This may well be a perennial question from some of our students struggling with those tricky algebra problems on a warm afternoon. To give them a succinct answer; Mathematics has given us most of the things we have in our Modern world although sometimes, I admit, it does take time for the maths to find its use.

In the Ancient world, a similar question was asked - what have the Romans ever done for us? ... which received the begrudging answer ... *“better sanitation, medicine, education, wine, public order, irrigation, roads, a fresh water system and public health ... and peace”* ... but their counting system was mainly additive without a zero or negative numbers, which made multiplication and division difficult. The superior Hindu-Arabic system which we use today is a positional system, containing zero, negative numbers and fractions as well as irrational numbers (√2) and more recently, imaginary numbers (√(-2)).

The Ancient Greeks saw their world through Geometry with numbers to the fore. Euclid studied prime numbers over 2,000 years ago and gave a clever proof that there were an infinite number of them. Mathematicians throughout the centuries have explored the primes and a search for a pattern in their distribution continues today. That may seem pointless except for the fact that prime numbers are now used to protect our credit card numbers and our computer passwords. If order is found in the prime numbers, there would be chaos.

And in the 18th Century, Swiss Mathematician Leonard Euler solved the famous puzzle of the Seven Bridges of Königsberg which led to the theory of networks. That now forms the basis of the internet, genetics and disease control. Understanding networks combined with Matrix theory, which was developed by Arthur Cayley in the 19th Century, form a major part of the algorithms behind search engines like *Google*.

In the 19th Century, James Maxwell found a link between magnetism, electricity and light and his famous equations gave us the solutions for Micro Waves, WiFi, Radio, TV and Mobile Phones. Around the same time, Florence Nightingale who was the founder of modern nursing, was also a pioneer in Statistics. She noticed that during the Crimean War, far more soldiers were dying from illnesses caused by poor hygiene than on the battlefield. She used her Rose diagrams to explain these statistics to her political masters in a visual and meaningful way with the subsequent saving of countless lives.

Mathematics has even resonated through our Music! Pythagoras, who studied the lengths of the sides of right angled triangles, also studied the lengths of strings that gave musical notes and discovered that the ratio of their lengths were always simple fractions, for example, 1 : 9/8 : 5/4 : 4/3 : 3/2 : 5/3 : 15/8 : 2. This is just for the Just Scale (C Major). Other scales are a little more complicated but mathematicians came up with the Well Tempered Scale (12 notes per octave) with the frequency of all notes in the same successive proportion of ^{12}√(2) which is about 1.059. This allows musical scores to be readily transposed between keys.

A major problem for the early seafarers was the problem of locating a ship’s longitude and was finally solved by John Harrison in the 18th Century with his marine chronometer. Many calculations were still needed to produce tables, called Ephemerides and these then needed a 22 step calculation on board the ship to pinpoint the location. Charles Babbage designed his calculating machine (called a difference engine) in order to simplify the process and, although it was never built, that laid the foundation for the modern computer.
Fourier’s heat equation led to him developing the Fourier series which meant that any type of wave equation could be solved. Lord Kelvin (of Kelvin temperature scale fame) used these ideas to develop a tidal computer and similar machines were used to predict the tides for the D Day landing in World War 2. The tides in the English Channel are complex and even thwarted Julius Caesar’s invasion of Britain in 55 BCE. These days the Fourier series forms the basis of modern electronics. In 1917, Austrian mathematician Johann Radon developed formulae for looking at the shadows of objects, which 60 years later are used in medical scanning machines, now routinely used as diagnostic tools.

And even when we do make mistakes, Mathematics has given us error correcting codes which were based on the ideas of Evariste Galois, a French mathematician in the 19th Century. These are now used in Mobile Phones, Digital TVs and Radios, Satellites, CD’s and Bar Codes.

So, apart from their development of Calculators, Computers, the Internet, Mobile phones, TV and Radio, Music and saving lives; what has Mathematics ever done for us?

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