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Calculus: Homework Help Resource13 chapters | 107 lessons

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Lesson Transcript

Instructor:
*Cory Haley*

In this lesson, we will explore the life and work of German mathematician and philosopher, Gottfried Wilhelm von Leibniz. In particular, we'll identify his major contributions to the field of mathematics, published books and technological accomplishments.

**Gottfried Wilhelm von Leibniz**, a German mathematician and philosopher, was born July 1, 1646 in Leipzig, Germany. At age 15, he enrolled at the University of Leipzig, where his father used to teach, and earned both a bachelor's and master's degree in philosophy. He received his Doctor of Law from the University of Altdorf.

Gottfried Leibniz discovered infinitesimal calculus, a distinction he shared with Sir Isaac Newton. However; each one made this discovery alone, not while working together. **Infinitesimal calculus** is the branch of mathematics that is concerned with differentiation, integrations, and limits of functions. Leibniz also developed the law of continuity and transcendental law of homogeneity, cutting-edge theories that were not published or used in mathematics until the 20th century.

Gottfried Leibniz was the first person to provide a description of a pinwheel calculator and invented a device of his own. The Leibniz wheel consisted of a cylinder with a set of teeth of increasing lengths. It was used in automatic mechanical calculators until the development of the electronic calculator in the mid-1970s.

Gottfried Leibniz's major contribution to mathematics was his discovery of the** binary numeral system**, or the base-2 system, which we find today in computers and related devices. The binary numeral system is a way of writing numbers using only two digits: 0 and 1. You're probably used to working with the base-10, or decimal system, which consists of the numbers 0-9. Instead of using ones, tens, and hundreds, which are powers of 10, consecutive digits in the binary number system are expressed as a power of two.

For example, let's take the number 14 (decimal number system/base-10) and use a very simple process to convert it to a binary number. This involves dividing the initial number and the subsequent quotients by two until you reach zero.

Let's start by creating a table. In the first column, write the dividend, which in this example is 14, and the divisor, which is two, and perform the operation. Next is the quotient and remainder in the second and third columns. If the dividend is an even number, such as 14, you'll end up with a whole number as a quotient, no remainder, and your result will be zero. If the dividend is an odd number, you'll have a remainder, and your result will be one, or in the case of the integer one, a fraction.

Dividend and Divisor | Quotient | Remainder | Binary Number |
---|---|---|---|

14/2 | 7 | 0 | 0 |

7/2 | 3 | 1 | 1 |

3/2 | 1 | .5 | 1 |

1/2 | 0.5 | NA | 1 |

If we write out the numbers found in the third column, beginning from the bottom up, we find that 1110 is the binary number for 14. We could also use a binary conversion table:

Decimal Number | Binary Number |
---|---|

0 | 0000 |

1 | 0001 |

2 | 0010 |

3 | 0011 |

4 | 0100 |

5 | 0101 |

6 | 0110 |

7 | 0111 |

8 | 1000 |

9 | 1001 |

10 | 1010 |

11 | 1011 |

12 | 1100 |

13 | 1101 |

14 | 1110 |

15 | 1111 |

Do you notice any patterns?

Gottfried Leibniz not only excelled in math, but also wrote two books on philosophy: *Theodicee* and *Discourse on Metaphysics*. *Theodicee*, which addressed issues related to God, human free choice, and the problem of evil, was published during his lifetime. *Discourse on Metaphysics* was released after his death. Leibniz's optimistic philosophical principles were based upon a belief in God, including faith that the best outcomes have already been pre-determined by a supreme being. He also believed that good ultimately prevails over evil and reason justifies existence.

In this lesson, we discussed the German mathematician and philosopher, **Gottfried Wilhelm von Leibniz**, and his contributions to math and thought. The **binary numeral system**, in which 0 and 1 are the only numbers, was one of his greatest discoveries. He also discovered **infinitesimal calculus**, the branch of mathematics that is concerned with differentiation, integrations, and limits of functions. His Leibniz wheel was used up until the 1970s, when it was replaced by the electronic calculator. A trained philosopher with both undergraduate and graduate degrees in this field of study, Leibniz was noted for his belief in God and sense of optimism, as expressed in his book, *Theodicee*.

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