Applications of Mathematics - Numeracy Program · Applications of Mathematics. Applications of...

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Transcript of Applications of Mathematics - Numeracy Program · Applications of Mathematics. Applications of...

Page 1: Applications of Mathematics - Numeracy Program · Applications of Mathematics. Applications of Calculus Calculus is used in: Physics, as the tool to embed motion and ... with sleight-of-hand

Please Note

These pdf slides are configured for viewing on a computer screen.

Viewing them on hand-held devices may be difficult as they require a “slideshow”mode.

Do not try to print them out as there are many more pages than the number ofslides listed at the bottom right of each screen.

Apologies for any inconvenience.

Page 2: Applications of Mathematics - Numeracy Program · Applications of Mathematics. Applications of Calculus Calculus is used in: Physics, as the tool to embed motion and ... with sleight-of-hand

Applications of MathematicsNumeracy Program

Adrian Dudek

Adrian Dudek

Applications of Mathematics

Page 3: Applications of Mathematics - Numeracy Program · Applications of Mathematics. Applications of Calculus Calculus is used in: Physics, as the tool to embed motion and ... with sleight-of-hand

Introduction

These slides hope to demonstrate that mathematics can be both applicable andinteresting.

Website: Slides, notes, worksheets.

http://www.studysmarter.uwa.edu.au

Email: [email protected]

Adrian Dudek

Applications of Mathematics

Page 4: Applications of Mathematics - Numeracy Program · Applications of Mathematics. Applications of Calculus Calculus is used in: Physics, as the tool to embed motion and ... with sleight-of-hand

What day of the week were you born on?

Some people remember what day of the week they were born on.

For those who don’t, there is a nifty little formula that you can use!

Alternatively, you could just look it up in a calendar... (the boring approach)

Adrian Dudek

Applications of Mathematics

Page 5: Applications of Mathematics - Numeracy Program · Applications of Mathematics. Applications of Calculus Calculus is used in: Physics, as the tool to embed motion and ... with sleight-of-hand

What day of the week were you born on?

Before we see this formula there are two little things we have to note.

Firstly, in higher mathematics, [x] means to round x down to the nearest wholenumber.

For example, [3.743] = 3.

Secondly, a mod n refers to the remainder when a is divided by n.

For example, 17 mod 5 = 2 and 33 mod 4 = 1.

Adrian Dudek

Applications of Mathematics

Page 6: Applications of Mathematics - Numeracy Program · Applications of Mathematics. Applications of Calculus Calculus is used in: Physics, as the tool to embed motion and ... with sleight-of-hand

What day of the week were you born on?

The formula is as follows.

W=(D+[2.6((M+9) mod 12) +2.4] - 2C + [5Y/4]+[C/4]-[((M+9) mod 12)/10]) mod 7

where

D is the day of the month from {0, 1, . . . 31}.M is the month from {1, . . . , 12}.C is the century, and Y is the year. So, for people born in 1989, we set C = 19and Y = 89.

W is the week day (Sunday = 0,Monday = 1,. . . ,Saturday = 6).

Adrian Dudek

Applications of Mathematics

Page 7: Applications of Mathematics - Numeracy Program · Applications of Mathematics. Applications of Calculus Calculus is used in: Physics, as the tool to embed motion and ... with sleight-of-hand

What day of the week were you born on?

Let’s use this formula. If somebody was born on the 28th of September 1989 , then

D = 28

M = 9

C = 19

Y = 89

Adrian Dudek

Applications of Mathematics

Page 8: Applications of Mathematics - Numeracy Program · Applications of Mathematics. Applications of Calculus Calculus is used in: Physics, as the tool to embed motion and ... with sleight-of-hand

What day of the week were you born on?

W=(D+[2.6((M+9) mod 12) +2.4] - 2C + [5Y/4]+[C/4]-[((M+9) mod 12)/10]) mod 7

W = (28 + [2.6(18mod12) + 2.4]− 38 + [111.25] + [4.75]− [(18mod12)/10])mod7

= (28 + [2.6× 6 + 2.4]− 38 + 111 + 4− [6/10])mod7

= (28 + [18]− 38 + 111 + 4− 0)mod7

= (28 + 18− 38 + 111 + 4− 0)mod7

= 123mod7

= 4

The number 4 corresponds to Thursday!

Adrian Dudek

Applications of Mathematics

Page 9: Applications of Mathematics - Numeracy Program · Applications of Mathematics. Applications of Calculus Calculus is used in: Physics, as the tool to embed motion and ... with sleight-of-hand

Question Time!

Question. As an eight year old, the mathematical prodigy Carl Gauss was able tocorrectly add up all of the whole numbers from 1 to 100 within a couple of seconds.

What was the answer he gave?

