Transcendental Numbers Trivia Quiz

Answer: all of these are descriptions of transcendental numbers, with varying degrees of precision

The most familiar transcendental numbers are pi (the ratio of a circle's circumference to its diameter) and e (Euler's number), but there are more!

A rational number can be written as a fraction with an integer in both the numerator and the denominator. They can also be written as a decimal number which either terminates (has no non-zero values after a certain number of decimal places) or repeats (has one or more digits which repeat endlessly as you write more and more decimal places to represent the number.)

Some real numbers which cannot be written exactly in rational form (called irrational numbers) are not transcendant, because they can be calculated as the solution to an algebraic equation. One such number is i, the square root of -1. Imaginary and complex numbers (such as 3i or 4+5i) also cannot be written in rational form, but that doesn't make them transcendant. A transcendental number, unlike a rational, irrational or complex number, cannot be found by solving a polynomial equation that has rational coefficients.

Answer: No

In 1873 Charles Hermite proved that the number e was transcendental. Pi was proven to be transcendental in 1882 by Ferdinand van Lindeman, ending over 2500 years of speculation.

Answer: No

The square root of 2 is a number which cannot be written exactly as a common fraction or decimal (either terminating or repeating), so it is irrational; however, it CAN be calculated as the solution to an algebraic equation (x^2=2), so it is not transcendental.

Answer: Yes

Napier's number is a less-used term to describe e, usually referred to as Euler's number. Both of these mathematicians worked extensively with e. John Napier developed a system of logarithms to the base e, usually called natural logarithms. Leonhard Euler worked more with the exploration of the exponential function f(x) = e^x, which has the useful (for mathematicians!) property that it is its own derivative.

The value of e can be approximated as 2.72 (to 2 decimal places). Using other number bases, it looks different again. Using base 2, e can be written as 10.100110 (to 5 decimal places); using hexadecimal numbers (base 16), it can be written as 2.B7E15 (to 5 decimal places).

Answer: Yes

This lovely number is written by concatenating (writing one after the other) the positive integers. Since they never run out, it makes sense that the number goes on forever. The proof that it is transcendental is, of course, somewhat more complex!

Answer: 2^(sqrt2)

David Hilbert first posed the theorem that 2^(sqrt2) was transcendental, then proved that it was. This has been more generally stated in the Gelfond-Schneider theorem: any number in the form a^b is transcendental if a & b are both algebraic numbers, a is not equal to either 0 or 1, and b is irrational.

Answer: 24

In Liouville's constant, the digits are 0 or 1. 1 only occurs when the place number is a factorial number.
1! = 1
2! = 2 x 1 = 2
3! = 3 x 2 x 1 = 6
4! = 4 x 3 x 2 x 1 = 24
5! = 5 x 4 x 3 x 2 x 1 = 120

Answer: e^(i*pi)

According to Euler's Identity, e^(i*pi) = -1. This is clearly NOT transcendental! The other three all are.

Answer: log(base e) of a, where a is algebraic and not equal to 0 or 1

Since log(base 10) of 10 is 1, it is not transcendental. Log(base 10) of any power of 10 will be an integer. Cos(a)may be transcendental, and is for most algebraic values of a; however, if a = 0, cos (a) = 1. Log(base 10) of a is transcendental for many values of a, but, as previously stated, not if a is a power of 10.

Answer: at least one of these is transcendental

It is (relatively) straightforward to prove that at least one of the sum and product must be transcendental, but not to show which one or ones.

For any two numbers a & b, (x-a)(x-b) = x^2 -(a+b)x +ab.
If all the coefficients of the expanded form on the right side of this equation are algebraic (not transcendental), then the polynomial is algebraic. Since all algebraic polynomials have algebraic roots, a & b must both be algebraic. But since we know that neither one is algebraic, this contradiction means that either their sum or their difference or both must be transcendental.