Maths in Everyday Life

Mathematics around us

Mathematics expresses itself everywhere, in almost every facet of life - in nature all around us, and in the technologies in our hands. Mathematics is the language of science and engineering - describing ourunderstanding of all that we observe.

This presentation explores the many wonders and uses of mathematics in daily lives. This exhibition is divided into nine areas focusing on different aspects of mathematics.

Arithmetic progression

What is Arithmetic progression?

Arithmetic Progression is useful in predicting

an event if the pattern of the event is known.

For Example,

If an asset costs ‘v’ when new, and is depreciated by ‘d’ per year, its value each year can be represented by an arithmetic progression

v, v-d, v-2d, ....

Arithmetic progression in nature

Old Faithful is a popular attraction at Yellowstone National Park, because the geyser produces long eruptions that are fairly predictable.

When tourists visit Old Faithful, they will see a sign that indicates an estimated time that the geyser will next erupt.

No one controls the geyser like an amusement park ride.

Its patterns over time have caused park rangers to develop predictable eruption times using an arithmetic sequence.

Old Faithful

The time between eruptions is based on the length of the previous eruption

If an eruption lasts one minute, then the next eruption will occur in approximately forty-six minutes (plus or minus ten minutes).

If an eruption lasts two minutes, then the next eruption will occur in approximately fifty-eight minutes.

This pattern continues based on a constant difference of twelve minutes, forming an arithmetic sequence of 46, 58, 70, 82, 94, ..

An eruption of n minutes will indicate that the next eruption, an, will occur in an = a1 + (n î 1)d minutes, where a1is the length after a one-minute eruption, and d is the constant difference of waiting time among eruptions that are a one-minute difference in time.

In this particular situation, the next eruption will occur in an = 46+(n î 1)12 minutes, if the previous eruption was n minutes long.

Trigonometry in daily life

What is Trigonomet

ry?And how is it used?

Trigonometry is the branch of mathematics that studies triangles and their relationships.

Primitive forms of trigonometry were used in the construction of these wonders of the world.

The Pyramids of Giza

Architecture

In architecture, trigonometry plays a massive role in the compilation of building plans.

For example, architects would have to calculate exact angles of intersection for components of their structure to ensure stability and safety.

Some instances of trigonometric use in architecture include arches, domes, support beams, and suspension bridges.

Architecture remains one of the most important sectors of our society as they plan the design of buildings and ensure that they are able to withstand pressures from inside.

For millenia, trigonometry has played a major role in calculating distances between stellar objects and their paths.

Jantar Mantar observatory

Astronomy

Astronomy has been studied for millennia by civilizations in all regions of the world.

In our modern age, being able to apply Astronomy helps us to calculate distances between stars and learn more about the universe.

Astronomers use the method of parallax, or the movement of the star against the background as we orbit the sun, to discover new information about galaxies.

Menelaus’ Theorem helps astronomers gather information by providing a backdrop in spherical triangle calculation.

Geologists had to measure the amount of pressure that surrounding rocks could withstand before constructing the skywalk.

Grand Canyon Skywalk

Geology

Trigonometry is used in geology to estimate the true dip of bedding angles. Calculating the true dip allows geologists to determine the slope stability.

Although not often regarded as an integral profession, geologists contribute to the safety of many building foundations.

Any adverse bedding conditions can result in slope failure and the entire collapse of a structure.

Fibonacci series around us

What are

fibonacci series?

Were introduced in The Book of Calculating Series begins with 0 and 1 Next number is found by adding the last two

numbers together Number obtained is the next number in the

series Pattern is repeated over and over

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, …

Fn + 2 = Fn + 1 + Fn

Who Was Fibonacci?

~ Born in Pisa, Italy in 1175 AD~ Full name was Leonardo Pisano~ Grew up with a North African education under the Moors~Traveled extensively around the Mediterranean coast~ Met with many merchants and learned their systems of arithmetic~Realized the advantages of the Hindu-Arabic system

Fibonacci Numbers

~ Were introduced in The Book of Calculating ~ Series begins with 0 and 1 ~ Next number is found by adding the last two numbers

together ~ Number obtained is the next number in the series ~ Pattern is repeated over and over

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, …

Fn + 2 = Fn + 1 + Fn

Fibonacci’s Rabbits

Suppose a newly-born pair of rabbits (one male, one female) are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month, a female can produce another pair of rabbits. Suppose that the rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. How many pairs will there be in one year?

