Download Isosceles: two sides/angles are equal

-1-
CHAPTER 1: REAL NUMBERS
1) The Pythagorean Theorem
a) The square
Area of a square:
A = s2
Where, A = area and s = side length
E.g.
A = (5 cm)2 = 25 cm2
5 cm
A = (b cm)2 = b2 cm2
b cm
When the area of a square is known, how can you
find the side length?
» by taking the square root of the area
s=
A
-2-
E.g.
s=
81cm2
= 9 cm
d
cm2
s=
d 2 cm2
= d cm
17
cm2
s=
17cm2
=
81
cm2
2
17
17
cm
≈ 4.123105626…
If A is not a perfect square number (1, 4, 9, 16, 25,
36, 49…) it is preferable to leave the answer in
radical form.
Squaring a number and taking the square root
are INVERSE operations.
» Taking the square root of a number means
finding the side length of a square whose area is that
number.
E.g.
Calculate
23 ≈ ______________
23
units2
each side ≈ 4.7958…
-3-
b) Triangles
Equilateral: all sides/angles are equal
Isosceles: two sides/angles are equal
Right: contains a right angle
Scalene: no sides equal
Isosceles right:
Area of a triangle
A = base x height = b x h
2
2
h
h
b
c) Disk
Area of a disk = r2
b
-4-
Examples
Find the area of the following figures:
1.
3 cm
8 cm
A=bxh
2
A=8x3
2
A = 12 cm2
2.
r
r = 5 cm
A = r2
A = 52 (use  key on calculator)
A ≈ 78.54 cm2
(78.539816339744830961566084581988…)
-5-
d) Right triangles
The longest side of a right triangle is called the
HYPOTENUSE; the two sides that form the right
angle are called the LEGS.
In a right triangle, the square of the length of the
hypotenuse is equal to the sum of the squares of the
lengths of the other sides.
» c2 = a2 + b2
With this formula (The Pythagorean Theorem) you
can calculate the length of any one side of a right
triangle when the other two lengths are known.
» c2 = a2 + b2
c=
a 2  b2
» a2 = c2 – b2
a=
c2  b2
» b2 = c2 – a2
b=
c2  a 2
-6-
c=
a 2  b2
c=
32  4 2
c=
9  16
c=
25 = 5
b=
c2  a 2
b=
132  112
b=
169  121
b=
48 ≈ 6.93
A triangle with sides that satisfy the relation
c2 = a2 + b2 must be a right triangle.
E.g.
Would a triangle with sides measuring 9
cm, 12 cm and 15 cm be a right triangle?
?
a2 +
b2
9
12
=
c2
?
2
+
2
=
152
?
81 +
144
225
=
=
225
225
YES! It is a right
triangle
-7-
E.g.
Would a triangle with sides measuring 7
cm, 24 cm and 26 cm be a right triangle?
?
a
2
+
b
=
2
c2
?
72 +
242
=
262
?
49 +
Sets of
Pythagorean
triples.
576
625
= 676
≠ 676
NO! It is not a
right triangle.
three numbers that
Theorem are called
satisfy the
Pythagorean
E.g. (9, 12, 15) form a Pythagorean triple because
92
81
+
+
122
144
225
= 152
= 225
= 225
Complete the following Pythagorean triple:
(15, 20, ?)
152
225
+
+
202
400
625
= ?2
= ?2
= ?2
625 = 25
» (15, 20, 25)
-8-
e) The Pythagorean Theorem and the Cartesian plane
2 points on a grid represent the end points of the
hypotenuse of a right triangle. The length of the
hypotenuse is calculated using the Pythagorean
Theorem.
E.g.
Workout p. 184 nos. 1-3, 6, 8, 9, 13, 15, 17, 20, 22 to 25, 27 to 29, 33, 34, 36, 37, 39, 41, 46, 49, 50, 51
Workbook, p. 4 act. 1 a, b, nos. 1 to 16, act. 2 a, b, nos. 17 to 19
-9-
2) Rational numbers
a) Sets of numbers
: the set of natural numbers = {0, 1, 2, 3, 4 …}
*
: the set of natural numbers excluding zero.