5050

Adrian Dudek

Applications of Mathematics

Page 10: Applications of Mathematics - Numeracy Program · Applications of Mathematics. Applications of Calculus Calculus is used in: Physics, as the tool to embed motion and ... with sleight-of-hand

Question Time!

Question. As an eight year old, the mathematical prodigy Carl Gauss was able tocorrectly add up all of the whole numbers from 1 to 100 within a couple of seconds.

What was the answer he gave?

5050

Adrian Dudek

Applications of Mathematics

Page 11: Applications of Mathematics - Numeracy Program · Applications of Mathematics. Applications of Calculus Calculus is used in: Physics, as the tool to embed motion and ... with sleight-of-hand

Topology

A very famous theorem in topology says

You can’t comb the hair on a coconut flat without creating a cowlick.

Adrian Dudek

Applications of Mathematics

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The Hairy Ball Theorem

This is called The Hairy Ball Theorem.

The theorem itself is actually quite hard to prove.

If you are super interested in this, then MATH3333 Introduction to GeometricTopology is a good place to start, before doing MATH7432: Algebraic Topology in

your honours year.

The proof is often presented at the end of the MATH7432 course, as it requires allprior material.

Adrian Dudek

Applications of Mathematics

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An Amazing Consequence

You can think of the Earth as a ball, with the wind patterns as hair.

The Hairy Ball Theorem dictates that there must be at least one point on the Earthwhere there is a tuft of wind.

This tuft corresponds to a cyclone, possibly a very small or very large one.

At any one time on Earth, there must be a cyclone somewhere.

Adrian Dudek

Applications of Mathematics

Page 14: Applications of Mathematics - Numeracy Program · Applications of Mathematics. Applications of Calculus Calculus is used in: Physics, as the tool to embed motion and ... with sleight-of-hand

Question Time!

Question. Which of the following books was written by a mathematician?

Winnie the Pooh

Alice in Wonderland

Oliver Twist

Sherlock Holmes

Adrian Dudek

Applications of Mathematics

Page 15: Applications of Mathematics - Numeracy Program · Applications of Mathematics. Applications of Calculus Calculus is used in: Physics, as the tool to embed motion and ... with sleight-of-hand

Question Time!

Question. Which of the following books was written by a mathematician?

Winnie the Pooh

Alice in Wonderland - Lewis Carrol

Oliver Twist

Sherlock Holmes

Adrian Dudek

Applications of Mathematics

Page 16: Applications of Mathematics - Numeracy Program · Applications of Mathematics. Applications of Calculus Calculus is used in: Physics, as the tool to embed motion and ... with sleight-of-hand

The Monty Hall Problem

Imagine you are on a game-show.

There are three doors, and you are to open one to reveal your prize!

Behind one of the doors, there is one million dollars.

Behind the other two doors, there is nothing.

Adrian Dudek

Applications of Mathematics

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The Monty Hall Problem

You choose a door to open, but before you see if you are right, the game-show hostopens another door, revealing nothing.

He then asks you if you would like to keep your option, or switch to the otherunopened door?

What should you do, to maximise your chances of winning one million dollars? Ordoes it not matter which door you choose?

Adrian Dudek

Applications of Mathematics

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The Monty Hall Problem

You should always switch. It doubles your chance of winning.

When this was first published in 1990 in a column in Parade Magazine, approximately10 000 readers, including 1000 of which had PhD’s, wrote to the magazine claiming

that switching was wrong, and that it didn’t matter.

Try it with a friend: You let him be the game-show host, and you can be thecontestant. Switch doors every time you play the game, and record your wins/losses.

You should find that you win 2/3 of the time.

Adrian Dudek

Applications of Mathematics

Page 19: Applications of Mathematics - Numeracy Program · Applications of Mathematics. Applications of Calculus Calculus is used in: Physics, as the tool to embed motion and ... with sleight-of-hand

Question Time!

Question. Which of the following is not a published mathematical theorem?

The British Flag Theorem

Doob’s First Martingale Convergence Theorem

The Stray Dingo Theorem

The Hairy Ball Theorem

The Cyclic Surgery Theorem

Adrian Dudek

Applications of Mathematics

Page 20: Applications of Mathematics - Numeracy Program · Applications of Mathematics. Applications of Calculus Calculus is used in: Physics, as the tool to embed motion and ... with sleight-of-hand

Question Time!

Question. Which of the following is not a published mathematical theorem?

The British Flag Theorem

Doob’s First Martingale Convergence Theorem

The Stray Dingo Theorem

The Hairy Ball Theorem

The Cyclic Surgery Theorem

Adrian Dudek

Applications of Mathematics

Page 21: Applications of Mathematics - Numeracy Program · Applications of Mathematics. Applications of Calculus Calculus is used in: Physics, as the tool to embed motion and ... with sleight-of-hand

Imaginary Numbers

If you were asked to solve the equation

x2 = 1

then you would find two solutions, x = 1 and x = −1.

