Course Outline
Our topics:
Algebra
Geometry
Functions and Graphs
Trigonometry
System of Linear Equations
Proportion, Direct & Inverse Variation
Analytic Geometry
Statistics
Calculus (Derivative and Integration)
The A&W 3/9 lb. Burger | A&W Restaurants https://www.youtube.com/watch?v=EMNqJQaf08E
https://www.cbc.ca/radio/undertheinfluence/how-failing-at-fractions-saved-the-quarter-pounder1.5979468
https://www.youtube.com/watch?v=YsEcpS-hyXw
Contar (povo que não conta:
https://www.bbc.com/portuguese/noticias/2016/01/160126_tribo_sem_numeros_mv)
Contar por traços – impraticável: como contar a população de uma cidade? Necessidade de símbolos.
Mathematics was invented or discovered?
All start with Arithmetic (addition, subtraction,…) and Geometry(points, lines, shapes…).
Babylonian geometry knew Pythagoras theorem long before Pythagoras.
China dealt with decimal fractions 1500 years before Europe.
The numbers we use are called Hindu-Arabic numerals they began in India about 3 centuries
B.C. They were introduced to the west by Arab merchants about 12th century.
Who started to concern about proofing were the Greeks.
The Babylonians got their number system from the Sumerians https://en.wikipedia.org/wiki/Sumer , the
first people in the world to develop a counting system. Developed 4,000 to 5,000 years ago, the Sumerian
system was positional — the value of a symbol depended on its position relative to other symbols.
(https://www.livescience.com/27853-who-invented-zero.html )
All geometry you know from high school came from there.
Trigonometry (study of triangle) starts with the interest of how Earth moves around the Sun 300 B.C.
"The Elements" is the most influential Mathematics text ever. Euclid based his geometry on that
axiomatic method with deductive reasoning.
Pythagoras Theorem = Book 1 Proposition 47
History of Algebra...in just 5 minutes
https://www.youtube.com/watch?v=f97mGPgBL_Q
Símbolos romanos: I, V, X, L, C, D, M – no ZERO (103 in Roman numerals is CIII. The number 99 is XCIX.
You try adding CIII + XCIX. It’s absurd. https://www.vox.com/science-andhealth/2018/7/5/17500782/zero-number-math-explained )
ZERO concept came from India around fifth century AD. (https://www.livescience.com/27853-whoinvented-zero.html )
Cross-multiplication
Which means X3 – X1/2 + X = 0
1º. através de palavras (retórica)
2º. Através de abreviações (exemplo Paciloi): cubo meno censo(Raiz quadrada) plus cosa(seria
equivalente a xis) equale zero. X3 – X1/2 + X = 0
3º. Álgebra simbólica – como conhecemos agora.
Descartes: He wanted to see if we can study Euclid’s geometry from an algebraic point of view. That is the
Method.
Fermat's Last Theorem
Archimedes https://en.wikipedia.org/wiki/Archimedes
Archimedes' other mathematical achievements include deriving an approximation of pi; defining and
investigating the spiral that now bears his name; and devising a system using exponentiation for
expressing very large numbers. He was also one of the first to apply mathematics to physical
phenomena, founding hydrostatics and statics. Archimedes' achievements in this area include a
proof of the principle of the lever,[10] the widespread use of the concept of center of gravity,[11] and
the enunciation of the law of buoyancy.[12] He is also credited with designing innovative machines,
such as his screw pump, compound pulleys, and defensive war machines to protect his native
Syracuse from invasion.
The most widely known anecdote about Archimedes tells of how he invented a method for
determining the volume of an object with an irregular shape. According to Vitruvius, a votive crown
for a temple had been made for King Hiero II of Syracuse, who had supplied the pure gold to be
used; Archimedes was asked to determine whether some silver had been substituted by the
dishonest goldsmith.[30] Archimedes had to solve the problem without damaging the crown, so he
could not melt it down into a regularly shaped body in order to calculate its density.
In Vitruvius' account, Archimedes noticed while taking a bath that the level of the water in the tub
rose as he got in, and realized that this effect could be used to determine the crown's volume. For
practical purposes water is incompressible,[31] so the submerged crown would displace an amount
of water equal to its own volume. By dividing the mass of the crown by the volume of water
displaced, the density of the crown could be obtained. This density would be lower than that of gold
if cheaper and less dense metals had been added. Archimedes then took to the streets naked, so
excited by his discovery that he had forgotten to dress, crying "Eureka!" (Greek: "εὕρηκα, heúrēka!,
lit. 'I have found [it]!').[30] The test on the crown was conducted successfully, proving that silver had
indeed been mixed in.[32]
To do Calculus must a genius (Archimedes). The Calculus (collection of rules) not necessarily
the others mentioned on the slide.
The 3 areas of mathematics in antiquity:
https://math.stackexchange.com/questions/280530/can-you-provide-me-historical-examples-of-puremathematics-becoming-useful