Gottfried Wilhelm Leibniz: Mathematical Biography

Leibniz was born in 1646 in the city of Leipzig, Germany. His father was a professor of Moral Philosophy at the University of Leipzig and died when little Gottfried was only 6 years old. After the death of his father, Leibniz took to reading from his father’s library learning about all sorts of subjects.

Leibniz became a law student at the University of Leipzig when he was only 15 years old, graduating with his bachelors degree at the age of 17. At the age of 20 he received a doctorate from the University of Altdorf, Nuremburg and began his career as counsel to Kings and Princes. During various diplomatic missions, Leibniz met many mathematicians who sparked an interest in this area, so reading papers and journals, Leibniz taught himself mathematics.

In 1673, duty took him to London where he joined the Royal Society after demonstrating his adding machine.

Leibniz became bitter at the latter stages of his life due to the honors bestowed upon Newton rather than him and eventually died in 1716 in obscurity. He never married and his funeral was attended only by his secretary.

This is a discovery credited to both Leibniz and Newton who seemed to discover the same things at around the same time.

In 1675, Leibniz first used the integral notation when finding the area under a function. The symbol was modified from an elongated “S”. Similarly, he invented the notation we use for differentials: ^d/_dx for example. Much of Leibniz’s work involved investigating infinite series which included his invention of his harmonic triangle. He also managed to find correlations between sums and differences, and tangent lines. We can see how this probably led to area and tangent problems we are familiar with from previous calculus classes.

Leibniz developed and used this extensively in his work with infinite series which was prompted by a mathematical challenge from Huygens in 1672. From

¹/₁   ¹/₂   ¹/₃   ¹/₄   ¹/₅   ¹/₆

¹/₂   ¹/₆   ¹/₁₂  ¹/₂₀  ¹/₃₀  ...

¹/₃   ¹/₁₂  ¹/₃₀  ¹/₆₀  ...

¹/₄   ¹/₂₀  ¹/₆₀  ...

¹/₅   ¹/₃₀  ...

¹/₆   ...

we can see that each term not in the first row is made from the numbers above and to the right: For example, ¹/₂₀ (r4,c2) = ¹/₁₂-¹/₃₀ (r3, c2 & 3).

Also, you could make up ¹/₂₀ by summing all the numbers starting below and going out to infinity to the right. So these are all infinite series. Only for the first row does the infinite series diverge (it is the harmonic series.)

Although the system had already been discovered, its use was far from widespread and it was Leibniz who resurrected the idea. Being the simplest notation possible for writing numbers, it forms the building blocks of all computers. Very basically, there are two symbols, 1 (for “on”) and 0 (for “off”) with the former indicating a filled space and the latter an empty space. Numbers can be represented as the following table shows.

Leibniz was one of the greatest producers of mathematical notation that we still use today. As well as inventing the differential notation remarked upon earlier, he was also one of the first to use the dot between products, and also one of the first to use the standard form a:b = c:d for ratios. Furthermore, he was a huge influence on the general acceptance of the = sign being pushed by Descartes.

Calculus by Anton, Bivens & Davis

A History of Mathematics by Carl Boyer

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purdue.edu

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