Zero and the number system

This is the story of a very important number, but a number that wasn't always a number. In fact it was much less than a number until relatively recently. This is the story of Zero and it's a story that takes a tortuous and meandering route through 1,500 years of human history.

Today we enjoy zero in all its glory where it takes on two roles: The first is as a placeholder within our positional number system. Zero notes an absence of a value and it allows us to create huge numbers without the need to create new digits. So we know 30 is larger than 3 and 300 is larger than 30 and 3.

The second use of zero is as a number in its own right, the middleman between positive and negative one and enjoying nearly all the same benefits as other numbers.

We can subtract, add and multiply by zero but dividing by zero just doesn't work.For example, you can't divide 1 chicken by no chickens. You might suggest that the answer is infinity, but it's not, because infinity isn't really a number, it's a concept.

So with that much of information let’s begin the story….

The ancient Indian seem to be less concerned about military organization unlike Romans who like conquering stuff, unlike Romans the Indians devised a system that could cope with vast numbers they developed a different symbol for every number from one to nine.

The numbers we use today began life in India as early as 500 BC but then around 1,500 years ago someone (Aryabhata or Brahmagupta who knows) came up with a stupendous incredible extraordinary idea the biggest revolution in numbers that would change the world, invented an entirely new number

First time in human history someone had made nothing number, you might say what's all the fuss about I mean what's a wonderful about inventing a symbol that means nothing. But when you team 0 up with other numbers the results were spectacular, with just 10 digits you could make numbers infinitely large as well as infinitely small the Romans couldn't do that this enabled Indian science to storm ahead, Indian astronomers for example were centuries ahead of the others. Indian scientists worked out that the Earth spins on its axis and calculated the diameter of the globe and they were less than one percent off all this was possible because of 0.

The number system were a sensation and their fame soon began to spread across the globe. It made their way across the deserts of Arabia to take on one of the most sophisticated societies of the age in what is now Iraq. When Islam was little more than a hundred years old Baghdad was ruled by the great caliph.

The Arabs were great mathematician and were up to perform complex arithmetic but was unable not that they were incapable but it was just their number system holding them back.

But one day there arrived in the court an ambassador from India he had to present the great caliph with a gift of some sort but the Caliph was a man of infinite riches. It was hard to know what to give I mean I love Arabia t-shirt was not going to do the job.

The Ambassador however have thought long and hard about this and decided to present the Caliph with the greatest gift he could think off the gift of numbers actually we don't know for certain exactly how in the Islamic world adopted this number system but the Ambassador story is my favorite.

The Muslim scholars were bowled over by this new number system and they started performing brand new tricks with this new number system like quadratic equation, algebra, logarithmic and so more which enabled science mathematics and astronomy to reach new heights in the Middle East. The Indian number system became a smash hit throughout the Islamic world but on the other side of the Mediterranean christian Europe was still in the static grip of the old army of Roman and being Romans they weren't going to give way to the feisty newcomers that easily you have seen 300.

A showdown between the two systems was inevitable and when it came it could shape the destiny of the Western world. The beginning of the end the Roman numerals started on the shores of North Africa. A lot Muslim traders have been quick to adopt Indian number system and code for their business dealings by the end of the 12th century. The Indian numerals were in common use and it was in the business cities that the young man Fibonacci son of an Italian diplomat based in Algeria first notice the art of the Indians nine symbols and zero and when he grew up he decided to take them home in 1202.

Fibonacci wrote a book all about calculation, he is now regarded as one of the greatest mathematicians of all time and his book was pretty much a showcase for Indian number system and calculations on of his book of was aimed specifically at merchants showing them how useful is the New system for say calculating their profits. His book became a must-read for merchants.

Unfortunately ordinary people felt comfortable with the old numbers after all they'd live with them part of a thousand years. People have their suspicious that merchants are trying to cheat them using the new number system. The distrust went right to the top, in 1299 the city of Florence actually banned merchants from using the new numbers and account they had to use Roman numerals.

But no number was treated with more suspicion than zero it was a sign which creates confusion and difficulties.

The roman numerals were able to withstand this time.

