Somerset Times

Add Infinitum – Don’t Mention the War!




Somerset Times Edition

Week 6,
Term Four, 2016

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Next Monday will mark the 300th anniversary of the death of Gottfried Leibniz, the famous German philosopher and mathematician. He is remembered these days for the development of Calculus and the controversy and acrimony caused over whether he had plagiarised the idea from the great English physicist and mathematician, Sir Isaac Newton.

But what is Calculus all about and why is it important? The simple answer is “Ask the Year 11s” as they have been studying the topic. The word Calculus comes from the Latin for ‘small pebble used for counting’ but is essentially the study of change. Imagine that a Year 11 student drives from Mudgeeraba to Brisbane (a distance of 80km) in 1 hour. Their average speed would therefore be 80km/h although during the journey the speedometer would register different speeds both above and below that value. The speedometer gives an instantaneous speed but in an ‘instant’ the car is travelling 0 km in 0 hours so its speed at that time is really 0/0 km/h. Try asking Siri “What is zero divided by zero?” She will explain using cookies and friends that it doesn’t make sense.

Generic - Add Infinitum

How small a time interval should be used to get a meaningful answer when calculating speed? The problem of dividing by the infinitely small seemed insurmountable to mathematicians for centuries. Calculus bypasses this challenge by allowing us to consider values approaching zero and these are called Limits.

Now back to the war between Newton and Leibniz. Both of them and their disciples argued about who got there first, who should take the credit and whether ideas had been plagiarised. Newton arrived at his discoveries in the 1660s (15 years before Leibniz) but didn’t publish his results. He communicated some of his work to Leibniz but Leibniz came up with his own method and the notation that we use today. It was finally decided that the Royal Society should investigate the matter and make a judgement. They came down on the side of Isaac Newton but this was no surprise, as Newton was president of the Royal Society at the time, and it was he who actually signed the report! So Newton got his way in the end but at the same time caused a rift between English and European Mathematicians which began a degree of isolation for years to come.

But did Newton really invent (or discover) Calculus? He famously wrote to Robert Hooke that “If I have seen further, it is by standing on the shoulders of giants”. Part of that phrase currently appears around the edge of a British ₤2 coin (a tribute to Newton’s time as Master of the Royal Mint). Interestingly, the original letter now resides in the Historical Society of Pennsylvania Museum in the United States of America. In the 3rd century CE, Archimedes used infinitesimals and the method of exhaustion to find the area, surface area and volume of shapes. This laid the foundation for integration. Isaac Barrow, who taught Newton, is credited with the discovery of the Fundamental theorem of Calculus before Newton (and Leibniz) formalised Calculus as a systematic body of knowledge.

The development of Calculus has been significant and also of enormous practical use in the fields of science, technology and economics. In 1938, English mathematician Mary Cartwright investigated noise in radar systems which led to work on non-linear differential equations and the development of chaos theory and the butterfly effect.

Calculus allows you to go from a variable rate of change to the total change and vice versa. This gives rise to its two main branches - differential and integral.

Integral Calculus is studied in Year 12.

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