The Scientific Revolution – Mathematics

Ajay Kumar, Lyndon Shi, Nicholas Voreas9-1

Gerolamo Cardano

Father of Complex Numbers

Gerolamo Cardano - Biography

• Born 1501• Unhappy childhood – illegitimate son• Inventor, astrologer, philosopher, algebraist,

physician• Known as the “Gambling Scholar” for his

gambling skills• Wrote more than 200 books on subjects that

interested him• Committed suicide (September 21, 1576)

Complex Numbers

An Introduction

Complex Numbers

• Also known as imaginary numbers

• A variable (i) stands for the square root of a negative number

• Always do the i part first when solving

Without Complex Numbers…

• No iPod

• No modern advancement

• No quantum theory

Francois Viete

Life

• French Mathematician• Lived 1540 to 1603• Father of modern symbolic algebra• Career as a lawyer, worked on mathematics in

spare time• Involved in politics, worked for Kings Henry III

& IV• Decoded messages sent to Philip II of Spain

Books

• First works – Canon mathematicus, Universalium Inspectionum Liber Singularis, (1579)– Trigonometric tables calculated to 9 decimal places– Collection of trigonometric formulas

• In artem analyticam isagoge (1591)– Algebraic notation– Use of letters for unknowns and constants– Vowels are unknowns, letters are constants

Trigonometry• Math of

triangles• Relationships

between sides and angles

Symbolic Algebra

John Napier

Life

• Scottish noble/landowner• Lives 1550 to 1617• Protestant• Attention to land (inventions)• Theologian/astronomer, worked on

mathematics in spare time• Invented logarithms, Napier’s Bones, decimal

point

Books

• The Plaine Discovery of the Whole Revelation of St. John (1593)– Against Papacy

• A Description of the Wonderful Canon of Logarithms (1614)– Explained inventions– Logarithmic tables– Led to base 10

Logarithms

• Exponential form• How many of one number do we multiply to

get another

log2(8) = 3

Napiers Bones

• Multiplication tables

Blaise Pascal

Pascal’s Triangle

Blaise Pascal

• Born: June 19, 1623 (Clermont)• Was kept away from mathematics at an early age –

led to curiosity on the subject• Easily mastered properties of geometry by

experimenting himself• Invented “arithmetic machine” – could

add/subtract• Spent some time studying religion• Died: August 19, 1662 (Paris)

Pascal’s Triangle

Isaac Newton

• Born Dec. 25, 1642 in Woolsthorpe, England• Father died before he was born, mother moved away• Grew up with his uncle• Attended Trinity College at the University of Cambridge in

1661, received bachelor of arts in 1665• In 1669, appointed professor of mathematics at Trinity

College, and elected to the Royal Society in 1672• Elected to Parliament in 1691, warden of the mint in 1696• Died March 20, 1727

Newton’s Development of Modern Calculus

• First version of Newton’s calculus published in 1665-6

• Seemed to be derived from ideas of motion

• Considered variables changing with time

• His calculus was geometrical, as opposed to analytical

• Used “infinitesimals”, infinitely small but not zero

• Later replaced by notions

• Applied his version of calculus to general physics

• Included the laws of differentiation and integration, second and higher derivatives, and the notion of an approximating polynomial series

• Calculus today is used in many ways, some including physical sciences, engineering, computer science and statistics

Gottfried Wilhelm Leibniz

• Born in 1646• Early years influenced by moral and religious views of

mother• Attended University of Leipzig at age 14 in 1661• Studied philosophy, mathematics, rhetoric, Latin, Greek

and Hebrew• Graduated with bachelor’s in 1663, got master’s degree in

philosophy the same year• Death on November 14, 1716

Development of the Binary System• Developed by Gottfried Wilhelm Leibniz

• Belief that all logic can be translated from a verbal representation to an absolute mathematical condition

• Ideas were repelled, Leibniz dropped the idea for about 10 years

• Hope revived when the Book of Change was published, and he found confirmations of his ideas within this book

• If such things as yes/no, on/off and male/female could be reduced to straightforward propositions, why couldn’t logic and thought?

• Went out to define his binary system

• Transposed numbers into seemingly infinite rows of ones and zeroes

• At the end of his life he began to believe that his binary number were quasi-religious mysticisms

• Claimed that it portrayed creation, with one being God, and zero being void

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California Berkeley Math Department. 02 Dec. 2012 <http://math.berkeley.edu/~robin/Viete/index.html>.

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