But as business grows a new organization developed you know banks lending and borrowing stuff and with that comes the new concept of simple and compound interest.

Now this is the time the Indian number system strikes back.

Suppose I lend you some 10 pounds at half percent compound interest a month how much do you owe me at the end of the year. You don’t want to calculate that using roman numerals.

The old Roman system was simply no match for the Indian numerals and finally the Indian number system was adopted.

The Indian numerals made European navigation easier to calculate their latitude and so that they can cross the great ocean out of sight of land that's how they stumbled on America and the new numbers became the vocabulary of modern banking and modern numbers as we know.

But the biggest contribution of Indian numerals only 1 and 0 is yet to come.

Though sophisticated number system the human error can’t be avoided.

Columbus thought he got to India and in fact he got to the west indies because of human error in calculation of navigation.

But one man was determined to stop this human error and how you do that simple remove human, I mean built a machine, it all happened around 16 century.

A man named Gottfried Wilhelm Leibniz the one who hates Newton. He developed this new system more comfortable to machines to eliminate human error. So he got rid of the other numbers and developed a system using just one and zeros.

it's called the binary system.

It's perfect for a machine because machine doesn't really care too much about how big the numbers gets it can keep track of very long numbers and it is a very efficient way of adding numbers that's why machines love to put numbers into binary. It seemed the digital age was ready to take over the world but unfortunately he never built any sort of such machine.

It would have to wait another two hundred and sixty-five years before anyone could step into the line.

Meet Colossus the world's first working binary computer. Colossus is electronic in 1 & 0 and was created during the Second World War and installed in Britain's code-breaking center at Bletchley Park. It was used to crack the enemy's code before the Germans had even sharpen their pencils.

The technology that started here in Bletchley has changed our world forever and the use of binary system is the biggest contribution of zero.

Fermat's Last Theorem

“A problem worthy of attack proves its worth by fighting back”

And that's what Fermat's last theorem was doing, it was fighting back every time any mathematician tries to solve it.

So we are talking about Fermat's last theorem I suppose the place to start is with Fermat, Pierre de Fermat, he was a 17th-century mathematician living and working in France. Not working as a mathematician but working as a judge. Every evening he'd go home and math was his hobby, one evening he was looking at an equation which looks a bit like Pythagoras' equation

And he was looking for whole number solutions to that equation, and there are lots of solutions

and so on...

Now Fermat asked himself the question "What if I change this equation so instead of it being x squared, what about if it's x cubed or x to the fourth power?

Are there solutions to that equation? So in general we're talking about

where n is bigger than 2. Does that equation have any whole number solution? He thought about it for a while and couldn't find any whole number solutions, and then he went one step further being cocky he believed he found an argument, he believed he found a proof that showed without any doubt whatsoever there were no whole number solutions. So this is kinda weird, because we have one equation x^2 + y^2 = z^2 that has not just one solution, it actually has an infinite number of solutions. And then we have an infinite number of equations: x^3 + y^3 = z^3,x^4 + y^4 = z^4.....an infinite number of equations which apparently have no solutions.

And Fermat discovered it's proof, and he wrote in the margin of a book he was reading that evening, called the Arithmetica by Diophantus, that “I have a truly marvelous proof but this margin is too narrow to contain.

In other words: "I know how to prove that this equation has no solutions, but I don't have the space to write it down." And then he drops dead.

It very much was a secret proof which he never wrote down

After his death his son rediscovered this book which had this marginal notes "I have a truly marvelous proof which these margins are too narrow to contain". In fact the book is full of these little annoying notes: "I can prove this but I gotta go to feed the cat, I can prove this but I gotta go and wash my hair."

So Fermat was quite annoying in this respect. So his son published a new version of the book the book with all of Fermat's little notes printed in the text. And people would look at these notes and they were like "Fermat said he can prove this, let's try!"

And one by one people rediscovered the missing proofs. And every case where Fermat said "I have a proof", he was right, there was a proof, except in this one example here Fermat's last theorem, it is called Fermat's last theorem because it was the last one that anybody could actually find the proof for. And of course because it's the last one that anyone can prove it's the most precious one, it's the one that's the most desirable.

