LLG
Advanced Math and Science Pilot Class
Mathematics, Grade 10
2013 – 2014
Paris – Abu Dhabi
Chapter 1 : Set of numbers, rational numbers
I. Sets of numbers :
 ℕ = {0, 1, 2, 3, … } is the set of the natural numbers or positive whole numbers or positive integers.
 ℤ = {… , −3, −2, −1, 0, 1, 2, 3, … } is the set of the integers, positive and negative whole numbers.
 𝔻 is the set of the decimal numbers. A decimal number is a number which has a finite decimal writing.
𝑎
 ℚ is the set of the rational numbers. A rational number is a ratio 𝑏 of two integers 𝑎 ans 𝑏 with 𝑏 ≠ 0.
Property : Any rational number can be written as a terminating or recurring decimal.
 ℝ is the set of all the real numbers (all numbers known in grade 10).
Definition : An irrational number is a real number which is not a rational number.
Property : The real numbers whose decimal writing are infinite and admit no pattern are irrational numbers.
Symbols : ∈ is read “belongs to” or “is an element of” ; ⊂ is read “is a subset of” or “is included in”.
1
3
1
3
1
3
Example : 0 ∈ ℕ, ∈ ℚ, ≈ 0.333333 … so ∉ 𝔻. 𝜋 and √2 are irrational numbers.
Property : ℕ ⊂ ℤ ⊂ 𝔻 ⊂ ℚ ⊂ ℝ
Example : Give the smallest set to which each following number belongs :
84
𝑎 = − 14
𝑒=−
25
√100
𝑏=
𝑓=
𝜋
𝑐 = 3.33333
3
4.1×10−3
𝑔 = 1.6666666 ….
5×10−4
𝑑=−
60√2
8
ℎ = −6√3 + √2
II. Rational Numbers :
2.1. Definition :
o Let 𝑎 and 𝑏 be two integers, 𝑏 ≠ 0, we call quotient of 𝑎 and 𝑏 the number such as the following
𝑎
equality is true : 𝑏 × … … = 𝑎. We denote it 𝑏 .
𝑎
As a consequence, we have : 𝑏 × 𝑏 = 𝑎 .
o
o
𝑎
A rational number is a fraction 𝑖𝑒 a ratio 𝑏 where 𝑎 and 𝑏 are integers (positive or negative whole
numbers ) and 𝑏 ≠ 0.
𝑎 is called the numerator and 𝑏 the denominator.
Equality of two fractions : For all 𝑎, 𝑏, 𝑐, 𝑑 with 𝑏 ≠ 0 and 𝑑 ≠ 0, we have :
𝑎
𝑏
Eg :
−45
3
𝑐
= 𝑑 If and only If 𝑎 × 𝑑 = 𝑏 × 𝑐
= −15 because …………………............... and 12 × 3 = 9 × 4 so ……………………………
From that property, we deduce that for all 𝑎, 𝑏, 𝑘 with 𝑏 ≠ 0 we have :
𝑘×𝑎
𝑎
=
𝑘×𝑏
𝑏
We will use this property either to simplify a fraction or to find common denominators by multiplying or dividing
the numerator and denominator of a quotient by the same number.
 A fraction is called reduced when its denominator and numerator have no common divisor (or common
factor).
Eg : The reduced form of
45
6
420
is ………………………………….. and 360 =…………………………………………………………………………
Any fraction has a unique reduced form.
2.2. Addition and subtraction :
For all 𝑎, 𝑏, 𝑐, 𝑑 with 𝑏 ≠ 0 and 𝑑 ≠ 0, we have :
a
b
c
a+c
b
b
+ =
and
a
b
c
ad
d
bd
+ =
+
cb
bd
=
ad+cb
bd
 Be careful not to add denominators and numerators !
2.3. Multiplication :
For all 𝑎, 𝑏, 𝑐, 𝑑 with 𝑏 ≠ 0 and 𝑑 ≠ 0, we have :
a
b
c
ac
d
bd
× =
and
a
𝑎𝑐
b
𝑏
𝑐× =
2.4. Reciprocal and division :
For all 𝑎, 𝑏, 𝑐, 𝑑 with 𝑏 ≠ 0 and 𝑑 ≠ 0, we have :
1
o The reciprocal of 𝑏 is also denoted 𝑏 −1.
o
The
𝑎
reciprocal of 𝑏
𝑏
𝑏
is 𝑎,
𝑎 −1
also denoted (𝑏 ) .
To divide by a non-zero number is equivalent to multiply by its reciprocal :
𝑎
𝑏 =𝑎×𝑑
𝑐 𝑏 𝑐
𝑑
1
 Be careful not to confuse reciprocal (𝑎)and opposite (−𝑎) of a number 𝑎