SLEs

Investigate some of the approximations to π which have been used.

11 Mathematics CTopics - 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)

Structure of the Real Number System including:• rational numbers• irrational numbers

Simple manipulation of surds

Subject Matter

Unit 1

Real and Complex Number Systems I

Real

Rational (a/b) Irrational (a/b)

Integers Non-integers SurdsTranscendental

(…-2,-1,0,1,2,…) (¼,-3.453,43/17,…) (5,5.17,-17…) (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 qAll integers, terminating decimals or recurring decimals are rational                                                        

-5 -4 -3 -2 -1 0 1 2 3 4 5

i

Get real!

i

Get real!Get rational!

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

5.223

22754

26

236

625.623672..

3

25333

425

425

41

ge

baabba

ba

80

516

54

surdaasWrite

335

53353254

5932534516

45751280

Simplify

335

53353254

5932534516

45751280

Simplify

)53)(53()(

)32)(53()(

b

aSimplify

MODEL

4553539

)53)(53()(

1552336

)32)(53()(

b

a

Page 21 Ex 1.4

2325

234

23

)(

)(

)(

:

c

b

a

atorsmindenorationalwithfollowingtheofeachExpress

223

22

23

23)(

a

7)23(4

29)23(4

2323

234

234)(

b

722137

2232515

29)23)(25(

2323

2325

2325)(

c

Page 26 Ex 1.5

Inequalities

ModelModel Solve and graph

32

7321

532)(

xxc

xb

xa

482532)(

xxxa

102022121

7321

xxx

xb

10

011

22223232

00

32

0xwhensolutionnotherefore1x0therefore

x

xbutxxxxxxxx

xConsiderxConsider

xxc

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 = 32-8 = -6 = 64-2 - 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+4 = 10 2x = -14 or 2x = 6 x = -7 or 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 and 3-x < 4 -x > -7 and -x < 1 x < 7 and x > -1

(d) 3x-2 1 3x-2 -1 and 3x-2 1 3x 1 and 3x 3 x and x 13

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 | ≤ 5-7xi.e. 16 + 4x ≥ -(5-7x) and 16 + 4x ≤ 5 - 7x 16 + 4x ≥ -5 + 7x and 16 + 4x ≤ 5 - 7x 21 ≥ 3x and 11x ≤ -11 x ≤ 7 and x ≤ -1

Page 34 Ex 1.6 9

Page 3 Ex 1.1

Symbol Meaning

is an element ofb M means that b is an element of the set M where M = {a,b,c,…}

for all2x 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 existsA : bA means there exists a set A such that b is an element of A

* A defined binary operatione.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