1

RATIONAL AND IRRATIONAL

NUMBERS CALCULATION OF THE
REAL NUMBERS

HISTORICAL NOTE
Counting was certainly one of the first daily 1
Objectives
activities of Man ; therefore , first , whole numbers
were used , followed by rational and finally by
irrational numbers , in order to measure some 1. Distinguish between a
geometric magnitudes . rational number and an
Around 3000 B.C and in the Middle east , counting irrational number .
was done using chips . 2. Know some irrational
Around 2000 B.C , the system of additive numbers .
numeration appeared in Egypt . 3. Perform the operations
Around 300 B.C , Pythagoras studied the properties of calculation of real
of irrational numbers  and 2 . numbers .
In the 8th century , Arabs used decimal fractions .
In the 15th century , El-kashi described decimal
numbers .
Around 1484 Chuquet used negative numbers .
Around the end of the 19th century , Lindeman ,
Canton , and Dedekind specified the nature of real
numbers .

PLAN OF THE CHAPTER

COURSE
1. Rational numbers
2. Irrational numbers
3. Know some irrational numbers
«God created whole
numbers , 4. Real numbers
Man created the rest». 5. Operations of the real numbers
KRONECKER 6. Power of a real number
7. Insertion of a rational

EXERCISES AND PROBLEMS

TEST

7
 1 The rational numbers

Definition
a
A rational is a number that can be written in the form of  where a is an integer and b is a non-zero
b
integer.

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• The decimal –3.2 is a rational since it can be written in the form:
10
 
a
where a is an integer and b is a non-zero integer .
b
Every decimal is a rational.
7
• The natural number 7 is a rational since it can be written in the form of: .
1
Every natural number is a rational.

• The number 3.666.... where 6 is a repetitive, is called an infinite periodic number; it is a rational that is written
11
.
3
The number 12.3141414... where 14 is repetitive is called an infinite periodic number; it is a rational that is
12,191
written .
990
Any periodic number is a rational.

• The infinite non-periodic number 3.1415927... is an approximation of π ; it is not a rational since it cannot be
a
written in the form where a and b are non-zero whole numbers.
b
Any infinite non-periodic number is not a rational.

Application 1
Name among the following the rational numbers.
7 ; – 8.3 ; 7.636363… ; 2.15 ; 3.7654317...
3

0 ;
–4 ; –8 ; 10 ; 60 .
9 10

8
Remark :
The number 3.6666 ... which is also written as 3.6 is a number whose decimal part is unlimited and periodic .
The period is 6 , we say that it is of length 1 .
6 2 11
=3+=3+=.
3.6
9 3 3
1 is a number whose decimal part is unlimited and
The number 1.21 21 21 .... which is also written as 1.2
periodic .
21 40
1 = 1 +  =  .
The period is 21 , we say that it is of length 2 and 1.2
99 33
In general , a number whose decimal part is unlimited and periodic and of length l , is written as :

Period
integral part + 
992
1 ...3
9
l times

Application 2
Write the following numbers in the form of a fraction :
 ; b = 2.1
a = 1.7 3 ; c = 4.5888... ; d = 0.1
31 .

2 The irrational numbers

The irrational numbers are the numbers that are not rational , that is to say the numbers that cannot be written in the
a
form of a fraction  where a is a number and b is a non-zero number .
b

EXAMPLE
π 1 – 5
2 ; –3 ; π ; –  ; 3 + 2 ;  ; 2.30 300 30003 .... ; 2.747854912 ... are irrational numbers .
2 2
Such number can be represented only by an approximation (approximate value) .

Application 3
1
Among the numbers –  ; 17 ; 4 ; 9 ;
2
22
 ; π + 3 ; 0.51 ; 14.3 ; π + 7 ; – 3.4 1111 ...
7
Indicate those that are rational and those that are irrational .

9
 3 Know some irrational numbers

By using the calculator , we find :

• 2 = 1.414 213 562 ....

The decimal part of 2 is an unlimited and non-periodic sequence , then 2 is an irrational number and therefore
this number is represented by an approximation.
1.4 is an approximate value of 2 to the nearest 10–1 by default (or to the nearest tenth).
1.42 is its approximate value of 2 to the nearest 10–2 by excess (or to the nearest hundredths).
1.414 is its approximate value to the nearest 10–3 by default (or to the nearest thousandths).

