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# Theorems Trivia

## Theorems Trivia Quizzes

8 Theorems quizzes and 80 Theorems trivia questions.
1.
Pythagoras's Crazy Theorem
Multiple Choice
10 Qns
How much do you know about the theorem of Pythagoras? This quiz will involve questions about the theorem, determining side lengths, and will also have questions about Pythagorean Triples. All non-integral solutions will be in radical form.
Average, 10 Qns, XxHarryxX, Jun 01 23
Average
XxHarryxX
Jun 01 23
4324 plays
2.
Fermat's Last Theorem
Multiple Choice
10 Qns
Fermat's last theorem remained unsolved for centuries until it was proven in the year 1995. Have fun and thanks for playing.
Average, 10 Qns, Matthew_07, Oct 05 16
Average
Matthew_07
1306 plays
3.
Benford's Law
Multiple Choice
10 Qns
Benford's law gives a neat mathematical description about the frequency distribution of leading digits of numerical figures that appear in large data sets. The law, albeit incomprehensible at first, has found many useful applications in real-world data.
Average, 10 Qns, Matthew_07, Aug 24 20
Average
Matthew_07
Aug 24 20
127 plays
4.
Famous Theorems in Math #1
Multiple Choice
10 Qns
This is my first quiz about some famous mathematical theorems.
Difficult, 10 Qns, Mrs_Seizmagraff, Jun 18 08
Difficult
Mrs_Seizmagraff
3365 plays
5.
Pythagorean Theorem
Multiple Choice
10 Qns
The Pythagorean theorem (or Pythagoras' theorem) describes the relationship between the lengths of three sides of any right-angled triangle. This quiz tests your knowledge on the history, proof and application of this famous theorem. Enjoy!
Average, 10 Qns, Matthew_07, Jan 20 22
Average
Matthew_07
Jan 20 22
952 plays
6.
Basic Math by the Definition
Multiple Choice
10 Qns
This quiz involves well known concepts from basic mathematics, but the actual definitions and theorems are probably not well known to non-mathematicians. I give the definition or theorem, you tell me what's being described. Good Luck!
Average, 10 Qns, rodney_indy, Jul 18 07
Average
rodney_indy
1787 plays
7.
Quod Erat Demonstrandum
Multiple Choice
10 Qns
This quiz is a mishmash of proofs, interesting theorems, and other mathematical marvels. No calculations are involved.
Tough, 10 Qns, redsoxfan325, Apr 29 10
Tough
redsoxfan325
547 plays
8.
The Square Root of the Problem
Multiple Choice
10 Qns
Title comes from kyleisalive in the Author Challenges. This quiz walks the quiz taker through a proof that the square root of 3 is irrational. The proof is one that I wrote while taking a course in the history of mathematics.
Average, 10 Qns, hlynna, Jan 14 17
Average
hlynna
555 plays

#### Theorems Trivia Questions

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

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.

2. Fermat's last theorem states that there is no solution for the equation x^n + y^n = z^n, where n is any integer greater than 2. Why was this theorem called the LAST theorem?

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.

3. What is the formula for the Pythagorean Theorem? (^2 means squared)

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.

4. According to the Pythagorean Theorem, the square of the hypotenuse of a right triangle is equal to what?

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.

5. The Pythagorean Theorem only applies to right-angled triangles. However, there is a more general "law" that governs all triangles in a relationship similar to that of the Pythagorean Theorem. What is the name of this law?

From Quiz Famous Theorems in Math #1

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.

6. Intuitively speaking, if the leading digits of all numbers in a data set are distributed uniformly, what is the expected frequency of each of the nine numbers (1-9)?

From Quiz Benford's Law

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%.

7. In trigonometry, the Pythagorean theorem is considered a special case of a more generic law. In other words, this mathematical law is a more general formula that can be used for all types of triangles. What is the name of this law?

From Quiz Pythagorean Theorem

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.