And the more that people try, the more they fail, the more wonderful it becomes. And this goes on for decades, it goes on for centuries. Right through to the 20th century were people desperate to rediscover what Fermat's proof might have been.

By the time we get to 20th century it's quite clear that this is an incredibly complex problem. Its simple to jot down in a few scribbles what the question is, it's easy to describe the problem.The proof is clearly profound and probably beyond Fermat's reach to be honest.

Some people say Fermat was just fooling around, that it was just a trick, that he left something in his book that he knew would trouble subsequent generations -

But you know there will be always someone who would rise up to the challenge.

And it starts with a ten-year-old child, a child called Andrew Wiles not the one in the above picture though, he's growing up in Cambridge, he went to the library, he got a book called “The last Problem” by E.T. Bell. And the book is all about Fermat's last theorem. And little Andrew Wiles, age 10, decided that he was going to rediscover the missing proof because he was a bright ten-year-old who can understand the problem. But a bright ten-year-old doesn't realize what they're letting themselves in for,

He tried, he talked to the school teachers about the problem, he talked to his A-level teachers about the problem, he goes to university talked to his undergraduate lecturers about the problem. He has a PhD and still this problem is obsessing him.

Yes that’s him, he was about in his late 30's by this time he was a Princeton professor there was something called the Taniyama-Shimura conjecture and trust me you don’t want to get into that. Which had been proposed in the 1950's. So a conjecture is an idea that we don't know whether it's true or not, but somebody is putting it on the table. Somebody proved that there was a link between these two conjectures.

It is much as if you could prove most of the Taniyama-Shimura conjecture you would get Fermat's last theorem for free. So somehow Fermat's last theorem is embedded in this other conjecture.

And Andrew Wiles' childhood passion, his childhood obsession is reignited  because he thinks the Taniyama-Shimura conjecture is worth a go. Wiles didn't tell anybody about it. He worked on in complete secrecy

He started not attending committee meetings, he started going to his office less and less, he started to focus on this problem. Once again not because it was the Taniyama-Shimura conjecture but because it would give him Fermat's last theorem for free. And for 7 years he worked in complete secrecy and at the end of 7 years he suddenly realized that he had Taniyama-Shimura and if he had Taniyama-Shimura, he had a proof of Fermat's last theorem.

He went to Cambridge, he presented his proof on a black board, it was a three-part lecture, the world cheered, he was the front page of the New York Times, he was on CNN he was everywhere. But the sting in the tale is that in any mathematical proof you have to have it checked. You have to have it refereed and published, and when he went through the checking process somebody found a mistake.

Wiles assumed that he could fix it, but the more he tried to unravel this problem the worse it became. And it became a huge embarrassment, you know you've been loathed as the greatest mathematician of the 20th century, you are a hero figure and now you have to admit you made a mistake.

And it took a whole year, but at the end of that year Andrew Wiles working with Richard Taylor managed to fix the proof.

It was like that the Fermat’s last theorem was fighting back. ”A problem worthy of attack proves its worth by fighting back” And that's what Fermat's last theorem was doing, it was fighting back, but Wiles proved that he was too good.

And of course what Wiles proved is that Fermat was right, this equation

For n bigger than 2, has no whole number solutions.

Euler constant...e!

e!

We have all used this mathematical constant in our math classes. Do you know how it came into existence and why it’s so famous?

The big, famous constant, e!, it's one of the famous mathematical constants, one of the most important, goes along with pi, golden ratio, and square root of two, etc.

So e is an irrational number, and it's equal to 2.718281828……. and goes on.

The problem with e is that it’s not defined by geometry. Now pi, is a something that is defined by geometry, right, it's the ratio of a circle's circumference and its diameter.

And it's something the ancient Greeks knew about. And a lot of mathematical constants go back to the ancient Greeks .But e is different, e is not based on a shape, it's not based on geometry.

It's a mathematical constant that is related to growth, and rate of change, but why is it related to growth and rate of change?

Let's look into its history, how it came to existence and where it was first used.

So we're going to go back to the seventeenth century, and this is Jacob Bernoulli, and he was interested in compound interest and earning interest on his money.

