Theorems Trivia

1. Apart from the Newcomb-Benford law and the law of anomalous numbers, what other alternative name is Benford's law known by?

From Quiz Benford's Law

Answer: The first-digit law

Benford's law provides the frequency distribution of the first digits of all the numbers that appear in big data sets. The alternative name of Benford's law, namely the first-digit law, implies that the first digits of all the numbers are taken into consideration in producing the frequency distribution that can be explained by the law.

From Quiz Fermat's Last Theorem

Answer: Because this theorem was the last of Fermat's theorems to be proved or disproved.

All of Fermat's other theorems have been proven or disproven by other mathematicians. Fermat's last theorem remained unproved from 1637 to 1995. Fermat died in 1665 and many mathematicians were very interested in this theorem.

From Quiz Pythagoras's Crazy Theorem

Answer: a^2 + b^2 = c^2

In the formula, a and b represent the shorter sides of a right triangle and the c represents the hypotenuse. Here's an example: You have shorter sides 3 and 4. What is the final side? Plug it into your equation: 3^2 + 4^2 = c^2. 9 + 16 = c^2. 25 = c^2. 5 = c.

From Quiz Famous Theorems in Math #1

Answer: The sum of the squares of the two other sides

The Pythagoreans were a strict secret society in Ancient Greece. It is not known whether Pythagoras himself discovered this famous theorem, since the Pythagoreans (even after his death) attributed all results to him.

From Quiz Famous Theorems in Math #1

Answer: The cosine law

The cosine law states that c^2=a^2+b^2-2ab*cos C (where capital C is the angle). Note that this reduces to the Pythagorean Theorem when C=90 degrees, because cos 90 = 0. There are sine and tangent laws, they describe ratios in the triangle. There is no "triangle law" that I know of.

From Quiz Benford's Law

Answer: 11.1%

In general, a numerical figure can start with 1, 2, ..., 9, giving a total of nine possible combinations. If each of these nine numbers is equally likely to appear as the leading digit of a number, then the probability of each of the nine digits (1, 2, ..., 9) appearing as the leading digit in a number is given by 1/9 = 11.1%.

From Quiz Pythagorean Theorem

Answer: Law of cosines

The law of cosines is given by the following equation: c^2 = a^2 + b^2 - 2abcos gamma, where a, b, c are the lengths of the 3 sides of a triangle and gamma is the angle opposite side c. Notice that when gamma = 90 degrees, cos 90 = 0. The equation simplifies to c^2 = a^2 + b^2, which is the formula of the Pythagorean theorem.

From Quiz Basic Math by the Definition

Answer: The set of rational numbers

The rational numbers include numbers such as 3/5 and 1 7/8 = 15/8. Note that any integer is also a rational number - for example, 3 = 3/1. The number (the square root of 3) is not a rational number. Any rational number can be expressed as a decimal that either terminates or has a repeating part.

From Quiz Fermat's Last Theorem

Answer: Lawyer

Initially, Fermat refused to publish his work on his theorem. It was a wise idea of his son, Samuel to collect his father's mathematics works.

From Quiz Famous Theorems in Math #1

Answer: The "reduction method" for the solution of the general cubic equation

The method was for the general solution of polynomial equations of degree 3. The general solution for degree 4 curves was found not long after, and in the 1800s Abel proved that equations of degree 5 and higher cannot be solved in general. The "exhaustion" method was used by Archimedes (a primitive version of the integral calculus), "fluxions" was Sir Isaac Newton's invention, and the "Erlangen Programme" was stressed by the German Felix Klein in the late 1800s to unify group theory and geometry.

From Quiz Benford's Law

Answer: 1

The probability of the number 1 to appear as the leading digit in a big data is 0.301. The theoretical probability values for the remaining eight digits (2, 3, ..., 9) are 0.176, 0.125, 0.097, 0.079, 0.067, 0.058, 0.051, and 0.046, respectively. It can be observed that a greater digit would result in a lower probability value. Let d denote the leading digit of a number. A mathematical function that captures this relationship is given by P(d) = log (base 10) (1+1/d). It can also be proven that the summation of all the possible values of d, namely 1, 2, ..., 9, over the probability mass function, is 1.