: the set of integers = {… -3, -2, -1, 0, 1, 2, 3, 4 …}
+: the set of positive integers = {0, 1, 2, 3, 4 …}
-: the set of negative integers = {… -3, -2, -1, 0}
*
: the set of integers excluding zero.
Other types of numbers:
- Fractions:
½
¾
-2/3
…
- Decimals:
0.2
0.375
-2.25
…
x 100%
DECIMAL
NUMBER
PERCENT
n%
n ÷100
a÷b
FRACTION
a/b
reduction
of n
100
- 10 -
When you solve an equation, the solution can be
shown on a number line, or by using a Venn diagram.
E.g.
3x + 6 = -3
3x + 6 – 6 = -3 – 6
3x = -9
3x = -9
3
3
x = -3
-4 -3 -2 -1 0 1 2 3 4 5 6
We can write that –3 

Workbook, p. 9 act. 1a) – d), act. 2a) – d)
b) Rational numbers ( )
-
- 11 -
They are numbers that can be written as a
fraction a/b, where a 
and b  *.
E.g.
5=
-2 =
0.3 =
the set of rational numbers
+ = the set of positive rational numbers
- = the set of negative rational numbers
*
= the set of rational numbers excluding zero


» Natural numbers are included in integers which are
included in rational numbers.
- 12 -
E.g.
Complete the following by using , ,  or :
5 ____
____
2/3 ____
-
____
-4 ____
*
_____
0.68 ____
_____
+
+
Workbook, p. 10, nos. 1 to 3 + p.11, act.3 a)-c)
Rational numbers can also be defined this way:
» numbers with repeating decimals whose period
can be zero or non-zero. A bar is placed over the
digit(s) of the first period after the decimal point.
_
E.g.
4 = 0.4444… = 0.4
9
__
7 = 0.63636363… = 0.63
11
_
16 = 0.6400000… = 0.640 = 0.64
25
_
5 = 5.0000… = 5.0 = 5
1
- 13 -
» It is not necessary for these numbers to be
written in the form a/b for them to be rational, but it
MUST be possible to write them in this form.
Workbook, p. 11, nos. 4 a) – g), act.4 a) – b)
c) Method to determine the corresponding fraction of
a rational number
__
METHOD
a = 1.12
EXAMPLE
b
1. Write an equation expressing
the number in decimal notation
2. Multiply both sides of the
equation by the power of 10
that moves the decimal point:
- after one period
- before the first period
3. Subtract the second
equation from the first
4. Solve the equation to
find the fraction
n = 1.121212…
100n = 112.1212…
n=
1.1212…
-
100n = 112.1212…
n=
1.1212…
99n = 111
99n = 111
99
99
n = 111 = 37
99
33
- 14 -
E.g.
1.
_
6.16
n
= 6.166666…
2. 100n
10n
= 616.66666…
= 61.66666…
3.
90n
= 555
4.
90n = 555
90
90
= 37
6
____
0.629
1.
n
= 0.629629629…
2. 1000n = 629.629629…
n = 0.629629…
3.
999n = 629
4.
999n = 629
999 999
Workbook, p. 13, nos. 5 to 8
= 17
27
- 15 -
d) Representing a rational number on the number line
A rational number can be represented approximately
on the number line by rounding up or down the
decimal form of the number.
E.g.
Show 726/160 on the number line.
The decimal form is 726 ÷ 160 = 4.5375
If we round (up or down) to the nearest tenth
we get:
4.5 < 4.5375 < 4.6
4.0
5.0
We could decide to round to the nearest
hundredth:
4.53 < 4.5375 < 4.54
4.50
Workbook, p.13-14, act.5
4.55
4.60
- 16 -
To represent a rational number precisely on the
number line (for example 11/6), follow these steps:
1. We have: 11/6 = 1 + 5/6
2. Knowing that 1 < 11/6 < 2, draw a half-line
starting at 1 on the number line (the angle between
this half-line and the number line is not important).
0
1
2
3
3. Scale this line using 6 sections, and complete a
triangle by connecting the last section (the 6th one)
to the 2, on the number line.
0
1
2
6
3
- 17 -
4. From the 5th section on the half-line, draw a line
parallel to the line that was last drawn to complete
your triangle. The intersection of this line and the
original number line is the point corresponding to
11/6.
