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Dr. Hannah Fry takes us on a whistle-stop tour around the mathematics of success, to help us understand how to get more of what we want in our own lives. From the best way to bag a budget dinner or keep the kids quiet, to averting nuclear Armageddon and negotiating global climate change agreements.

Without us noticing, modern life has been taken over. Algorithms run everything from search engines on the internet to satnavs and credit card data security - they even help us travel the world, find love and save lives. Professor Marcus du Sautoy demystifies the hidden world of algorithms. By showing us some of the algorithms most essential to our lives, he reveals where these 2,000-year-old problem solvers came from, how they work, what they have achieved and how they are now so advanced they can even programme themselves.

Writing itself is 5,000 years old, and for most of that time words were written by hand using a variety of tools. The Romans were able to run an empire thanks to documents written on papyrus. Scroll books could be made quite cheaply and, as a result, ancient Rome had a thriving written culture. With the fall of the Roman Empire, papyrus became more difficult to obtain. Europeans were forced to turn to a much more expensive surface on which to write: Parchment. Medieval handwritten books could cost as much as a house, they also represent a limitation on literacy and scholarship.

No such limitations were felt in China, where paper had been invented in the second century. Paper was the foundation of Chinese culture and power, and for centuries how to make it was kept secret. When the secret was out, paper mills soon sprang up across central Asia. The result was an intellectual flourishing known as the Islamic Golden Age. Muslim scholars made discoveries in biology, geology, astronomy and mathematics. By contrast, Europe was an intellectual backwater.

That changed with Gutenbergâ€™s development of movable type printing. The letters of the Latin alphabet have very simple block-like shapes, which made it relatively simple to turn them into type pieces. When printers tried to use movable type to print Arabic texts, they found themselves hampered by the cursive nature of Arabic writing. The success of movable type printing in Europe led to a thousand-fold increase in the availability of information, which produced an explosion of ideas that led directly to the European Scientific Revolution and the Industrial Revolution that followed.

No such limitations were felt in China, where paper had been invented in the second century. Paper was the foundation of Chinese culture and power, and for centuries how to make it was kept secret. When the secret was out, paper mills soon sprang up across central Asia. The result was an intellectual flourishing known as the Islamic Golden Age. Muslim scholars made discoveries in biology, geology, astronomy and mathematics. By contrast, Europe was an intellectual backwater.

That changed with Gutenbergâ€™s development of movable type printing. The letters of the Latin alphabet have very simple block-like shapes, which made it relatively simple to turn them into type pieces. When printers tried to use movable type to print Arabic texts, they found themselves hampered by the cursive nature of Arabic writing. The success of movable type printing in Europe led to a thousand-fold increase in the availability of information, which produced an explosion of ideas that led directly to the European Scientific Revolution and the Industrial Revolution that followed.

Dr Hannah Fry explores a paradox at the heart of modern maths, discovered by Bertrand Russell, which undermines the very foundations of logic that all of maths is built on. These flaws suggest that maths isn't a true part of the universe but might just be a human language - fallible and imprecise. However, Hannah argues that Einstein's theoretical equations, such as E=mc2 and his theory of general relativity, are so good at predicting the universe that they must be reflecting some basic structure in it. This idea is supported by Kurt Godel, who proved that there are parts of maths that we have to take on faith.

Hannah then explores what maths can reveal about the fundamental building blocks of the universe - the subatomic, quantum world. The maths tells us that particles can exist in two states at once, and yet quantum physics is at the core of photosynthesis and therefore fundamental to most of life on earth - more evidence of discovering mathematical rules in nature. But if we accept that maths is part of the structure of the universe, there are two main problems: firstly, the two main theories that predict and describe the universe - quantum physics and general relativity - are actually incompatible; and secondly, most of the maths behind them suggests the likelihood of something even stranger - multiple universes.

We may just have to accept that the world really is weirder than we thought, and Hannah concludes that while we have invented the language of maths, the structure behind it all is something we discover. And beyond that, it is the debate about the origins of maths that has had the most profound consequences: it has truly transformed the human experience, giving us powerful new number systems and an understanding that now underpins the modern world.

Hannah then explores what maths can reveal about the fundamental building blocks of the universe - the subatomic, quantum world. The maths tells us that particles can exist in two states at once, and yet quantum physics is at the core of photosynthesis and therefore fundamental to most of life on earth - more evidence of discovering mathematical rules in nature. But if we accept that maths is part of the structure of the universe, there are two main problems: firstly, the two main theories that predict and describe the universe - quantum physics and general relativity - are actually incompatible; and secondly, most of the maths behind them suggests the likelihood of something even stranger - multiple universes.

We may just have to accept that the world really is weirder than we thought, and Hannah concludes that while we have invented the language of maths, the structure behind it all is something we discover. And beyond that, it is the debate about the origins of maths that has had the most profound consequences: it has truly transformed the human experience, giving us powerful new number systems and an understanding that now underpins the modern world.

Dr Hannah Fry travels down the fastest zip wire in the world to learn more about Newton's ideas on gravity. His discoveries revealed the movement of the planets was regular and predictable. James Clerk Maxwell unified the ideas of electricity and magnetism, and explained what light was. As if that wasn't enough, he also predicted the existence of radio waves. His tools of the trade were nothing more than pure mathematics. All strong evidence for maths being discovered.

But in the 19th century, maths is turned on its head when new types of geometry are invented. No longer is the kind of geometry we learned in school the final say on the subject. If maths is more like a game, albeit a complicated one, where we can change the rules, surely this points to maths being something we invent - a product of the human mind. To try and answer this question, Hannah travels to Halle in Germany on the trail of perhaps one of the greatest mathematicians of the 20th century, Georg Cantor. He showed that infinity, far from being infinitely big, actually comes in different sizes, some bigger than others. This increasingly weird world is feeling more and more like something we've invented. But if that's the case, why is maths so uncannily good at predicting the world around us? Invented or discovered, this question just got a lot harder to answer.

But in the 19th century, maths is turned on its head when new types of geometry are invented. No longer is the kind of geometry we learned in school the final say on the subject. If maths is more like a game, albeit a complicated one, where we can change the rules, surely this points to maths being something we invent - a product of the human mind. To try and answer this question, Hannah travels to Halle in Germany on the trail of perhaps one of the greatest mathematicians of the 20th century, Georg Cantor. He showed that infinity, far from being infinitely big, actually comes in different sizes, some bigger than others. This increasingly weird world is feeling more and more like something we've invented. But if that's the case, why is maths so uncannily good at predicting the world around us? Invented or discovered, this question just got a lot harder to answer.

Alan Turing's breakthroughs, his story and tragic destiny, gives us a chance to look at the Second World War from a different angle.