Course description
Two Mathematical Innovations – one Greek, one Indian
Pi & Zero
Pi
From the first year of secondary school to postgraduate studies at universities around the world, there is nowhere to hide from Π. It is a nightmare for maths-hating teenagers and a never-ending mystery for professors who teach it in the daytime and dream about it at night. Since Archimedes tried to ‘square the circle’ in third century B.C.E. Sicily, at that time a part of the Greek Empire, mathematicians have wondered about this number. Nowadays, supercomputers can calculate Π to 1.4 trillion places, although we only need the first thirty-two digits to work out the size of the universe. So, why does Π continue to excite scientists and bore school kids more than two thousand years after Archimedes first played with the idea of this strange number?
The Ancient Greeks thought in lines and squares. They tried to calculate the area of a circle by working out what it could look like as a square. In his book, Archimedes described how he drew a larger rectangular hexagon outside a circle and a slightly smaller one inside it. He then halved the length of the sides until each polygon had 96. He did this because he wanted to get closer to what a circle looked like. When each polygon had 96 sides, he calculated that the value of Π was between 31/7 or about 3.1429, and 310⁄71, around 3.1416. Archimedes was very nearly right – but only very nearly, not exactly. He also showed that the area of a circle was equal to Π multiplied by the square of the radius of that circle. We can write this as πr2. Of course, if we cannot know the exact value of Π, we cannot know the exact area of a circle.
For more than 1500 years, nobody after Archimedes seemed very interested in Π. The first man to take up the subject again was Madhava, who lived from about 1340 to 1425 near the town of Cochin in southern India. He calculated Π arithmetically (not geometrically) as:
Π ÷ 4 = 1 – 1/3 + 1/5 – 1/7 + 1/9 … + (-1)n/2n + 1 …
What Madhava did was very special because he showed Π as an infinite series, one that could never end. (Don’t forget that the supercomputers are still calculating the value of Π after 1.4 trillion places!) In other words, Π is a number that can never have an exact value. So, Madhava moved mathematics from the finite to the infinite.
About three and a half centuries later on another continent, the Swiss mathematician, Lambert, showed in 1761 that Π would never repeat the same pattern of numbers and was, therefore, an irrational number. Obviously, then, we could never write it as a fraction.
Then, in 1882, the great German mathematician, Lindemann proved that Π was a transcendental number, which means that we cannot write an equation with values of x and whole numbers to show its value. So, we cannot use algebra to show the value of Π. This is different from Lambert’s finding that Π is irrational. For example, √2 is an irrational number because it will never repeat the same pattern of numbers as decimals, but it can be written in algebra as: x2 - 2 = 0. If Π is transcendental, then we can solve the Greek problem of trying to square the circle by saying that it can never be done!
So, Π is an infinite series if we write it in arithmetic, an irrational number and a transcendental one too. But we do not know its value and the more that mathematicians find out about numbers, the less we seem to know about it. As Ian Stewart says, the search for Π is like the drunk man who loses his keys and tries to find them in the light of a street lamp when all the time, they are lying in the dark just outside that circle of light. Perhaps one day, every mathematician hopes, she will step away from the place where the answer to Π is expected and find it in the dark … but only perhaps.
Zero
Zero is a necessary part of our lives in the twenty-first century. It is hard for us to imagine our world without it. But people have not always seen zero in the same way as they have looked at numbers like one, two or three. In fact, until the seventeenth century, many religious people saw zero as the work of the Devil. This is zero’s story from its first use 5,000 years ago in the Sumerian Empire to its importance in computer programming in the modern world.
The first time we know that zero was used was 3,000 BC in Sumer, an ancient civilisation in what is now Iraq. Zero was important for counting in business and the Sumerians used two diagonal lines to show it, like //. They were expert mathematicians in their age: for instance, they could calculate the area of triangles and the volume of cubes. They also made multiplication tables to make arithmetic faster. But they did not see zero as a number. It was more like a sign, similar to the decimal point we use today to show parts of one. It was never used on its own as a number.
Slowly, the two diagonal lines // changed to a circle: 0.
Nearly 3,000 years later, in 331 BC, Alexander the Great, the famous soldier - king of Macedonia (in northern Greece), conquered the land where the Sumerians had lived and worked. He took zero back to Greece, but it was soon forgotten. It is natural for us to ask why the Greeks did not take this very useful idea into their own mathematics. There are, perhaps, two main answers: first, the Greeks preferred to use lines, not numbers, in theircalculations. They calculatedby adding two lines like this:
-------------- + -------- = ----------------------
As we all know, the Greeks were especially interested in geometry – the mathematics of shapes – not numbers.
