11. Decimal & Recurring numbers
( (
'
9.. ..90.. ..0p a
eap ea
e ap
0....10
' (a
ea
ae
100
235
35'2
55
19
2
55
129
990
2322
...
...
990
232345
453'2
12. Numbers: Powers, surds and logarithms
Powers Roots Logarithms
mnmn
xxx
yxxy bbb loglog)(log
mn
m
n
x
x
x
yx
y
x
bbb logloglog
nmmn
xx
nmn m
xx xnx b
n
b loglog
10
x
xx 1
1log bb
n
n
x
x
1
nmn
yxyx )( nnn
yxyx
n
n
n
y
x
y
x
n
n
n
y
x
y
x
nn
xx
1
n mn
m
xx x
n
x b
n
b log
1
log
n nn yxyx b
x
x
a
a
b
log
log
log
10log b
11 x
x
x
bb log
1
log
x1log1
13. Numbers:
1 is equal to 2
Demostrar que 1=2
Partimos de una igualdad irrefutable: -2 = -2
( 1 - 3 ) = ( 4 - 6) Sumamos a cada miembro 9/4.
( 1 - 3 + 9/4 ) = ( 4 - 6 + 9/4 ) Sabemos que 9/4 = (3/2)2, luego
( 1 - 3 + (3/2)2 ) = ( 4 - 6 + (3/2)2 ) Recordando a Newton y su binomio
(12 - 2·1·3/2 + (3/2)2) = (22 - 2·2·3/2 + (3/2)2)
Resumiendo
( 1 - 3/2 )2 = ( 2 - 3/2 )2 Con lo cual haciendo la raíz cuadrada
( 1 - 3/2 ) = ( 2 - 3/2 ) y restando a ambos miembros 3/2
1 = 2
14. Numbers:
2 is equal to 3
Demostrar que 2 = 3
Algo indiscutible es que -6 = -6, luego:
4 - 10 = 9 - 15 Si a ambos miembros le sumo 25/4,
4 - 10 + 25/4 = 9 - 15 + 25/4 Podemos hacer las transformaciones
22 - 2 · 2 · 5/2 + (5/2) 2 = 32 - 2 · 3· 5/2 + (5/2) 2.
Con lo que también puedo expresarlo cómo:
( 2 - 5/2) 2 = (3 - 5/2) 2 Hallando la raíz cuadrada de ambos miembros
( 2 - 5/2) = (3 - 5/2) Sumándole a cada miembro 5/2
2 = 3 .
15. Numbers:
-1 is equal to 1
Demostrar que 1 = -1
-1 = -1 calculando las raíces cuadradas de ambos miembros
-1 = -1 Calculado el inverso de estas expresiones podemos escribir
1/-1 = -1/1 lo que equivale a :
1/-1 = -1/1 y multiplicando en cruz
1 1 = -1 -1
(1 )2 = ( -1) 2 y simplificando el cuadrado con la raíz
1 = -1 ¿Dónde está el error?
16. ¡La mayor toca el piano!
Dos hombres lógicos se encuentran por la calle
después de mucho tiempo. Uno de ellos,
cortésmente, le pregunta al otro.
- Y que es de tus tres hijas?
- Pues mira!, el producto de sus edades ya es 36
años, y su suma es igual al número del portal
de tu casa.
El hombre lógico piensa y le dice:
- Me falta un dato!
- Ah si!, ¡la mayor toca el piano!
Calcular las edades de las tres hijas del primer
hombre lógico.
17. El Problema del Alabardero
El esqueleto de un alabardero es encontrado en el
hoyo producido por la explosión de una bomba
durante la Primera Guerra Mundial, en el último
día de un mes. Sabiendo que el producto de la
longitud de la alabarda en pies ( 3 piés es
aproximadamente 1 metro), multiplicado por el día
del mes en que se encontraron los restos del
alabardero, multiplicado por la mitad de los años
que tenía el general que mandaba las tropas del
alabardero, multiplicado por el número de años
que llevaba muerto hasta que fue encontrado, es
471,569, se pide:
a) ¿Cuál es la longitud de la alabarda?
b) ¿Cómo se llamaba el general que mandaba las
tropas del alabardero?
c) ¿Cómo se llamaba la batalla?
18. Numbers:
is irrational2
Demostrate that is a irrational number .
Proof by contradiction or reductio ad absurdum (latin)
Let’s assume that is rational and let’s search a contradiction,
then, if is rational where m and n are primes between them.
Then
So n is even, then n2 is a multiple of 22, the m2 is even as well.
If m2 is even , m is even and so, m and n are evens .
Therefore can be simplified by 2. Then m and n are not primes between them
Contradiction.
2
2
n
m
Znm 2/,
evennevennmnmn 222
22
2
n
m
21. An arithmetic exercise
222
With 3 “2” and the arithmetic operations you need, can you obtain
the number 6 ?
With 3 “3” and the arithmetic operations you need, can you obtain
the number 6 ?
333
With 3 “4” and the arithmetic operations you need, can you obtain
the number 6 ?
With 3 “5” and the arithmetic operations you need, can you obtain
the number 6 ?
444
5
5
5
With 3 “6” and the arithmetic operations you need, can you obtain
the number 6 ?
666
22. An arithmetic exercise
7
7
7
With 3 “7” and the arithmetic operations you need, can you obtain
the number 6 ?
With 3 “8” and the arithmetic operations you need, can you obtain
the number 6 ?
333
888
With 3 “9” and the arithmetic operations you need, can you obtain
the number 6 ?