• 3 = 1.732 050 808 ....

The decimal part of 3 is an unlimited non periodic sequence , then 3 is irrational.
1.732 is an approximate value of 3 to the nearest 10–3 by default.

• 5 = 2.23606 7977 ....

5 is irrational and 2.2360 is its approximate value to the nearest 10–4 by default .

• π = 3.14 15 92 65 4 ... ; π is an irrational number and 3.14 is its approximate value to the nearest 10–2 by default .

1 + 5 1 + 5
•  = 1.6180 33 989 ... ,  is an irrational number and 1.618 is its approximate value to the nearest
2 2
10–3 by default .

Application 4
Give an approximate value for each of the following irrational numbers, to the nearest
10–2 by excess and another to the nearest 10–3 by default :
π 2 + 7
11 ;  ; – 14 ; π  ; 1 – 5 ;  ; π + 3 .
4 5

10
 4 The real numbers
Natural numbers , integers , decimals , rational and irrational numbers are all called real numbers .

5 Operations of the real numbers

The operations (addition , subtraction , multiplication and division) as well as the known rules of calculation on the
rational numbers are valid for real numbers .
If x and y are two real numbers , then :
• the sum x + y , the difference x – y , the product x × y (or x . y or xy) are real numbers .
x
• If y is not zero , then the quotient x  y , denoted also by  , is a real number .
y

Application 5
Find x + y , x – y , x × y and x  y in each of the following :
3 7
1° x = – 37 and y = 87 2° x =  6 and y = –  6 .
5 4

6 Power of a real number

The known properties of power of integers and rational numbers are also valid for the real numbers .
If a and b are two real numbers , m and n are natural numbers , then :


n
a0 = 1 (a ≠ 0) an = a
 (b ≠ 0)
a1 = a bn b
m
an = a × a × … × a (n ≥ 2) (an) = an.m
14243
n times an
m = an  am = an–m for a ≠ 0 and n > m
a
an × am = an + m an 1
m = m for a ≠ 0 and m > n
an × bn = (a × b)n a a –n

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EXAMPLES
3 2 5
• (3) × (3) = (3) = 93
π4
• = π4–2 = π2
π2
3
2 3 (2) 22
•


π
=
 
π3
=
π3
.

Application 6
Simplify the following :
2 3
1° (7) × (7) 3° π7  π3
π5
 × 2 .
1 3 1
2°  4° 
π 2

7 Insertion of a rational

3 7
Let us insert (place) a third rational between the two rationals  and  .
5 8
3 24 7 35
By reducing to the same denominator , we obtain  =  and  =  .
5 40 8 40
a
As a result , any rational of the form  with 24 < a < 35 is such that :
40
24 a 35
<<.
40 40 40
For a = 27 , as example ,
24 27 35 3 27 7
 <  <  or  <  <  .
40 40 40 5 40 8

Application 7
4 8
Insert a third rational between the two rationals  and  .
7 13

12
 Exercises and problems
Test your knowledge
1 Among these rational numbers , which are decimals ?

15 3 15 13 15 4 1 5 11
 ;  ;  ;  ;  ;  ;  ;  ; .
100 7 1000 5 7 20 6 18 50

2 Complete by rational or irrational .

7
1°  is a .......... number 5° 0.3 is a .......... number
8
2° 5 is a .......... number 6° 1.6
 is a .......... number
3° 16 is a .......... number 7° – 17 is a .......... number
2π is a .......... number
4°  8° 2.31 is a .......... number
5

1.7 112 –4 –2 13 3
3 Calculate the following quotients :  ;  ;  ;  ;  ;  .
3 5 7 3 2 10
Can you affirm that these numbers are rational ?

4 By using the calculator , give an approximation value to the nearest 10–3 by default for each of
the following numbers :


15
810 ; 270 ;  ; 7,9 56 ;  .
43 ; 12,4
2

5 By using a calculator , give an approximate value to the nearest 10–2 by excess for each of the
following numbers :
π
318 ; 1,0
15 ; 9,8
70 ;  .
3

6 5 is unlimited and periodic .

The decimal part of the number 0.2
a
1° Write this number in the form  , where a and b are two natural numbers .
b
5 a rational number ?
2° Is the number 0.2

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 Exercises and problems
a
7  in the form  where a and b are two natural numbers and b non-zero
1° Write the number 1.5
b
number .
2° Is 1.5 an irrational number ?

