What have mathematicians done for us? Chris Budd

What have mathematicians done for us? Chris Budd

Some common views on math and mathematicians Mathematics is completely useless Mathematicians are evil souless geeks All Mathematicians are mad!

Flight delayed after passenger becomes suspicious of equation -. News Sport Weather http://www.bbc.co.uk/news/world-us-canada-36240523 iPlayer TV Radio Find local news This can cause serious problems Home UK World US & Canada Business Politics Tech Science Health US Election 2016 Flight delayed after passenger becomes suspicious of equation 8 May 2016 US & Canada Share SPL We can't fly on together with suspicious maths: the flight was delayed for two hours An Italian economist says his flight was delayed after a fellow passenger saw him working on a differential equation and alerted the cabin crew. Guido Menzio was taken off and questioned by agents who did not identify themselves, after the woman next to him said she felt ill. 1 of 5 30/09/2016 10:34

And it is completely false The modern world would not exist without math Math lies at the heart of all modern technology

But still Too few people recognize that the high technology so celebrated today is essentially a mathematical technology Edward David, ex-president of Exxon R&D

Possible reasons for this 1. Mathematics is quite hard to define Today, no consensus on the definition of mathematics prevails, even among professionals. There is not even consensus on whether mathematics is an art or a science. A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable. Some just say, "Mathematics is what mathematicians do.” [Wikipedia] My own ‘definition’ is that math is

2. No one knows who mathematicians are Maxwell and the discovery of electromagnetic waves Electromagnetism, radio, WiFi,TV, radar, mobile phones, microwaves all come from the work of Maxwell!

The most famous ever female mathematician? Florence Nightingale Medical statistics

3. Math is vital but like the air we breath is often invisible

Now show how math has changed history 1. Early math, and the quadratic equation 1. Math and music 1. Math amazes the Internet 2. Math makes waves 3. Math saves lives 4. Math helps us communicate 5. Where next?

The virtuous circle of math Abstract math Good things happen here Applications of math

1. Early math and the quadratic equation Early people counted on their fingers This led to the natural numbers 1,2,3,4,5 . Numbers recorded on Babylonian cuneiform tablets

Important early mathematical problems I have 7 cows, the taxman takes 5, how many do I have now? I have 5 cows, the taxman takes 7, how many do I have now? I have 5 cows and 3 children. How many cows do they get? Gives us the integers, negative numbers and fractions 1, 0 , -1, -2 , 3/5, 7/8, 9/7, Leading to number theory, discrete mathematics, cryptography

Taxman again: I want you to double the amount of crops that you give me Area 2 Area 1 1 x

Quadratic equation Early calculation of the square root of 2 1.4142135623730950488 Irrational number . Led ultimately to mathematical analysis Similar looking equation:

2. Mathematical music Some musical notes sound better when played together than others The octave C to C The notes C and G (a perfect 5th) The notes C and E (a perfect 3rd)

Reason was discovered by Pythagoras Length of strings giving C and G, and C and E, were in simple fractional proportions C:C 2/1 C:G 3/2 C:E 4/3

A right-angled triangle opined My hypotenuse squared is refined For if any one cares It’s the sum of the squares Of my other two sides when combined!

Pythagoras invented the Just Scale . Sequence of notes with frequencies in simple fractional proportions 1 : 9/8 : 5/4 : 4/3 : 3/2 : 5/3 : 15/8 : 2

Problem: Keyboard instruments could only be tuned for one key Mathematicians invented a new Well Tempered scale with all notes in the same proportion A geometric progression of the semi-tone frequencies, ratio which works well in all keys

Mathematical ‘fluke’: Interval C:G on the well tempered scale Which is very close to 3/2 Well tempered scale sounds good Wouldn’t work any like so well if there were not 12 semi-tones in the scale

3. Math amazes the Internet One of the earliest myths which features both math and computer science is the story of Theseus and the Minotaur

Draw a labyrinth starting from a seed

Labyrinths on the underground Mark Wallinger

Later became the Puzzle maze eg. Hampton Court Solved by Euler, who developed the theory of networks to do it

Maze Network

Facebook Network

Understanding networks, combined with Matrix Theory (due to Cayley) and eigenvectors Wolfram also forms a major part of the algorithms behind

4. Math makes waves Fourier was studying the temperature T of a heated bar Heat equation IDEA: Express T in terms of simpler building bricks Solve a set of simple problems in terms of these

Astonishing idea Build T out of sines and cosines Fourier Series

Now use Fourier series to make up any shape of wave

Example: Computing the tides using Kelvin’s tidal computer Used in D-Day Now the basis of the modern synthesizer and most of modern electronics

Babbage’s difference engine Designed to compute the tables for maritime navigation Ideas behind it led directly to the modern computer

4. Mathematicians save lives Studied shadows cast by objects Radon 1917 Can you reconstruct a shape just by knowing its shadows?

Shadow Object (find using ideas from Fourier)

Modern CAT (Computerised Axial Tomography) scanner implements this and related formulae to look inside you. Also used to X-ray mummies Detect land mines Save bees

6. Math communicates

Error correcting codes. Used to store the numbers 0,1,2,3,4,5,6,7 and other data in such a way that any errors can not only be detected but corrected.

Work by representing the numbers by codes which are as different as possible so we can still tell the right answer even if it has mistakes in it Invented in the 1940s by Shannon and Hamming in the Bell Labs Using very fancy maths (Galois theory)

Examples of Error correcting codes. Hamming (7,4) code 1950: Uses three parity digits Can correct and detect a single bit error

Original message x (d1,d2,d3,d4) Transmitted code y (p1,p2,d1,p3,d2,d3,d4) Linear code y Gx

Receive code z instead of y Magic of the method: Calculate d Hz If d (0 0 0) then z has no errors Otherwise d reversed is the binary digit of the error

Reed-Solomon code 1960 Polynomials over finite fields

And finally . This brief overview of what mathematicians have done for us is meant mainly to whet your appetite. There many more applications of mathematics to the modern world, and the latest developments in math are likely to lead to even newer technologies You are the people who will lead us forward in this!