Page 16
Ex 1.3
Surds
e.g .
3
ab 
a
72 
36  2

36

6
54 
3
a
b
b
2
2
27 3 2
 33 2

6.25 
a
b
6



1
4
25
4
25
4
5
2
 2.5
Write as a surd 4 5
 16 5
 80
Simplify
80  12 
75 
45
 16 5  4 3  25 3  9 5
 4 5  2 3 5 3 3 5
 5 3 3
Simplify
80  12 
75 
45
 16 5  4 3  25 3  9 5
 4 5  2 3 5 3 3 5
 5 3 3
MODEL
Simplify (a ) (3  5 )( 2  3 )
(b) (3  5 )(3  5)
(a ) (3  5 )( 2  3 )
 6  3 3  2 5  15
(b) (3  5 )(3  5)


93 5  3 5 5
4
Page 21
Ex 1.4
Express each of the following with rational denominators :
(a)
3
2
(b)
4
3 2
(c )
5 2
3 2
3
(a)
2
3
2


2
2
3 2

2
4
(b)
3 2
4
3 2


3 2 3 2
4(3  2 )

92
4(3  2 )

7
5 2
(c )
3 2
5 2 3 2


3 2 3 2
(5  2 )(3  2 )

92
15  5 2  3 2  2

7
13  2 2

7
Page 26
Ex 1.5
Inequalities
Model
Solve and graph
(a) 2 x  3  5
1 2 x
b 
7
3
2 x
c 
3
x
(a) 2 x  3  5
2x
8
x
4
1 2 x
b 
 7
3
1  2 x  21
 2 x  20
x   10
c 
2 x
 3
x
Consider x  0
2  x  3x
x0
Consider
2  x  3x
2
 2x
2
 2x
1
 x
1
 x
but x  0
therefore 0  x 1
 0  x 1
therefore no solution when x  0
Page 34 Ex 1.6
1 (second column)
3(first column)
Graphing Inequalities
• Solve and graph
• x – 10 < 4x – 2 ≤ 2x + 8
x – 10 < 4x – 2 ≤ 2x + 8
x – 10 < 4x – 2 and 4x – 2 ≤ 2x + 8
-3x <
8 and 2x
≤
10
x
> -2 ⅔ and x
≤
5
Page 34
Ex 1.6
2 c-j
Absolute Value
-3 = 3
2-8 = -6 = 6
4-2 - 5-9 = 2 – 4 = -2
Page 34
Ex 1.6 4
Graphing Absolutes
Solve and graph:
(a) 2x+4 = 10
(b) x-3 ≥ 4
(c) 3-x < 4
(d) 3x-2≤ 1
(a) 2x+4 = 10
 2x+4 = -10 or
2x = -14 or
x = -7 or
2x+4 = 10
2x = 6
x = 3
(b) x-3 ≥ 4
 x-3 ≤ -4 or x-3 ≥ 4
x ≤ -1 or x ≥ 7
(c) 3-x < 4
 3-x > -4
-x > -7
x < 7
and 3-x < 4
and -x < 1
and
x > -1
(d) 3x-2  1
 3x-2  -1 and
3x  1 and
1
x  3 and
3x-2  1
3x  3
x 1
Page 34 Ex 1.6
5a-c,7,8
Solve and graph
| 16+4x | ≤ 5-7x
(P35 No 9h)
If the question said:
| 16+4x | ≤ 3
then you would say:
i.e. 16 + 4x ≥ -3 and 16 + 4x ≤ 3
Solve and graph
| 16+4x | ≤ 5-7x
(P35 No 9h)
| 16+4x | ≤ 5-7x
i.e. 16 + 4x ≥ -(5-7x) and 16 + 4x ≤ 5 - 7x
16 + 4x ≥ -5 + 7x and 16 + 4x ≤ 5 - 7x
21
≥
x
≤
3x and
7
and
11x ≤ -11
x ≤ -1
Page 34
Ex 1.6 9
Page 3
Ex 1.1
Symbol
Meaning

is an element of
b  M means that b is an element of the set M where M = {a,b,c,…}

for all
2x is even x where x is a positive integer
:
such that
{x: x is even} means the set of all x such that x is even

there exists
A : bA means there exists a set A such that b is an element of A
*
A defined binary operation
e.g. x*y = 2x+y – 3
 4*5 = 2x4 + 5 – 3 = 10
Laws of Addition on the set of integers (J)
• Closure Law
• The sum of any two integers results in another
number which is also an integer.
•  a,b ∈ J,
a+b=c
where c ∈ J
• Commutative Law
• The order in which numbers are added does not
alter their sum.
•  a,b ∈ J,
a+b=b+a
• Associative Law
• No matter how numbers are associated in addition,
it does not alter their sum.
•  a,b,c ∈ J,
(a + b) + c = a + (b + c)
• Identity Law of Addition
• 0 is the identity element for addition.
• When 0 is added to any number, the sum is the
same as that number.
•  a ∈ J,
a+0 = 0+a = a
• Additive Inverse Law
• For every integer, a, there exists another unique
number, -a, such that they add to give 0 (the
identity element)
•  a ∈ J,  -a ∈ J
a + -a = 0
Page 7
Ex 1.2