11/6
0
1
2
5
6
E.g. Locate 12/7 on the number line.
Workbook, p.14, act.6 1)-4), nos. 9 - 12
3
- 18 -
3) Irrational numbers
a) The set of irrational numbers (
|
)
It is possible to imagine numbers that have nonperiodic decimals, and that are non-terminating:
E.g.
»
»
»
0.123456789101112131415…
impossible to cancel the period
impossible to write it as a fraction (a/b)
it is not a rational number!
Classic examples:
 ≈ 3.1415926535…
(et pis après…)
1
1
2
1
2 ≈ 1.4142135623…
1
» there is no rational number whose square is
2 or 
» 2 , no period appears
the decimal form is only an
approximate value of 2
- 19 -
Numbers that have non-terminating,
repeating decimals are irrational numbers.
non-
These numbers are the result of square roots,
cube roots, etc. which cannot be taken precisely.
Workbook, p.16, act.1 a)-d), p.17, nos. 1-2
b) Locating an irrational number on the number line
Gaps in the number line correspond to irrational
numbers. Each point on the line corresponds to one,
and only one, rational or irrational number.
E.g.
Locate 13 on the number line
Use the perfect squares: 1, 4, 9, 16, 25, 36, 49,
64, 81, 100, 121, 144, 169, …
Find two perfect squares that when added or
subtracted will give you 13.
13 =
49
13 =
2 2  32
This means that the legs of the right triangle are 2
and 3, and the hypotenuse is 13 .
- 20 -
13
0
1
2
2
3
4
5
13
E.g.
Locate
21 on the number line
Use the perfect squares: 1, 4, 9, 16, 25, 36, 49,
64, 81, 100, 121, 144, 169, …
Find two perfect squares that when added or
subtracted will give you 21.
21 =
25  4
21 = 52  2 2
This means that the legs of the right triangle are 2
and 21 , and the hypotenuse is 5.
- 21 -
0
1
2
Workbook, p.17, act.2 3)-4), p.17, nos. 3-10
3
4
5
- 22 -
4) Real numbers ( )
a) The set of real numbers ( )
The set of all rational and irrational numbers is
called the set of real numbers.
 |
-9/10
2
-15

17/5
-1
½
0
8
-6
0.57
1
5
-2/3
- 3
2 5
-2.54
= the set of all positive real numbers
- = the set of all negative real numbers
*
= the set of real numbers excluding zero
+
Workbook, p.21, act.1, p.22, nos. 1-3
b) Real number line
Every real number corresponds to a unique
point on the number line. This is why the number line
is called the real number line.
Workbook, p.23, act.2 a)-b), nos. 4-3
- 23 -
5) Intervals
a) Intervals as line segments
Any set of real numbers that is represented by a
segment on the real number line is called an interval.
E.g.
[2, 5[
lower limit upper limit
0
1
2
3
This interval is closed on
the left, and open on the
right (which means that
2 is included but not 5).
4
5
6
7
This interval can also be shown using the setbuilder notation:
{x 
 2  x  5}
- 24 -
Intervals
1) Closed
0
1
2) Closed on the
left, open on the
right
0
1
3) Open on the
left, closed on
the right
0
{x 
[2, 5]
1
4) Open
2
3
4
{x 
[2, 5[
2
3
4
3
4
1
2
3
5
{x 
]2, 5[
0
5
{x 
]2, 5]
2
5
4
5
 2  x  5}
6
7
 2  x  5}
6
7
 2  x  5}
6
7
 2  x  5}
6
7
- 25 -
b) Interval represented by a ray
Any set of real numbers that is represented on
the real number line by a ray is called an interval.
E.g.
Represent on the number line the set of
numbers that are greater than or equal to 2.
[2, +∞ [
lower limit plus infinity
0
1
2
3
This interval is closed on
the left, and there is no
upper limit.
