Mathematica Historia
What famous mathematician was well known for his work with triangles.
Pythagoras
What is the ancient mathematical theory that organizes and systematizes the principles of geometry, presented in 13 books and serving as the foundation for much of Western mathematics?
What is Euclidean Geometry?
integral of e^xdx
e^x+C
You draw one card from a standard 52-card deck. Given that the card is a red card, what is the probability that it is a heart?
What is 1/2?
This method assumes the opposite of the statement you want to prove, then shows that this assumption leads to an impossible or contradictory solution.
What is Proof by Contradiction?
Which famous person was often credited with being a mathematician in the early 20th century but was actually a very well-known physicist.
Albert Einstein
What is the theorem that connects differentiation and integration, showing that they are inverse processes, and was made by Isaac Newton and Gottfried Wilhelm Leibniz?
What is the Fundamental Theorem of Calculus?
Integral of ln(x) with respect to x
xlnx - x +C
In a group of 23 people, what is the approximate probability that at least two people share the same birthday?
50.7%
This method is used to rigorously prove the limit of a function at a point. It involves showing that for every ϵ>0\epsilon > 0ϵ>0, there exists a δ>0 such that if 0<∣x−c∣<δ, then ∣f(x)−L∣<ϵ
Proof by Epsilon-Delta
These two mathematicians are often credited with the independent development of calculus in the 17th century, leading to a famous historical dispute over priority.
Who are Isaac Newton and Gottfried Wilhelm Leibniz?
What is the theory that no three positive integers a, b, and c can satisfy the equation a^n+b^n=c^n for any integer n>2, famously unsolved until 1994?
What is Fermat's Last Theorem?
what is the integral of sec(x)
ln|secx+tanx| + C
if you pick one of three doors, one of which has a car behind it and two of which have goats, what is the probability of winning the car if you switch doors after Monty reveals a goat behind one of the remaining doors?
What is 2/3?
This method involves proving a base case and then showing that if a statement holds for some n=k it must also hold for n=k+1.
Proof by Mathematical Induction
Often called the "father of algebra," this Persian mathematician wrote the book Al-Kitab al-Mukhtasar fi Hisab al-Jabr wal-Muqabala, from which the term "algebra" is derived.
Who is Al-Khwarizmi?
What is the mathematical theory that models randomness and uncertainty, first systematically developed in the 17th century, and is now fundamental in fields such as statistics, economics, and science?
What is Probability Theory?
integral of sin(x)/x from 0 to infinity, with respect to x
pi/2
Two envelopes contain a sum of money. One envelope contains twice the amount as the other. You pick one envelope at random and see that it contains $100. What is the probability that you have the envelope with the smaller amount, assuming you switch envelopes?
What is 2/3? (Two Envelopes Paradox.)
This method involves dividing a problem into a finite number of cases and proving each one individually. It was famously used to prove the Four-Color Theorem, which required checking a large number of configurations.
What is Proof by Exhaustion or Case Analysis?
This mathematician's work on the Poincaré conjecture, a major unsolved problem in topology for over a century, was finally solved by a reclusive Russian mathematician in 2003, who declined the Fields Medal and the $1 million Millennium Prize.
Who is Grigori Perelman?
What is the revolutionary theory that introduced the concept of different sizes of infinity and the idea of countable and uncountable sets?
What is Cantor's Set Theory?
integral from 0 to infinity, of e^-x^2
sqrt(pi/2)
A game is played where you flip a fair coin. If heads, you win $2, and the game stops. If tails, you double the amount to be won, and flip again. What is the expected value of this game?
Answer:
What is infinite? (St. Petersburg Paradox)
This proof method, used notably by Fermat, involves showing that any counterexample would lead to an even smaller counterexample, which leads to a contradiction because the process cannot continue indefinitely.
What is proof by Infinite Descent