So imagine you've got one rupee in the bank and you have a very generous bank and they're going to offer you 100 percent interest every year. Wow, thanks a lot, bank! so 100 percent interest, so it means after one year, You’ll have two rupees. So you've earned one rupee interest and you've got your original rupee. So, you now have two rupees.

What if I offered you instead fifty percent interest, every six months?

Now is that better or worse?

Well, let's think about it. You're starting with one rupee and then I'm going to offer you fifty percent interest every six months. So after six months, you now have one rupee and 50 paise and then you wait another six months and you're earning fifty percent interest on your total, which is another seventy-five paise. And you add that on to what you had so it's two rupee twenty-five paise

Better! So what happens if I do this more regularly?

What if I do it every month? I offer you one-twelfth interest every month, Is that better?

So, let's think about that. So after the first month, it's gonna be multiplied by one-twelfth, that's your interest and then you're adding that onto the original rupee that you've got.

Then for your second month you take that and multiply it again by the same value, and your third month you would multiply it again and again you actually do that twelve times in a year. So in a year, you'd raise that to a power twelve.

and you would get two rupees sixty-one paise.

So it's actually better. In fact, the more frequent your interest is the better the results. Let's start with every week. So if we do it for every week, how much better is that? What I'm saying is you're earning one over fifty two interest every week. And then after the end of the year you got fifty two weeks and you would have two rupees sixty nine paise.

So it's getting better and better and better. In general, you might be able to see a pattern happening here. In general it would look like this

So here n is equal to twelve if you do it every month, fifty two if you do it every week. If you did it for every day.

So it would get better if you did it every second, or every nanosecond. But Jacob Bernoulli wanted more than that, let’s see what he did.

What if every instant I'm earning interest. Continuous interest. What does that look like? That means if I take make n in the formula tend to infinity I would have continuous interest.

Now what is that? What is that value? And that's what Bernoulli wanted to know. He didn't work it out. He knew it was between two and three.

So fifty years later, Euler worked it out. You know Euler, he works everything out.

And the value was 2.718281828459... and so on.

You may have noticed we got very close when we did the compounding for each day. So making the interval smaller we will get closer and closer to e. Now Euler called this irrational number e. He didn't name it after himself, although it is now known as the Euler constant.

Euler proved that this was irrational. He found a formula for e which was a new formula. Not this one here, a different formula. And it showed that it was irrational.

This is a fraction that goes on forever, continuous fraction. Because there's a pattern, and that pattern does hold and if the fraction goes on forever it means it's an irrational number If it didn't go on forever, it would terminate, and if you terminate you can write it as a fraction. And he also worked out the value for e. He did it up to 23 decimal places. To do that, he had a different formula

It's a nice formula, if you're happy with factorials. Factorials means you're multiplying all the numbers up to that value. So if it was four factorial, it'd be four times three times two times one.

Okay so far so good but why is e a big deal? It's because e is the natural language of growth. Okay, let's draw a graph y equals e to the power of x.

So if you took a point on this graph, the value at that point is e to the power x. And this is why it's important. The gradient of the curve at that point is e to the x. And the area under the curve which means the area under the curve all the way down to minus infinity is also e to the x.

And it's the only function that has that property. So it has the same value, gradient, and area at every point along the line.

So at one, the value is e because it's e to the power one. The value is 2.718, the gradient is 2.718 and the area under the curve is 2.718.

The reason this is important then, because it's unique in having this property as well, it becomes the natural language of calculus. And calculus is the maths of rate of change and growth and areas. And if you're interested in those things, if you write it in terms of e, then the maths becomes much simpler. Because if you don't write it in terms of e, you get lots of nasty constants and the maths is really messy. If you're trying to deliberately avoid using e, you're making it hard for yourself. It's the natural language of growth.

And of course e is famous for bringing together all the famous mathematical constants with this formula,

Euler's formula

So there we have all the big mathematical constants in one formula brought together. We've got e, we've got i, square root of minus one, we've got pi of course, we've got one and zero and they bring them all together in one formula.

which is often voted as the most beautiful formula in mathematics.