4
5
6
7
This interval can also be shown using the setbuilder notation:
{x 
 x  2}
- 26 -
Intervals
1) Unlimited on the
left, and closed on
the right
-2
-1
0
2) Unlimited on the
left, and open on
the right
-2
-1
0
3) Closed on the
left, and unlimited
on the right
-2
-1
-1
1
2
3
1
2
0
1
2
3
1
2
4
{x 
3
4
{x 
]2, +∞[
0
4
{x 
]- ∞, 2[
[2, +∞[
4) Open on the
left, and unlimited
on the right
-2
{x 
]- ∞, 2]
3
4
 x  2}
5
6
 x  2}
5
6
 x  2}
5
6
 x  2}
5
6
- 27 -
By convention, the empty set (), any single
element, and the set of real numbers ( , are all
considered intervals.
Workbook, p.24, act.1, + p.25, act.2, p.26, nos. 1-7
- 28 -
6) Natural number exponents
a) Power of a real number
Expressions such as 219 or ma are called
exponential expressions. Such expressions are
written according to convention.
When a number is written in exponential notation
it is in exponent form.
(base)(exponent) = (power)
E.g.
exponent
23 = 8
base
power
Some exponent rules:
1. For any base m and any integral exponent a > 1
ma = m • m • m … • m
(a times)
73 = 7 • 7 • 7
x4 = x • x • x • x
(-5)2 = (-5) • (-5) = 25
(-5)3 = (-5) • (-5) • (-5) = -125
» ma is negative when the base “m” is negative
and the exponent “a” is odd.
- 29 -
2. For base m and exponent 1
m1 = m
91 = 9
y1 = y
3. For base m ≠ 0 and exponent 0
m0 = 1
Workbook, p.28, act.1, nos. 1-11
50 = 1
n0 = 1
(n ≠ 0)
- 30 -
Workbook, p.30, act.2 a), b), c)
b) Multiplying two powers with the same base
The exponent of the product of two powers with
identical bases is the sum of the powers’ exponents.
The base m is not 0 and the exponents a and b
are integers.
ma • m b = ma + b
E.g.
5 2 • 53 =
52 + 3 = 55
Same
base
Add
exponents
3 4 • 35 =
34 + 5 = 39
24 • 2-2 =
24 + -2 = 22
10-2 • 10-3
=
10-2 + -3 = 10-5
IMPORTANT
AVOID THESE COMMON MISTAKES!
52 + 53 ≠ 55
24 • 23 ≠ 212
42 • 53 ≠ 205
Workbook, p.30, nos. 1 to 3
» not a multiplication!
» do not multiply the exponents!
» not the same base!
- 31 -
Workbook, p.31, act.3 a), b), c)
c) Dividing two powers with the same base
The exponent of the quotient (÷) of two powers
with the same base is the difference between the
powers’ exponents. The base m is not 0 and the
exponents a and b are integers.
ma ÷ mb = ma – b or ma = ma – b
mb
E.g.
55 ÷ 52 =
55 - 2 = 53
Same
base
Subtract
exponents
34 ÷ 31 =
34 - 1 = 33
26 ÷ 2-3 =
26 - -3 = 29
10-3 ÷ 10-1 = 10-3 - -1 = 10-2 (explain***)
105 = 105 - 3 = 102
103
IMPORTANT AVOID THESE COMMON MISTAKES!
56 – 52 ≠ 54» not a division!
56 ÷ 52 ≠ 53
» do not divide the exponents!
42 ÷ 22 ≠ 22
» not the same base!
Workbook, p.32, nos. 4 to 6
- 32 -
Workbook, p.32, act.4, a), b), c)
d) Power of a power
To find the power of a power, you need to multiply
the exponents.
(ma)b = ma x b
E.g.
(55)2 = 55 x 2 = 510
Multiply
exponents
(34)3 = 34 x 3 = 312
((-4)3)5 = (-4)3 x 5 = (-4)15
((103)2)4 = 103 x 2 x 4 = 1024
Workbook, p.33, nos. 7 to 10
- 33 -
Workbook, p. 33, act.5, a), b)
e) Power of a product
To find the power of a product, you need to
distribute the exponent.
(m • n)a = ma • na
E.g.
