4 ESO Academics - Unit 01 - Exercises 1.2. Irrational Numbers, Real Numbers

1. UNIT 01 September
1. Describe this picture where every triangle is a right triangle.
2. The Great Pyramid (or the Pyramid of Keops) is a square pyramid. Approximately, the
height of this pyramid and the side of the base are 146 m and 230 m respectively.
Prove that:
𝒄𝒄
𝒂𝒂
≈ 𝚽𝚽 𝐚𝐚𝐚𝐚𝐚𝐚 𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩 𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩 𝐨𝐨𝐨𝐨 𝐭𝐭 𝐭𝐭𝐭𝐭 𝐛𝐛𝐛𝐛𝐛𝐛𝐛𝐛 ≈ 𝟐𝟐𝟐𝟐𝟐𝟐
Axel Cotón Gutiérrez Mathematics 4º ESO Exercises 4.1.2.1

2. UNIT 01 September
3. Complete the following table and check that the value of:
�
𝟏𝟏
𝟏𝟏 + 𝒏𝒏
�
𝒏𝒏
approaches e as n gets bigger and bigger.
Axel Cotón Gutiérrez Mathematics 4º ESO Exercises 4.1.2.2

3. UNIT 01 September
4. Are these numbers irrationals?
𝑎𝑎) √4
𝑏𝑏) √9 + √4
𝑐𝑐) √4 + √1
𝑑𝑑) √5
𝑒𝑒) √14 + 2
𝑓𝑓) √9 + 4
5. Order from highest to lowest the following numbers and classify them.
𝟑𝟑. 𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 …; 𝟑𝟑. 𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 … ; 𝟑𝟑. 𝟒𝟒𝟒𝟒𝟒𝟒𝟒𝟒𝟒𝟒𝟒𝟒𝟒𝟒𝟒𝟒 …; √𝟑𝟑𝟑𝟑
𝟑𝟑
; �
𝟐𝟐𝟐𝟐𝟐𝟐
𝟐𝟐𝟐𝟐
; 𝟑𝟑. 𝟒𝟒𝟒𝟒𝟒𝟒 … ;
√𝟏𝟏𝟏𝟏; 𝟑𝟑. 𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏… ; 𝟐𝟐𝟐𝟐; 𝟑𝟑. 𝟎𝟎𝟎𝟎𝟎𝟎
6. In a Equilateral triangle which kind of number is:
a) The height, if the side measures 10 cm.
b) The area, if the side measures 3 cm.
c) The height and the area if the side measures √𝟑𝟑 cm
Axel Cotón Gutiérrez Mathematics 4º ESO Exercises 4.1.2.3

4. UNIT 01 September
7. Represent the numbers on the real number line.
𝟓𝟓
𝟕𝟕
; −
𝟓𝟓
𝟕𝟕
;
𝟐𝟐𝟐𝟐
𝟑𝟑
; 𝟑𝟑. 𝟕𝟕𝟕𝟕; 𝟒𝟒. 𝟑𝟑�; 𝟐𝟐. 𝟏𝟏𝟔𝟔;� √𝟐𝟐; √𝟔𝟔; √𝟏𝟏𝟏𝟏; √𝟏𝟏𝟏𝟏
8. Represent the numbers on the real number line.
𝟐𝟐√𝟔𝟔; 𝟏𝟏 + √𝟑𝟑; 𝟑𝟑𝟑𝟑
Axel Cotón Gutiérrez Mathematics 4º ESO Exercises 4.1.2.4