With 3 “1” and the arithmetic operations you need, can you obtain
the number 6 ?
999
)!111(
24. REAL STRAIGHT LINE
TRUE or FALSE?
1. You can write all decimal numbers as a fraction.
2. All real numbers are rational numbers.
3. Any irrational number is a real number.
4. There are integres (or whole) numbers that they are
irrationals.
5. Exist real numbers that they are irrationals.
6. Any decimal number is rational.
7. Every irrational number has infinite decimal
significative digits.
8. All rational numbers have infinite figures that they
repeat.
9. All rational numbers can be written by fractions.
10. A recurring number has a sequence of decimal digits
27. ACCURACY
1 significant digits
• Marks or grades in an High school examination
• He is on his fifties.
2 significant digits
• Age: He is 23 years old NOT he is 23 years, 2 months and 21 days old.
• Cooking: 357 gr of flour, we say 350 gr.
• Distance of a journey: there are 3437.70 Km from Madrid to Moscow but we
say 3500 Km.
• Area of a garden: If it is 337 m2, we would say 350 m2
• Weight of people : He weigh 82 kg, NOT 82,32 Kg
• Temperature : It is 23º degree, NOT 23,12º degrees
• Geology: Dinosaurs lived from 160 to 65 millions years ago
3 significant digits
• Height of people: He is 1’76 m tall NOT 1.80 m
• Measure in biological works: measure of a shell 25.6 cm NOT 26 cm.
• Accurate measures with a rule: we say 67,5 cm NOT 70 cm.
4 or more, significant digits
• Trigonometric ratios: sin, cos, tan, etc.
• Logarithms
• Really scientific works
28. ESTIMATING
Estimate the value of the following arithmetic expressions:
1
180
170
360
40130
2.35.56
9.418.127
2
50
100
510
6040
13.596.9
2.6168.40
1
28
10
22
100
88.113.2
6.98
33
29. ROUNDING & ERRORS
Rounding a real number is to replace it by a rational number with a finite number
of decimal digits
BASIC Method Rounding
E.g.: Round 7.45839 with 2 decimal places
7.4 5 8 39
Last digit Decider
Round-up : If decider is 5 or more = 7.4 6
Round down :If decider were 4 or less = 7.4 5
Absolute Error (or Discrepancy) Ea = │Actual value –
Calculated value │
Relative Error Er = Ea /Actual value
30. SCIENTIFIC NOTATION
According to legend, A long
time ago chess was invented
by Grand Vizir Sissa ben
Dahir and given to King
Sirham of India. The king
offered him a reward and he
requested the following:
“Jusn one grain of wheat on the
first square of the chessboard then
put two on the second squared,
four on the next, then eight, and
continue, doubling the number of
grains on each successive
sequence until every square on the
chessborad is reached.”
31. SCIENTIFIC NOTATION
Mean Distance from Earth to the Sun
Ordinary number
149,597,870 Km
Rounding 3 s.f.
150,000,000 Km
Standard form
1,5 108 = 1,50+E08
32. SCIENTIFIC NOTATION
BIG NUMBERS: the googol.
The number was devised by the mathematics teacher Edward Kasner in 1939 but
the name was coined by his 9 years old nephew Milton Sirotta.
The googol number is represented by a digit 1 followed of 100 zeros:
1 googol = 10100 = 10000 ...(100...0000.
Although is easy to overcome this value using your imagination, e.g. :
1 googolplex 10googol
Black holes are presumed to evaporate because they faintly give off Hawking
radiation; if so, a supermassive black hole would take about a googol years to
evaporate
33. SCIENTIFIC NOTATION
1. Minimum distance between Earth and Mars
2. Mass Atomic unit
3. Distance between Polar star and The Sun
4. Average distance between Saturn and the
Sun
5. Avogadro’s number.
6. Proton radius
7. Electric charge of electron
8. Light speed
9. Distance from Earth to Moon
10.One Googol
11.Spanish life expectancy
12.A billion in the USA.
13.Grains of sand on Doniños beach (a
quadrillion)
14.The total amount of grains of wheat that Sissa
ben Dahir requested to King Sirham
a) 4.1·1015 Km
b) 109
c) 2.53·109 seconds
d) 3·108 m/s
e) 6.023 ·1023
f) 5.9·1010 m
g) 3.84·108 m
h) 264 -1 = ≈1.84·1019
i) 10100
j) 1.43·109 Km
k) 1.6·10-19 C
(coulombs)
l) 8·10-16 m
m) 1.66·10-27 Kg
n) 1018
34. GOLDBACH Conjecture
Christian Goldbach (Prussian mathematician , 1690 –1764)
“Every even integer greater than 2 can be written as the sum of two primes .”
4 = 2 + 2
6 = 3 + 3
8 = 3 + 5
10 = 3 + 7 = 5 + 5
12 = 5 + 7
14 = 3 + 11 = 7 + 7
......
35. PASCAL’S Triangle (Tartaglia’s triangle)
TARTAGLIA (Italy, 1499-1557) & PASCAL (France, 1623 –
1662)
Can you guess any properties?
• 1 the first and the last.
• Sucesión números naturales 1,2,3,4..... en la 2º y penúltimo
términos.
• Es simétrico.
• Cada término es la suma de los dos que figuran encima.
1 1
1 1 2
1 2 1 4
1 3 3 1 8
1 4 6 4 1 16
1 5 10 10 5 1 32
1 6 15 20 15 6 1 64
1 7 21 35 35 21 7 1 128
1 8 28 56 70 56 28 8 1 ... 256