From Quiz Basic Math by the Definition

Answer: The remainder

The statement above is actually a theorem called the division algorithm. It is just a mathematical statement of division of positive integers. For example, 23 divided by 7 has a quotient of 3 and a remainder of 2: 23 = 7*3 + 2 7 is the divisor, 23 is the dividend. Note that the remainder must be non-negative and smaller than the divisor.

From Quiz Fermat's Last Theorem

Answer: 4

For the case n = 14, Dirichlet managed to provide a proof for this by the year 1832. For the case n = 7, it was successfully proved by Lame and Lebesgue.

From Quiz Pythagorean Theorem

Answer: James A. Garfield

In his proof, Garfield divided a trapezoid into 3 sections, namely 2 identical right-angled triangles of equal area and another isosceles triangle. By performing some algebraic manipulation, he arrived at the equation a^2 + b^2 = c^2.

From Quiz Quod Erat Demonstrandum

Answer: the parallel postulate

Euclid's five postulate's were (generally speaking): 1. A line can be created by connecting two points. 2. Lines can be extended indefinitely. 3. A circle can be formed from a point [i.e. center] and a radius. 4. All right angles are congruent. 5. If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles [i.e. less than 180°], then the two lines intersect at some point. Euclid could not prove this statement, but reasoned that it had to be true, so he left it as a postulate. Many mathematicians have attempted to prove this over the years, mainly because it is complex compared to the other four, which are very basic. It has not been proved, but a proof has been written proving that it cannot be proved one way or another. Two branches of geometry (hyperbolic and elliptical), both under the shell of non-Euclidean geometry were created, and Einstein actually discovered that when talking about very large spaces (like the universe), Euclidean geometry does not hold and in fact it is hyperbolic geometry that is more applicable. Cool, huh?

From Quiz Basic Math by the Definition

Answer: A square root of b

This is the definition of a square root. Note that any positive real number will have two square roots. For example, the number 49 has square roots 7 and -7 since 7^2 = 49 and (-7)^2 = 49.

From Quiz Famous Theorems in Math #1

Answer: Elliptic curves & modular functions

The method of infinite descent was used by Fermat in order to solve lower degree specific cases of the theorem, and is essentially a proof by contradiction. This method was proven not to work in general. Direct proofs also exist for specific cases (I have seen n=3 and n=4) but no general direct proof is known. Wiles finally cracked Fermat's enigma using modular forms of elliptic curves.

From Quiz Basic Math by the Definition

Answer: The additive inverse of a

For example, 3 is the additive inverse of -3 since -3 + 3 = 0. If x is a real number, the additive inverse of x is denoted -x. The first part of the definition asserts the existence of an additive inverse (this is usually an axiom for certain algebraic structures such as Abelian groups, rings, or vector spaces). The second part states that the additive inverse is unique. Here is a proof of uniqueness: Suppose c is another real number which satisfies a + c = 0. Then c = 0 + c = (b + a) + c since b + a = 0 = b + (a + c) by associativity = b + 0 = b.

From Quiz Fermat's Last Theorem

Answer: Sophie Germain

Sophie German was born in 1776 in Paris, France. She was a student to Lagrange, a great mathematician.

From Quiz Famous Theorems in Math #1

Answer: The four-colour theorem

To this day there are mathematicians that debate the validity of the proof, as it cannot be manually checked by man. The computer program used, however, can be checked and reproduced. Rolle's theorem is a simple result from the calculus, the marriage theorem is a combinatorial result that describes matchings, and Kuratowski's theorem is a test for planarity of graphs.