What if instead you were asked to solve

x2 = −1.

Well, no number squares to −1 right?

Adrian Dudek

Applications of Mathematics

Page 22: Applications of Mathematics - Numeracy Program · Applications of Mathematics. Applications of Calculus Calculus is used in: Physics, as the tool to embed motion and ... with sleight-of-hand

Imaginary Numbers

Historically, mathematicians have pretended that there is a number x which satisfiesx2 = −1.

They call this number i , and then continued to extend the theory surrounding thisnumber i .

The term “imaginary number” was coined for i , in an initial attempt to be derogatoryto those who studied it.

Adrian Dudek

Applications of Mathematics

Page 23: Applications of Mathematics - Numeracy Program · Applications of Mathematics. Applications of Calculus Calculus is used in: Physics, as the tool to embed motion and ... with sleight-of-hand

Imaginary Numbers

The imaginary number i is essential for a range of applications in science andengineering.

Particularly, any device with an alternating current relies fundamentally on the notionof an imaginary number.

Adrian Dudek

Applications of Mathematics

Page 24: Applications of Mathematics - Numeracy Program · Applications of Mathematics. Applications of Calculus Calculus is used in: Physics, as the tool to embed motion and ... with sleight-of-hand

Question Time!

Question. For a fraction, what is the name of the line which separates the numeratorfrom the denominator?

The vinculum

Adrian Dudek

Applications of Mathematics

Page 25: Applications of Mathematics - Numeracy Program · Applications of Mathematics. Applications of Calculus Calculus is used in: Physics, as the tool to embed motion and ... with sleight-of-hand

Question Time!

Question. For a fraction, what is the name of the line which separates the numeratorfrom the denominator?

The vinculum

Adrian Dudek

Applications of Mathematics

Page 26: Applications of Mathematics - Numeracy Program · Applications of Mathematics. Applications of Calculus Calculus is used in: Physics, as the tool to embed motion and ... with sleight-of-hand

Ciphers and Cryptography

Mathematics is used to encode information for the purpose of making it more secure.

Ciphers have been used throughout history to transmit information.

The problem: How do we make ciphers that are difficult to break?

Information is easily shared between two parties if they both share a key to unlock it.

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 254 9 14 19 24 3 8 13 18 23 2 7 12 17 22 1 6 11 16 21 0 5 10 15 20 25E J O T Y D I N S X C H M R W B G L Q V A F C P U Z

Adrian Dudek

Applications of Mathematics

Page 27: Applications of Mathematics - Numeracy Program · Applications of Mathematics. Applications of Calculus Calculus is used in: Physics, as the tool to embed motion and ... with sleight-of-hand

Ciphers and Cryptography: RSA Cryptosystem (1978)

Let’s say you want to send a box of information to a friend, but you want to do sowithout meeting up with him/her beforehand and swapping keys for your padlocks.

You and your friend each have a different padlock and corresponding key.

How can you get the contents of your box to your friend, keeping the informationinside of it locked to anybody who intercepts it along the way?

Hint: Do not worry about the cost of sending it back and forth.

Adrian Dudek

Applications of Mathematics

Page 28: Applications of Mathematics - Numeracy Program · Applications of Mathematics. Applications of Calculus Calculus is used in: Physics, as the tool to embed motion and ... with sleight-of-hand

Ciphers and Cryptography: RSA Cryptosystem (1978)

The solution to this problem goes as follows:

1 Padlock the box and send it to your friend.

2 Your friend adds his padlock to the box and sends it back.

3 Remove your padlock, and send it back.

4 Your friend removes the padlock, and now has the information it contains.

In reality, we lock our information using prime numbers.

Adrian Dudek

Applications of Mathematics

Page 29: Applications of Mathematics - Numeracy Program · Applications of Mathematics. Applications of Calculus Calculus is used in: Physics, as the tool to embed motion and ... with sleight-of-hand

Question Time!

Question. True or False?

0.9999999 · · · = 1

True

Adrian Dudek

Applications of Mathematics

Page 30: Applications of Mathematics - Numeracy Program · Applications of Mathematics. Applications of Calculus Calculus is used in: Physics, as the tool to embed motion and ... with sleight-of-hand

Question Time!

Question. True or False?

0.9999999 · · · = 1

True

Adrian Dudek

Applications of Mathematics

Page 31: Applications of Mathematics - Numeracy Program · Applications of Mathematics. Applications of Calculus Calculus is used in: Physics, as the tool to embed motion and ... with sleight-of-hand

Calculus

Calculus is arguably the single most important discovery/invention in all of history.

[People often debate about whether maths is discovered or invented.]

