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MATH IN DAILY LIFE

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MATH IN DAILY LIFE

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  1. Mathematics in our daily life A presentation byA.MADHAN

  2. HOW MATHS CAN SAVE YOUR LIFE AND HELP YOU SEE THE WORLD IN A DIFFERENT WAY A.Madhan KENDRIYA VIDYALAYA MEG & CENTRE

  3. SOME COMMON VIEWS OF MATHEMATICS • MATHS IS HARD • MATHS IS BORING • MATHS HAS NOTHING TO DO WITH REAL LIFE • ALL MATHEMATICIANS ARE GEEKS!!!! BUT THE TRUTH IS THAT MATHS IS IMPORTANT IN CRIME DETECTION, MEDICINE, FINDINGLANDMINES AND EVEN DISNEYLAND !!!!!!!!!

  4. The modern world would not exist without maths With maths you can tell the future and save lives Maths lies at the heart of art and music DID YOU KNOW THAT -

  5. MATHS AND CRIME A short mathematical story • Burglar robs a bank • Escapes in getaway car • Pursued by police • GOOD NEWSPolice take a photo • BAD NEWS Photo is blurred

  6. Original Blurred

  7. SOLUTION Take the photo to a mathematician g(x) Blurring h(x) = f(x)*g(x) Original f(x) • Maths gives a formula for blurring convolution • By inverting the formula we can get rid of the blur

  8. Processed image : Image Processing

  9. MATHS AND PICTURES • PICTURES AND IMAGES ARE ALL AROUND US • TV • DVD • COMPUTER GRAPHICS • SPECIAL EFFECTS IMAGES ARE STORED AS NUMBERS WITH THESE NUMBERS WE CAN PROCESS THE PICTURES BY USING MATHEMATICS

  10. SOME APPLICATIONS PRODUCING THE PICTURES IN THE FIRST PLACE TRANSMITTING THE PICTURES WITHOUT MISTAKES Error Correcting Codes

  11. SOME MORE APPLICATIONS DEBLURRING ORIGINALS FINDING THINGS HIDDEN IN AN IMAGE Edges BrainsLandmines

  12. MATHS AND MEDICINE Modern medicine has been transformed by methods of seeing Inside you without cutting you open! • Ultra sound: sound waves • MRI: magnetism • CAT scans:X rays ALL USE MATHS TO WORK!!

  13. WHAT IS A CAT SCAN?? CAT = Computerised axial tomography Based on X-Rays discovered by Roengten X-Rays cast a shadow GOOD for looking at bones BAD for looking at soft tissue

  14. USING MODERN MATHS WE CAN DO A LOT BETTER Modern CATscanner CAT scanners work by casting many shadows with X-rays and using maths to assemble these into a picture

  15. USING SIMPLE MATHEMATICS, WE CAN SAVE OUR SOLDIERS LIVES Land mines are hidden in foliage and triggered by trip wires Trip wires are well hidden – can they be quickly and safely detected??

  16. Find the trip wires in this picture

  17. Digital picture of foliage is taken by camera on a long pole Image intensity f(x,y) Trip wires are like X-Rays R(ρ,θ) f(x,y) • Radon transform • • y ρ • x θ Points of high intensity in R correspond to trip wires Isolate points and transform back to find the wires

  18. Mathematics finds the land mines! Used by the Canadian Peace keeping forces

  19. And now for the most interesting part!!! GUESS WHAT?

  20. The Fibonacci Sequence and Phi

  21. What Is the Fibonacci Sequence? • The sequence begins with one. Each subsequent number is the sum of the two preceding numbers. • Fib(n) = Fib(n-1) + Fib(n-2) • Thus the sequence begins as follows: • 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144….

  22. Fibonacci’s Rabbits • Fibonacci applied his sequence to a problem involving the breeding of rabbits. • Given certain starting conditions, he mapped out the family tree of a group of rabbits that initially started with only two members. • The number of rabbits at any given time was always a Fibonacci number. • Unfortunately, his application had little practical bearing to nature, since incest and immortality was required among the rabbits to complete his problem.

  23. 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 our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was... How many pairs will there be in one Year?

  24. solution • At the end of the first month, they mate, but there is still one only 1 pair. • At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field. • At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field. • At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs. If we continue this pattern, we would get 377 pairs of rabbits in one year.