Fibonacci’s Rabbits Continued

End of the first month = 1 pair End of the second month = 2 pair End of the third month = 3 pair End of the fourth month = 5 pair 5 pairs of rabbits produced in one

year

1, 1, 2, 3, 5, 8, 13, 21, 34, …

Fibonacci Numbers in Nature

Fibonacci spiral is found in both snail and sea

shells


Sneezewort (Achillea ptarmica) shows the

Fibonacci numbers


Plants show the Fibonacci numbers in the arrangements of their leaves

Three clockwise rotations, passing five

leaves Two counter-clockwise rotations


Pinecones clearly show the Fibonacci spiral


A

Lilies and irises = 3 petals

Black-eyed Susan’s = 21 petalsCorn marigolds = 13 petals

Buttercups and wild roses = 5 petals


The Fibonacci numbers are found in the arrangement of seeds on flower heads

55 spirals spiraling outwards and 34 spirals spiraling inwards


Fibonacci spiral can be found in cauliflower


The Fibonacci numbers can be found in pineapples and bananas

Bananas have 3 or 5 flat sides Pineapple scales have Fibonacci spirals in sets of 8, 13, 21

Golden Section

What is golden

section ?

Represented by the Greek letter Phi Phi equals ± 1.6180339887 … and

± 0.6180339887 … Ratio of Phi is 1 : 1.618 or 0.618 : 1 Mathematical definition is Phi2 = Phi + 1 Euclid showed how to find the golden

section of a line

< - - - - - - - 1 - - - - - - - >

A G B

g 1 - g

GB = AG or 1 – g = g

so that g2 = 1 – g

Golden Section and Fibonacci Numbers

The Fibonacci numbers arise from the

golden section The graph shows a line whose gradient is

Phi First point close to the line is (0, 1) Second point close to the line is (1, 2) Third point close to the line is (2, 3) Fourth point close to the line is (3, 5) The coordinates are successive Fibonacci

numbers

Golden Section and Fibonacci Numbers

The golden section arises from the Fibonacci numbers Obtained by taking the ratio of successive terms in the

Fibonacci series Limit is the positive root of a quadratic equation and is

called the golden section

Golden Section and Geometry

Is the ratio of the side of a regular pentagon to its diagonal The diagonals cut each other with the golden ratio Pentagram describes a star which forms parts of many

flags

European Union

United States

Golden Section in Nature

Arrangements of leaves are the same as for seeds and petals

All are placed at 0.618 per turn Is 0.618 of 360o which is 222.5o

One sees the smaller angle of 137.5o

Plants seem to produce their leaves, petals, and seeds based upon the golden section

Golden Section in Architecture

Golden section appears in many of the

proportions of the Parthenon in Greece Front elevation is built on the golden section

(0.618 times as wide as it is tall)


Golden section can be found in the Great pyramid in Egypt Perimeter of the pyramid, divided by twice its vertical

height is the value of Phi


Golden section can be found in the design of Notre Dame in Paris

Golden section continues to be used today in modern architecture

United Nations Headquarters

Secretariat building

Golden Section in Art

Golden section can be found in Leonardo da Vinci’s artwork

The Annunciation


a

Madonna with Child and SaintsThe Last Supper


Golden section can be seen in Albrecht Durer’s paintings

Trento Nurnberg

Golden Section in Music

Stradivari used the golden section to place the f-holes in his famous violins

Baginsky used the golden section to construct the contour and arch of violins


Mozart used the golden section when composing music Divided sonatas according to the golden section Exposition consisted of 38 measures Development and recapitulation consisted of 62 measures Is a perfect division according to the golden

section


Beethoven used the golden section in his famous Fifth Symphony

Opening of the piece appears at the golden section point (0.618)

Also appears at the recapitulation, which is Phi of the way through the piece

Fibonacci Numbers and Golden Section at Towson

Fibonacci Numbers and Golden Section at Towson

Thank You-Rachit Bhalla

Education

Maths in Everyday Life