5. UNIT 01 September
9. Classifies these numbers:
𝟓𝟓. 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎… ; −𝟐𝟐𝟐𝟐; √𝟒𝟒𝟒𝟒, 𝒆𝒆;
−𝟏𝟏𝟏𝟏
𝟐𝟐
; √𝟏𝟏𝟏𝟏; 𝟓𝟓𝟓𝟓. 𝟗𝟗𝟗𝟗𝟐𝟐�; √𝟕𝟕 + 𝟐𝟐; � 𝟗𝟗 + √𝟔𝟔
10. A fence is 22 cm long to the nearest centimetre. Find an upper bound for the absolute
and relative errors. (Remember: An upper bound for the absolute error is half a unit of
the last significant figure in the approximate value).
11. Consider the golden number 𝝓𝝓. Round it to two decimal places and find an upper
bound for the absolute and relative errors.
Axel Cotón Gutiérrez Mathematics 4º ESO Exercises 4.1.2.5

6. UNIT 01 September
12. Use a reasonable number of significant figures to express the following quantities. Find
an upper bound for the absolute and relative errors in each case.
a) Listeners of a radio program: 843,754
b) Price of a car: 28,782 €
c) Time it takes light to travel a distance: 0.0375 seconds.
d) Council expenses: 48,759,450 €
13. Using the calculator, write √𝟖𝟖 in decimal form and its approaches, rounding and
truncation to thousandths.
14. Find the absolute error by rounding 4.7569 to hundredths. Which one is the relative
error?
Axel Cotón Gutiérrez Mathematics 4º ESO Exercises 4.1.2.6

7. UNIT 01 September
15. Find the absolute and the relative error by rounding:
a) 𝟏𝟏𝟏𝟏. 𝟒𝟒𝟒𝟒𝟒𝟒𝟒𝟒 to thousandths.
b) √𝟏𝟏𝟏𝟏 to tenthousandths.
c)
𝟐𝟐
𝟑𝟑
to tenths.
d) 𝟑𝟑. 𝟏𝟏𝟏𝟏𝟏𝟏 to thousandths.
16. Find two different approaches of 1.45 with the same relative error.
Axel Cotón Gutiérrez Mathematics 4º ESO Exercises 4.1.2.7

8. UNIT 01 September
• Prove that the triangles 𝑨𝑨𝑨𝑨𝑨𝑨 and 𝑨𝑨𝑨𝑨𝑬𝑬 are similar. (Hint: inscribe the pentagon in a
circumference and remember that: angles on same arc are equal, and if two triangles
have two of their angles equal, the triangles are similar).
• Let “𝒅𝒅” and “𝒂𝒂” be the diagonal and the side of the pentagon respectively. Prove
that 𝑭𝑭𝑭𝑭 = 𝒅𝒅 − 𝒂𝒂 .
• Since
𝒅𝒅
𝒂𝒂
= 𝚽𝚽
prove that
𝟏𝟏
𝚽𝚽 − 𝟏𝟏
=
𝒂𝒂
𝒅𝒅 − 𝒂𝒂
• Since the triangles 𝑨𝑨𝑨𝑨𝑨𝑨 and 𝑨𝑨𝑨𝑨𝑨𝑨 are similar, we can write that:
𝑨𝑨𝑨𝑨
𝑪𝑪𝑪𝑪
=
𝑨𝑨𝑨𝑨
𝑭𝑭𝑭𝑭
⇒
𝒅𝒅
𝒂𝒂
=
𝒂𝒂
𝒅𝒅 − 𝒂𝒂
Therefore:
𝚽𝚽 =
𝟏𝟏
𝚽𝚽 − 𝟏𝟏
• In order to find the value of 𝚽𝚽, solve the previous equation.
Axel Cotón Gutiérrez Mathematics 4º ESO Exercises 4.1.2.8

9. UNIT 01 September
http://www.math-play.com/rational-and-irrational-numbers-game/rational-and-
irrational-numbers-game.html
http://staff.argyll.epsb.ca/jreed/math9/strand1/munchers.htm
Axel Cotón Gutiérrez Mathematics 4º ESO Exercises 4.1.2.9

10. UNIT 01 September
http://www.sheppardsoftware.com/mathgames/decimals/scooterQuestDecRound.htm
Axel Cotón Gutiérrez Mathematics 4º ESO Exercises 4.1.2.10