Different Types of Numbers Trivia Quiz

Answer: Yes, you can arrange 25 points into a 5 by 5 square.

You can arrange 25 points into a square with 5 points on a side, and more generally you can rearrange x^2 points into a square with x points on a side. This is why these numbers are called square numbers!

Square numbers also have the property that they x^2 is the sum of the first x odd numbers. For example, 25=1+3+5+7+9.

Answer: 15

All triangular numbers have the form 1+2+3+...+n for some whole number n. For example, 15=1+2+3+4+5. This sum can be simplified by the formula 1+2+3+...+n=(n)(n+1)/2.

Just like with square numbers, you can arrange a triangular number of points into an equilateral triangle. More generally, one can define pentagonal numbers, or hexagonal numbers, or similar numbers for a shape with an arbitrary number of sides.

Answer: Every positive integer is a product of primes.

Every positive integer is a product of primes in a unique way. For example, 30=2*3*5, and 16=2*2*2*2. The number 1 is considered to be an empty product of primes, while a prime number is considered to be a product of one prime. This theorem goes by the name of the Fundamental Theorem of Arithmetic.

The number 2 is the unique even prime number.

In general, the only way for the sum of two prime numbers to be prime is if one of the primes is 2. In that case, q and q+2 are called twin primes. For example, 3 and 5 are twin primes.

Answer: If p is a Fermat prime, you can construct a regular p-gon with a straightedge and compass.

There are only 5 known Fermat primes, 3, 5, 17, 257, and 65537, and there are believed to be only finitely many such primes. Being able to construct a regular n-gon with a straightedge and compass is related to the factors of that number being either 2 or a Fermat prime. But don't try constructing a 65537-gon on your own, as it requires a lot of intermediate steps.

Answer: A perfect number

A perfect number is a number where the sum of all the proper factors (so don't include the number itself) equals the number. So for example, 28 has proper factors 1, 2, 4, 7, and 14, and 1+2+4+7+14=28.

The first few perfect numbers are 6, 28, 496, and 8128. Notice they all factor as 2^(n-1)*p where p is the prime number (2^n)-1. Such primes ( that are one less than a power of two) are known as Mersenne primes, and all even perfect numbers come from a Mersenne prime. It is unknown if there are any odd perfect numbers.

Answer: 6 factorial

In general, n! is pronounced n factorial (and is not shouted loudly).

Notice that if you ask 6 people to stand in a line, there are 6! ways for them to do so, since there are 6 ways to pick a first person, 5 ways to pick a second person, 4 ways to pick a third person, and so on. Factorials show up frequently in counting problems.

Answer: A complex number

A complex number is a number of the form a+bi, where i is the square root of -1. A square root of -4 is then just the complex number 2i (or -2i). While these numbers seem complex, you can understand them fairly easily by just understanding how to add, subtract, multiply, and divide 1 and i.

Special cases of complex numbers are real numbers, which have the form a, and imaginary numbers, which have the form bi.

A more complicated number system is the quaternions, which are numbers of the form a+bi+cj+dk, with i^2=j^2=k^2=-1 and ij=k. These have the unusually property that it matters what order they are multiplied in; while ij=k, ji=-k. They can be used to describe the rotations of objects in 3 dimensional space.

Answer: 12321

A palindromic number is just a number that looks the same when read backwards or forwards, just like a normal palindrome.

There are also palindromic primes, which are primes which are also palindromic numbers. It is not known if there are infinitely many palindromic primes.

Answer: golden ratio

The ratio of adjacent Fibonacci numbers converges to the golden ratio (1+sqrt(5))/2, which is approximately 1.618. For example 5/3 =1.66..., while 34/21=1.619...

Fibonacci originally used the Fibonacci numbers to model the breeding habits of rabbits!

Answer: A number that is not the root of a polynomial with integer coefficients.

A transcendental number is a number that can't be written as a root of an equation with integer coefficients. A number that can't be represented as a ratio of two integers is called an irrational number, which is equivalent to not having a repeating decimal expansion. All transcendental numbers are irrational, but some irrational numbers are not transcendental.

For example sqrt(2) is irrational, but it satisfies the equation x^2-2=0, so it is not trascendental.