Given a relationship between two or more quantities, calculus involves findingexact expressions for the rate of change of a quantity in terms of the other

quantities.

It holds the ability to embed motion and change into the equations we use. Before itsbirth, mathematics was very static.

Adrian Dudek

Applications of Mathematics

Page 32: Applications of Mathematics - Numeracy Program · Applications of Mathematics. Applications of Calculus Calculus is used in: Physics, as the tool to embed motion and ... with sleight-of-hand

Applications of Calculus

Calculus is used in:

Physics, as the tool to embed motion and flow into equations.

Chemistry, to compute reaction rates.

Biology, to model population growth.

Economics, to model compound interest.

and more . . .

Adrian Dudek

Applications of Mathematics

Page 33: Applications of Mathematics - Numeracy Program · Applications of Mathematics. Applications of Calculus Calculus is used in: Physics, as the tool to embed motion and ... with sleight-of-hand

Applications of Calculus

The discovery of calculus is credited to two people: Isaac Newton, and GottfriedLeibniz.

Apparently, Newton discovered first, although Leibniz published first.

The name calculus (latin, small stone used for counting) was chosen by Leibniz.

Isaac Newton chose to call it the Science of Fluxions, which evidently did not stick.

Adrian Dudek

Applications of Mathematics

Page 34: Applications of Mathematics - Numeracy Program · Applications of Mathematics. Applications of Calculus Calculus is used in: Physics, as the tool to embed motion and ... with sleight-of-hand

Question Time!

Question. Which two Australian Universities tied for highest rank in the 2010Australian Research Council report on pure mathematics?

ANU and UWA

Adrian Dudek

Applications of Mathematics

Page 35: Applications of Mathematics - Numeracy Program · Applications of Mathematics. Applications of Calculus Calculus is used in: Physics, as the tool to embed motion and ... with sleight-of-hand

Question Time!

Question. Which two Australian Universities tied for highest rank in the 2010Australian Research Council report on pure mathematics?

ANU and UWA

Adrian Dudek

Applications of Mathematics

Page 36: Applications of Mathematics - Numeracy Program · Applications of Mathematics. Applications of Calculus Calculus is used in: Physics, as the tool to embed motion and ... with sleight-of-hand

Randomness

The study of randomness is an intriguing area of mathematics.

Is any event truly random?

Einstein: “God does not play dice with the universe”

Stephen Hawking: “Not only does God play dice, but he sometimes confuses us bythrowing them where they can’t be seen”

Adrian Dudek

Applications of Mathematics

Page 37: Applications of Mathematics - Numeracy Program · Applications of Mathematics. Applications of Calculus Calculus is used in: Physics, as the tool to embed motion and ... with sleight-of-hand

Randomness

Persi Diaconis dropped out of high school and left home at the age of 14 to travelwith sleight-of-hand legend Dai Vernon.

Persi later returned to school and by the age of 29 had a PhD in MathematicalStatistics from Harvard University.

Through sheer persistence, Persi trained himself to control the flip of a coin.

Adrian Dudek

Applications of Mathematics

Page 38: Applications of Mathematics - Numeracy Program · Applications of Mathematics. Applications of Calculus Calculus is used in: Physics, as the tool to embed motion and ... with sleight-of-hand

Randomness

Persi Diaconis is known for determining how much shuffling is actually needed to makea deck of cards completely random.

We define a shuffle as taking a deck of cards, splitting it into 2, and then interlacingthese two parts together, so that the cards alternate on average.

It takes seven shuffles to randomise a deck.

Adrian Dudek

Applications of Mathematics

Page 39: Applications of Mathematics - Numeracy Program · Applications of Mathematics. Applications of Calculus Calculus is used in: Physics, as the tool to embed motion and ... with sleight-of-hand

Question Time!

Question. In a randomly selected group of 23 people, what is the probability that atleast 2 people have the same birthday?

2%

8%

23%

35%

51%

Adrian Dudek

Applications of Mathematics

Page 40: Applications of Mathematics - Numeracy Program · Applications of Mathematics. Applications of Calculus Calculus is used in: Physics, as the tool to embed motion and ... with sleight-of-hand

Question Time!

Question. In a randomly selected group of 23 people, what is the probability that atleast 2 people have the same birthday?

2%

8%

23%

35%

51%

Adrian Dudek

Applications of Mathematics

Page 41: Applications of Mathematics - Numeracy Program · Applications of Mathematics. Applications of Calculus Calculus is used in: Physics, as the tool to embed motion and ... with sleight-of-hand

Using STUDYSmarter Resources

This resource was developed for UWA students by the STUDYSmarter team for thenumeracy program. When using our resources, please retain them in their original

form with both the STUDYSmarter heading and the UWA crest.

Adrian Dudek

Applications of Mathematics