8 Insert a third rational number between the two given numbers .

2 4 5 13 11 7
1°  and  2°  and  3°  and  .
3 5 9 15 18 9

9 Insert a third decimal between the two given decimals .

1° 2.3 and 2.4 2° 3.14 and 3.15 3° – 3.72 and – 3.69 .

10 Insert a decimal number between the two given rationals :

1 5 7 14 11 17
1°  and  2°  and  3°  and  .
3 6 12 9 28 723

11 Write in the form of one exponent :

2 3
   
3 3
23 × 25 – × – (– 3)2 × 34 (3)4 × (3)5 × (3)2
5 5

108 × 10–4 103 × 104 × (102)–3 π2 × π5 10–1 × 10–3 × 104

57  53 π7  π3 104  10–2 π3 × π5 × π2 .

For seeking
12 Among the following numbers , indicate those which are rational and those which are irrational :
0
1° x = 0.4 2° y = 0.40 400 4000 4 .... 3° z = 7.6666 .... 4° t = 7.61 611 61116 ....

13 Calculate and give the result in a fractional form :

5
+
1° 0.3 3° 8 × 4.9 5° 14.62  3
9
2 11 99
2 + 
2° 1.1 4°  – 0.5 6° 62.14   .
99 3 8

14
 Exercises and problems

14 Insert a rational and a decimal between the two 19 a and b are real numbers . Reduce the
given rationals : following :
1° 3 a2 × 5 a
1° 0.3 and 1.6

2° – 7 ab2 × 27 a2 b3
2° 0.7 and 1.4
 3° (– 3 a2) × (3 . a)
1 3
3° 0.33 and 1.1
1 . 4°  a2 b3 ×  a . b
3 7
5° 2 a3 × 32 a2 b
6° 0.3 a3b2 × 1.5 a2 b2 .
15 Write in scientific notation each of the following
numbers :

1° 0.03 5° 2107.34 × 10–2 20 Write in the form of an irreducible

4 fraction :
2° 13.25 6° 
5 8 5 213 4
213
 
(3.4)3 × 
18
; (2.1  
5)6   .
99
3° 403 7°  .
16
4° 15.14 × 10–3
21 Consider the plane of an orthonormal system of axes
x′Ox and y′Oy .

16 ABC is a right triangle at A such that y

AB = 2 cm and AC = 3 cm .
Calculate the exact value of BC then give an
approximate value of BC to the nearest 10–2 by default . 1 A

x′ O 1 2 x

17 LAC is a semi equilateral triangle so that : A =

90° ; C = 30° and LC = 3 .
y′
Calculate the exact values of AC and AL then give an
1° Calculate OA .
approximate value of AL to the nearest 10–2 by excess .
2° Place , by using only the compass , the point B on x′Ox
B
axis such that : O = 5 .

18 A circular solid has an area of 78.5 cm2 . 22 Calculate the exact value of A and B then
give an approximate value to the nearest 10–2 by
Give an approximate value of its radius R to the nearest default :
10–2 by default , take π = 3.14 . A = x4 – 2x3 + 3 x2 – 4x + 2 for x = 32 .
2
B = 5x2 +  x for x = 5 + 3 .
3

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 T E S T

1 Find , among the following numbers , those that are rational and those that are irrational .
1 + 5 7 π π
18 ;  2 ; – 36 ; 8 × 8 ; – 
5 ;  ; –. (4 points)
3 2

2 By using the calculator , give an approximate value to the nearest 10–3 by default for each of the following numbers
:
1575 ; 19683 . (2 points)

3 Calculate and give the result in the form of a fraction :

7
 + .
1.3 (4 points)
3

4  and 1.3
Insert a rational number between 1.3 1 . (4 points)

 
5 MER is a semi-equilateral triangle such that MER = 90° , EMR = 60° and EM = 3 .

Calculate the exact values of ER and MR , then give an approximate value for MR to the nearest 10–2 by default .
(3 points)

6 KIM is a right triangle at I so that KI = IM = 2 . (3 points)

1° Calculate KM .

A
2° Then place , by using only the compass , on the x′Ox axis the point A so that O = 22 .

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