8. Let Q be the set of all numbers of the form a/b where a and b are both integers and b cannot be 0. What is the set Q called?

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.

9. It is interesting to find out that Pierre de Fermat was only an amateur mathematician. What was his main profession?

From Quiz Fermat's Last Theorem

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.

10. Moving on several hundred years, in Italy in the 1500s there arose a great dispute between two leading mathematicians named Cardano and Tartaglia over a new method. What was this method?

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.

11. Benford's law, albeit counter-intuitive, states that this digit is the most likely to appear as the leading digit in any numbers contained in a big data set. Which lonesome number is being described?

From Quiz Benford's Law

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.

12. Let a and b be positive integers. Then there exist integers q and r which satisfy a = b*q + r with r greater than or equal to 0 and less than b. What is r called?

From Quiz Basic Math by the Definition

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.

13. For the equation x^n + y^n = z^n, Euler proved that for the case where n = 3, there were no integer solutions. Fermat himself provided the proof for a case where n = ?

From Quiz Fermat's Last Theorem

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.

14. The 20th President of the United States provided an algebraic proof of the Pythagorean theorem. Who is he?

From Quiz Pythagorean Theorem

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.

15. When Euclid published his text, "Elements", he included five postulates. One of those postulates has been disputed and actually led to the study of non-Euclidean geometry. Which one was it?

From Quiz Quod Erat Demonstrandum

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?

16. Suppose a and b are real numbers (b positive) which satisfy a^2 = b. What is a called?

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.

17. Fermat's Last Theorem is undoubtedly the most famous theorem in all of mathematics, first being proposed in the 1600s but not fully solved until the mid 1990s. What method was used to finally solve this centuries-old problem?

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.

18. Let a be a real number. Then there exists a real number b which satisfies: a + b = 0. Furthermore, if c is another real number which satisfies a + c = 0, then c = b. What is this number b called?

From Quiz Basic Math by the Definition

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.

19. Which female mathematician, who took on a man's pseudonym, namely Monsieur Le Blanc, also attempted to prove Fermat's last theorem?

From Quiz Fermat's Last Theorem

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

20. Another famous theorem was first conjectured in the 1800s, but was not solved until 1976 in a highly controversial way: the proof depends on the use of a computer. To which theorem am I referring?

From Quiz Famous Theorems in Math #1

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.

21. This is a type of sequence of numbers in which the difference between successive terms is constant.

From Quiz Basic Math by the Definition

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.

22. By 1993, Fermat's last theorem was proved to be valid for all n from 3 to 4,000,000 by computers. Eventually, Fermat's last theorem was proven in the year 1995 by a mathematician. Who is he/she?

From Quiz Fermat's Last Theorem

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

23. An example of a deceptively simple-stated theorem with an extraordinarily difficult proof is the Jordan curve theorem. What is the fundamental result of this theorem?

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".

24. So, I have sqrt3=a/b. I'll do some algebra! I multiply both sides by b, then square both sides. Now I have 3b^2=a^2. Now I have two cases to consider. One is that a^2 is even. What is the other?

From Quiz The Square Root of the Problem

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

25. Most people have heard of Fermat's Last Theorem, but fewer people have heard of his Little Theorem. What is it?

From Quiz Quod Erat Demonstrandum

Answer: If p is prime then, for any integer a, ap = 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.

26. Which function f has the following definition for a real number x? If x is greater than or equal to 0, then f(x) = x, otherwise f(x) = -x.

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.

27. The proof of Fermat's last theorem was published in May 1995 in which mathematical journal?

From Quiz Fermat's Last Theorem

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

28. The sets, {3,4,5}; {5,12,13}; and {8, 15, 17} are all examples of what? (The numbers are the side lengths of the triangles.) Please give your answer in plural form.

From Quiz Pythagoras's Crazy Theorem

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.

29. A positive integer can be written uniquely (up to order of the factors) as a product of powers of distinct primes. What is this theorem called?

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 Apr 13 2024 5:46 AM
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