The second reason was that the Greeks thought philosophy was much more important than something as ordinary as business and, so, the use of zero for adding money in the east made them look at it as unintellectual.
The Romans followed the Greeks in their intellectual interests and, besides, they did not use position to show the value of numbers. What does that mean? Well, the Romans used the following symbols:
1 = I
5 = V
10 = X
50 = L
100 = C
500 = D
1,000 = M
They added three Xs together to make 30; 40 was L with an X before it, meaning that 10 must be subtracted from 50. So, XL was 40 and LX was 60.
By contrast, the Sumerians, and, as we will see, the Indians and Arabs later in history used the position of numbers to show how much their value was. In this way, 30 shows 3 tens and no units; 33 is 3 tens and three units. It’s the position of the number 3 that shows us how much it’s worth. It also makes arithmetic easier. 33 in Roman numbers is XXXIII. Think about multiplying XXXIII by LXVIII!
So, the Sumerian zero got lost for nearly 4,000 years until we find it again in India. The first written record comes from 865 AD, but it was probably used a couple of centuries before then. The Indians also gave us the original forms of the numbers we use today. They thought of zero as a number too, not just a sign. We know about this from the great Arab mathematician, Al Khwarizmi, who translated Indian maths into Arabic. This allowed advanced mathematics to happen in the Islamic world around the tenth century.
Of course, we need to ask why the Indians and Arabs accepted zero – or nothingness – when Europeans did not. This is, perhaps, because of the different religious beliefs in these countries. In Hinduism and Buddhism, the major religions of Asia, nothingness was a well-known idea. The Holy Quran also says that the world came from nothingness. In Christianity, however, nothingness is problematic because God is eternal and there is an idea that nothing can come from nothing. The English historian and priest, William of Malmesbury (1096 – 1143), for example, said that zero was dangerous Arab magic and the work of the Devil. Even after the great Italian mathematician, Fibonacci, used zero in his famous book of 1202, nobody paid any attention.
It was not until the late sixteenth century that zero began to appear in Europe. It was needed for three things: business; pure maths and astronomy; the study of the stars and planets – without astronomy, ships’ captains did not know where they were at sea or how to reach their destinations.
In the late seventeenth century, the calculus of Leibniz or Newton needed zero. Until that time, mathematics was static; it was interested in looking at what was there – for instance, triangles. Calculus is the mathematics of change. Let’s look at an example. A tennis player throws a ball into the air. The ball is obviously moving up and then down, but in the middle there must be a point where it does not move. That point is represented by zero. Newton and Leibniz wanted to look at ever-smaller periods of time and so infinitesimal change (or fluxion). But even at this late age, an Irish philosopher and man of religion, George Berkeley, attacked the idea of infinitesimal change and the mathematicians who worked in the area. To him, it was anti-Christian – the same criticism that William of Malmesbury had used 600 years before.
Nowadays, of course, we know that all numbers can be written in 0 and 1 (or binary) and that computer languages depend on this. In chemistry, there is a theoretical element called tetraneutron which has four neutrons and no protons or charge and so has a value of zero. In Kelvin’s measurement of temperature, 0 is the coldest point, while in Celsius it is not. In physics, zero point energy is the lowest possible level. Zero is an even number between two odd numbers 1 and – 1. Without zero, there would be no place between positive or negative. And so on and so on. In other words, zero is a necessary part of our understanding of science and the world around us.
But zero still causes problems. A silly example: because there is no year 0 in the Gregorian calendar, the new millennium started on 1 January 2000 after only 1,999 years, but should have started on 1 January 2001. The problems of zero are still with us even today. But we now know that you can make something from nothing!
If you want to watch some videos on this topic, you can click on the links to YouTube videos below.
If you want to answer questions on this article to test how much you understand, you can click on the green box: Finished Reading?
Videos :
1. Archimedes (1:51)
2. How to Calculate Pi, Archimedes' Method (5:01)
3. Pi (1:13)
4. Calculating Pi, Madhava (2:29)
5. Zero (4:07)
6. The History of Zero - Discovery of the Number 0 by Ancient India (4:46)
7. Muhammad ibn Musa al-Khwarizmi (4:42)
8. What is Zero? Getting Something from Nothing (3:53)