From Quiz Basic Math by the Definition

Answer: Arithmetic progression

An example of an arithmetic progression is 2, 9, 16, 23, 30, 37, ... The difference between any two successive terms in the sequence is 7.

From Quiz Fermat's Last Theorem

Answer: Andrew Wiles

Andrew Wiles grew up in Cambridge, England. He is a British mathematician and works as a professor in Princeton University, USA.

From Quiz Famous Theorems in Math #1

Answer: All closed curves have an "inside" and an "outside"

Seems obvious, doesn't it? The proof, however, is not. The reason is due to topological generalizations of the notions of "closed" and "open".

From Quiz The Square Root of the Problem

Answer: that a^2 is odd

Conveniently, I have two lemmas that deal with my two cases!

From Quiz Quod Erat Demonstrandum

Answer: If p is prime then, for any integer a, a^p = a (mod p).

This theorem is the motor behind the Fermat Primality Test. The Fermat Primality Test can say for certain that a number is not prime, but can only give a probability that something is prime. Say you want to see whether 319 is prime. Pick an a less than 319, say 101. So a=101 and p=319. 101^319 ≡ 182 (mod 319), so we can conclude that 319 is not prime (because our result was 182 and it should have been 101). It turns out that 11*29 = 319. Now let's test to see whether 317 is prime. We'll use 101 again. 101^317 ≡ 101 (mod 317). So far so good. Let's try another value for a, such as a = 251. 251^317 ≡ 251 (mod 317). Let's check one more, say a = 299. 299^317 ≡ 299 (mod 317). So we can conclude that 317 is probably prime, because Fermat's Test worked for three random, unrelated integers. It turns out that 317 is prime. However, it could have been the case that I was just lucky with my selections. There are times when a^p ≡ a (mod p) holds even when p is not prime. An example of this (from Wikipedia) is p=221 and a=38. The above equation is true even though 221 is composite. (It's 13*17.) Note that if a is larger than p, you will have to take into account the fact that there exists b such that a ≡ b (mod p) and 0 ≤ b ≤ p. For instance, take p=2 and a=3. 3^2 ≡ 1 (mod 2). This may raise a red flag, but remember that 3 ≡ 1 (mod 2), so by the transitive property, 3^2 ≡ 1 ≡ 3 (mod 2). For the other choices: "If x, y, z, and n are natural numbers, there are no x, y, z such that x^n+y^n=z^n for n≥3." - This is Fermat's Last Theorem. "If n≥2 and (n-1)! ≡ -1 (mod n), then n is prime." - This is Wilson's Theorem. "If a and p are relatively prime, then a mod p ≠ 0." - This is a true statement though it's nothing earth-shattering.

From Quiz Basic Math by the Definition

Answer: f(x) = the absolute value of x

This is the actual definition of absolute value! Note that if x negative, -x is positive. By the definition, |-2| = -(-2) = 2.

From Quiz Fermat's Last Theorem

Answer: "Annals of Mathematics"

The proof is 140 pages long. The theory of elliptic curves is used in this proof.

From Quiz Pythagoras's Crazy Theorem

Answer: right triangle&right triangles&Pythagorean Triples &pythagorean triplets& Pythagorean Triads&Pythagorean Triple &Pythagorean triplet& Pythagorean Triad

A Pythagorean Triple is a set of three integers that make the Pythagorean Theorem true. Multiples of Pythagorean Triples (like a 6-8-10 right triangle is a multiple of a 3-4-5 one because all the sides are multiplied by 2) will also prove the Pythagorean Theorem true.

From Quiz Basic Math by the Definition

Answer: The Fundamental Theorem of Arithmetic

For example, 120 = (2^3) * 3 * 5. The above theorem states that this is in fact the only way you can write 120 as a product of powers of primes (we consider 3 * (2^3) * 5 to be the same factorization of 120). This theorem is not obvious! For a proof, consult a book on Number Theory.

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Last Updated Nov 16 2024 5:46 AM

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