  25. Besides Incestuous Rabbits, What Good Does the Sequence Do Anybody? • The Fibonacci sequence has far more applications than immortal rabbits. • Fibonacci numbers have numerous naturally-occurring applications, ranging from the very basic to the complex geometric.

  26. Fibonacci Numbers in Nature • Many aspects of nature are grouped in bunches equaling Fibonacci numbers. • For example, the number of petals on a flower tend to be a Fibonacci number.

  27. Some F l ower Petal Examples • 3 petals: lilies • 5 petals: Buttercups, roses • 8 petals: Delphinium • 13 petals: Marigolds • 21 petals: Black-eyed Susana • 34 petals: Pyrethrum • 55 or 89 petals:Daisies

  28. Leaves and Branching Plants • Leaves are also found in groups of Fibonacci numbers. • Branching plants always branch off into groups of Fibonacci numbers.

  29. Think about yourself. You should have: 5 fingers on each hand 5 toes on each foot 2 arms 2 legs 2 eyes 2 ears 2 sections per leg 2 sections per arm We could go on, forever citing examples Think About Yourself....

  30. Deeper Applications: The Fibonacci Spiral • Fibonacci numbers have geometric applications in nature as well. • The most prominent of these is the Fibonacci spiral.

  31. Construction of the Fibonacci Spiral • The Fibonacci spiral is constructed by placing together rectangles of relative side lengths equaling Fibonacci numbers.

  32. Construction of the Fibonacci Spiral • A spiral can then be drawn starting from the corner of the first rectangle of side length 1, all the way to the corner of the rectangle of side length 13.

  33. Fibonacci Spiral in nature CauliflowerPine Cone

  34. Some More Fibonacci Applications: Music • Music involves several applications of Fibonacci numbers. • A full octave is composed of 13 total musical tones, 8 of which make up the actual musical octave.

  35. Fibonacci Numbers in Music

  36. F

  37. Now For the Good Stuff….PHI!! • One of the most significant applications of the Fibonacci sequence is a number that mathematicians refer to as Phi (F). • It looks very similar to Flux. In this case, F refers to a very important number that is known as the golden ratio.

  38. THE DIVINE NUMBER, THE GOLDEN RATIO, THE HOLY RATIO, all refer to this : 1.618 Approximately

  39. So What Is Phi Anyway? • Phi is defined as the limit of the ratio of a Fibonacci number i and its predecessor, Fib(i-1). • Mathematically, this number is equal to: or approximately 1.618034.

  40. Some Mathematical Properties of Phi • Phi can be derived by the equation: • With some fancy factoring and division, you get: • This implies that Phi’s reciprocal is smaller by 1. It is .618034, also known as phi (f).

  41. Some Geometric Properties of Phi • Is there anything mathematically definitive about F when used in geometry? You bet there is. • A rectangle whose sides are in the golden ratio is referred to as a golden rectangle. • When a golden rectangle is squared, the remaining area forms another golden rectangle!

  42. Mathematical Applications of Phi • WithoutF,in order to find any Fibonacci number, you would need to know its two preceding Fibonacci numbers. • But with Fat your service, you can find any Fibonacci number knowing only its place in the sequence!

  43. Finding the nth Fibonacci Number Using Phi: Binet’s Equation

  44. Natural Applications of Phi • Remember how flowers have leaves and petals arranged in sets of Fibonacci numbers? • This ensures that there are F leaves and petals per turn of the stem, which allows for maximum exposure to sunlight, rain, and insects.

  45. More Natural Applications of Phi • How about your body? • You have NO IDEA how many segments of the human body are related in size to each other by F!

  46. A Closer Look at Some Examples of the Body and Phi • The human arm: • The human finger:

  47. The World is No Stranger to Phi (Even if you were up to until a few mintues ago) • When used in dimensioning objects, it has always been thought that F produces the most visually appealing results. • Many marketers have used F in their products over the years to make them more attractive to you. • An extremely basic example: 3 x 5 greeting cards.

  48. There are numerous other applications of the Fibonacci sequence, Fibonacci numbers, and Fthat were not covered in this presentation—simply because there are far too many to list.

  49. “Mathematics expresses values that reflect the cosmos, including orderliness, balance, harmony, logic, and abstract beauty.” ― Deepak Chopra “Since the mathematicians have invaded the theory of relativity I do not understand it myself any more.” ― Albert Einstein

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