(5 • 2)2 = (5 • 2) • (5 • 2) = 5 • 5 • 2 • 2 = 52 • 22
(3 • 4)5 = 35 • 45
(4x)2 = 42 • x2 = 16x2
(3y2)4 = 34 • y8 = 81y8
Workbook, p.34, nos. 11 to 13
- 34 Workbook, p.34, act.6, a), b)
f) Power of a quotient
To find the power of a quotient, you need to
distribute the exponent.
a
m
 m
   a
 n
n
a
4
E.g.
c4
 c
   4
d
d
3
3
3
x
27
x
 
  
 4
64
3
 3x 
27 x 6
 3 
8y9
 2y 
2
 4x 


 3y 
2
3
2
 3x 
64 x 6 9 x 4 576x10 16x10
. 3  
. 6

3
9
27 y 4 y
108 y
3y 9
 2y 
Workbook, p.35, nos. 14 to 17
2
- 35 -
Workbook, p.36, act.1 a) – i)
7. Negative exponents
For any base m ≠ 0 and exponent a 
m-a = 1
ma
E.g.
1 (t ≠ 0)
t2
t-2 =
4-3 = 1 = 1
43 64
(-4)-2 = 1
=1
(-4)2 16
(-5)-3 = 1
=-1
(-5)3
125
2
42 32 9
4
   2  2 
3
4 16
3
2
2a 
 2x 
2 2 x  2 32 y 2 9 y 2
    2  2  2 2  2
3 y
2 x
4x
 3y 
Workbook, page 36, nos. 1 to 13
3 3
 2 3 a  9 
1
1

23 a 9 8a 9
Workbook, page 39, act. 1 a) – c)
- 36 -
8. Scientific notation
a) Product of a real number by a power of 10
Given a real number “a” and a natural number “n”:
1- To determine the product (a x 10n) the decimal
point is moved “n” decimal places to the right.
2- To determine the product (a x 10-n) the decimal
point is moved “n” decimal places to the left.
E.g.
3.459 x 102 = 345.9
125.24 x 10-2 = 1.2524
0.025493 x 105 = 2549.3
12.87 x 10-4 = 0.001287
Workbook, page 39, nos. 1 to 3
- 37 -
b) Scientific Notation
Scientific notation is an easy and practical
shortcut to writing very large or very small numbers.
Scientific notation uses the powers of 10.
A positive number in scientific notation is in the
form a x 10n where 1 ≤ a < 10 and n is an integer.
E.g.
45 000
=
4.5 x 104
235
=
2.35 x 102
1 220 000
=
1.22 x 106
0.025
=
2.5 x 10-2
0.000 045
=
4.5 x 10-5
- 2 315
=
-2.315 x 103
-0.0069
=
-6.9 x 10-3
DECIMAL
NOTATION
SCIENTIFIC
NOTATION
- 38 -
** Calculators (Learn how to use your calculator)
Key for scientific notation » E or EE or EXP
To enter
2.5 x 103 » 2.5 KEY 3
Display:
2.5 E 3
or 2.5 03 or
or 2.5 03 or 2.5 1003
Calculator display
2.5 EE 03
Answer
3.4575 1003
3.4575 x 103
2.154 -04
2.154 x 10-4
Prefixes
Deca
Hecto
Kilo
Mega
Giga
Tera
101
102
103
106
109
1012
Workbook, page 40, act.2 + nos. 4 to 10
Deci
Centi
Milli
Micro
Nano
Pico
10-1
10-2
10-3
10-6
10-9
10-12
- 39 -
9. Rational number exponents
a) nth root of a real number
The nth root of the real number “a” written as
is the unique real number “b” such that bn = a.
E.g.
N.B.
2
36  6
because 62 = 36
3
64  4
because 43 = 64
3
 125  5
4
81  3
4
 16 
n
n
a
because (-5)3 = -125
because 34 = 81
does not exist in
a does not exist when the index “n” is even
and “a” is negative.
Workbook, page 42, act.1 and 2 + nos. 1, 2
- 40 -
b) Rational number exponents
If “a” is a real number and “n” is a natural
number, we have:
1
n
a n a
E.g.
1
2
49  2 49  7
1
3
1000  3 1000  10
1
3
( 343)  3  343  7
1
4
( 625)  4  625  does not exist in
N.B.
1
n
a does not exist when the “n” is even and
the base “a” is negative.
Workbook, page 43, act.3 + nos. 3 to 6
Workbook, page 44, Evaluation 1, nos. 1 to 15
Workbook, page 3, Challenge 1, nos. 1 to 7