Mathematics

Arithmetics

123

23'0

Algebra0352 2 xx

Mathematics

Geometry

Analysis

dxexf x)(

x

y

x

P

Q

x+x

y=f(x)

x

y

Mathematics

Statistics

dxexf x)(

Arithmetics. Numbers

Naturals

Integers 3 – 5 = -2

Rationals(Decimals and recurring

numbers)

Irrationals

Real

Complex 3 - 4i

1,2,3,4,5,6,...

..., 5, 4, 3, 2, 1,0,1,2,3,4,5,....

,m

m nn

/ , / mn

I x m n Z x

I

/ , ; 1a bi a b R i

4 148 2 3

13

32 1'04 0,23

5

3, , , 2, 3, 5,..., 2e

Arithmetics. Numbers

Arithmetics. Numbers

Operations

BASIC OPERATIONS

+ Addition / plus Sum, addends, summand

- Substraction / minus Difference, minuend, substrahend (no common), terms

* Multiplication Product, multiple, multiplicand, multiplier

/ Division Quotient, ratio, dividend, divisor, remainder

^ Exponentiation; power Indices

square root; cube root Surds3

3

The most 10 unforgivable errors

1 Priority

2 Negative numbers

3 Powers y , besides

4 Parenthesis

5 Fractions

6 Simplifying , however,

7 Roots is not a real number, however

8 Mixed numbers

9 Binomials however

10 Calculus

24 16

3 4 3( 4) 12

20 3 4 20 12 8

2( 4) 16

(4 3) 4 3 1

4 3 4 3

2 2

4 3

4

3

55 4 3 3

4 5 5

2 8

3 8 2

23

23 3

3

2 2 2(4 3) 4 3 2 2 2(4 3) 4 3

4 4 4

3 5 3 5

2 2yy

x x

5 3 3 5

4 4 4

4 3 4 3

2 2

00

a

0

a logb a

Mixed fractions

4

289

4

17272

41

5

1

5,22

5

2

122

21

12

2

2

122

21

Decimal & Recurring numbers

( ('

9.. ..90.. ..0p a

eap eae ap

0....10'

(a

eaae

100

23535'2

55192

55

129

990

2322...

...990

232345453'2

Numbers: Powers, surds and logarithmsPowers Roots Logarithms

mnmn xxx yxxy bbb loglog)(log

mn

m

n

xx

x yxy

xbbb logloglog

nmmn xx nmn m xx xnx b

n

b loglog

10 x

xx 11log bb

n

n

xx

1

nmn yxyx )( nnn yxyx

n

n

n

y

x

y

x

n

n

n

y

x

y

x

nn xx

1n mn

m

xx xn

x bn

b log1

log

n nn yxyx b

xx

a

ab

log

loglog

10log b

11 x

xx

bb log1

log

x1log1

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 3/21 = 2

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 transformaciones22 - 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 .

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 escribir1/-1 = -1/1 lo que equivale a :1/-1 = -1/1 y multiplicando en cruz1 1 = -1 -1(1 )2 = ( -1) 2 y simplificando el cuadrado con la raíz1 = -1 ¿Dónde está el error?

¡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.

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?

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

mZnm 2/,

evennevennmnmn 222 22

2

n

m

Rational numbers: 3/5 y 10/3

Ejecuta applet

Irrational numbers

Run applet

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

55

With 3 “6” and the arithmetic operations you need, can you obtain

the number 6 ?666

An arithmetic exercise

7

77

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(

What root is bigger ?

242424 66

6 44 4096444444444

4 64 or 6

242424 44

4 66 12966666666

REAL STRAIGHT LINETRUE 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

that it is repeated indefinitely.

Absolute value

Exercises

0

0)(

xifx

xifxxxf

...32

...32

...)3(2

...32

1132...

1132...

11...

532...

Intervals

ACCURACY1 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

ESTIMATINGEstimate the value of the following arithmetic expressions:

1180

170

360

40130

2.35.56

9.418.127

250

100

510

6040

13.596.9

2.6168.40

128

10

22

100

88.113.2

6.9833

ROUNDING & ERRORSRounding 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

SCIENTIFIC NOTATIONAccording 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.”

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

SCIENTIFIC NOTATIONBIG 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

SCIENTIFIC NOTATION1. 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

GOLDBACH ConjectureChristian 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

......

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.

• Cada fila tiene un término más.

La suma de los términos es la sucesión 2 , 2 , 2 , 2 , 2 , 2 , ....

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

PASCAL’S Triangle (Tartaglia’s triangle)

El 1º término es

1 en los extremos.

Es simétrico.

Sucesión números naturales 1,2,3,4..... en la 2º y penúltimo términos.

Cada término es la suma de los dos que figuran encima

En cada fila se verifica que

1;10

m

mm

nm

m

n

m

mm

mm

11

n

m

n

m

n

m 1

1

0

0

1 1

0 1

2 2 2

0 1 2

3 3 3 3

0 1 2 3

4 4 4 4 4

0 1 2 3 4

5 5 5 5 5 5

0 1 2 3 4 5

6 6 6 6

0 1 2 3

6 6 6

4 5 6

0

1

2

3

4

5

6

1 2

2 2

4 2

8 2

16 2

32 2

64 2

1!00

0

n

n

n

nnnnnn

1....

210)11(2

PASCAL’S Triangle andFIBONACCI Sequence

1 1 2 3 5 8 13

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

LOGARITHMSJOHN NEPER (Scotland, 1550-1617)

abxa x

b lg

0

2

3

2

6

2

3

3

2

10

4

10

2 1 lg 1 0

2 8 lg 8 3

2 64 lg 64 6

3 27 lg 27 3

10 100 lg 100 2

10 10,000 lg 10,000 4

LOGARITHMSAPPLICATIONS

Growing Polulations

Compose Interest

C14

Earthquakes

pH

CALCULATOR

CALCULATOR

ARITHMETICS

THE END

MATHEMATICS