Excited About "e" Trivia Quiz

Answer: Euler

In mathematics, the letter e stands for "Euler's number". Leonard Euler (1707-1783) is generally considered one of the greatest and most influential mathematicians in history. He is responsible: for introducing the letter pi to represent the ratio of a circle's circumference to its diameter; for using the letter i to represent the square root of -1; the geometric labeling of angles in polygons with upper case letters, and the sides in lowercase letters; many of the notations used in calculus; establishment of the field of graph theory as he solved the Konigsberg bridge problem; and, relevant for this quiz, developing the current definition of the value of e.

He was not the first to encounter e, but he was the first to use e to represent the number.

Answer: compound interest

Bernoulli was investigating the effect of compounding interest at shorter intervals. In other words, instead of calculating the interest and adding it to the amount invested once a year, at the end of the year, one could cut the rate in half but process it twice a year. Using shorter and shorter compounding intervals led him to the theoretical process of continual compounding - even more often than once a second, which is mind-boggling in itself.

He found that there was a number which could be raised to the power of Rt, where R is the annual interest rate and t is the number of years, to produce the amount at any given time.

This number we now call e.

Answer: pi

Rational numbers can be written as the quotient of two integers (1/2 or 1/3, for example) and as a decimal which either terminates (1/2 = 0.5) or repeats forever (1/3 = 0.333...). Numbers for which this is not true are called irrational numbers. Some irrational numbers can be derived as the root of a polynomial expression. For example, i is the value of x for which x^2 + 1 = 0; it is also called the imaginary unit, as all numbers that can be written as the square root of a negative number can be written as a multiple of i.

Like e, pi (usually shown using the Greek letter which doesn't display well here) is transcendental, as there is no polynomial expression that can be solved to get its exact value. Both of these numbers must have their value approximated, and there are several techniques that can be used in each case.

Answer: 2.71828

The value of e has been calculated to more decimal places than the average person cares about - in 1978 Steve Wozniak used an Apple computer to obtain the first 116,000 digits following the decimal point. In 2020, the record was 31,415,926,535,897 decimal places.

For the sake of it, here is e, written to 50 decimal places:
2.71828182845904523536028747135266249775724709369995

1.41421 is approximately the square root of 2; 1.61803 is approximately the value of the golden ratio (that's for another quiz); 3.14159 is approximately the value of pi.

Answer: factorials

The factorial of a whole number whose value is n (written as n!) is found by multiplying that number by all the integers smaller that itself until you get to 1. (Our friend 0 creates problems here, so 0! is defined to be equal to 1. This makes mathematical sense, even if it seems arbitrary to the casual observer.) 2! = 2 x 1 = 2; 3! = 3 x 2 x 1 = 6; 4! = 4 x 3 x 2 x 1 = 24, and so on.

e = 1/0! + 1/1! + 1/2! + 1/3! + ...
The ellipsis at the end means you have to go on forever to get the exact value, but fewer terms can be used to get an approximation. The expression above gives 1+1+0.5+0.1666... which is roughly 2.66. Obviously we need to add some more calculations to improve our accuracy!

Answer: 1/n

This is actually the expression for which Bernoulli was trying to find the limiting value as n gets larger and larger, eventually reaching infinity. The expression approaches e more slowly than the sum of the reciprocals of the factorials of the whole numbers, so requires more calculation to get a good approximation.
(1+1/1)^1 = 2
(1+1/2)^2 = 2.25
(1+1/3)^3 = 2.370370...
(1+1/4)^4 = 2.44140625

Answer: Euler's identity

Mathematicians love this identity, as it involves all five of the most fundamental and intriguing numbers! It is actually a consequence of the relationship that Euler established between exponential and complex trigonometric functions: e^(ix)=cos(x)+i*sin(x). I won't even try to discuss this - it is worth a quiz of its own - but when the value of x is pi, it gives the identity in the question.

Quick look at the five terms:
*e is the subject of this quiz, and needs no further introduction
*i has already been described in passing as the square root of -1, and the fundamental unit for complex numbers
*pi is familiar from finding areas of circles (and so much more!)
*1 is the multiplicative identity for real numbers - multiplying any number by 1 give you the original number as the answer
*0 is the additive identity for real numbers - adding 0 to any number gives you the original number as the answer

Answer: The function is its own derivative and antiderivative.

The function f(x)=e^x is an exponential function, which means that, like any function in which some number is raised to the power of x, a graph with f(x) on the vertical axis and x on the horizontal axis forms a smooth increasing curve (not a spiral) which has no upper bound, since there is not any largest possible value; it does, however, have a lower bound, as the graph gets closer and closer to the x-axis as x gets more and more negative. The x-axis is therefore a horizontal asymptote. A vertical asymptote would mean there was a value of x for which e^x is meaningless, and that does not exist.

Enough of the definitions. What is important about using e as the base for your exponential function (as opposed to some other number, such as 10) is that differentiation and antidifferentiation processes are vastly simplified. Differentiation is used to determine how steeply the curve is trending upwards at any point, while antidifferentiation is used to find the area between the curve and the x-axis between two given points. Trust me, people do this in real life, not just in the classroom.

Answer: ln

The natural logarithm of any positive real number a can be defined without referencing the number e; it is the area under the curve y=1/x between 1 and a. It follows that the ln(e) = 1. This means that the two functions are inverses. If y = e^x, then x = ln(y). In other words, when e is raised to a certain power (x) the result is y. Exponential functions work out what the result will be for a given power; logarithmic functions work out what power is needed to get the result.

Fascinating, no? Well, John Napier (1550-1617) thought logarithms were incredibly interesting, and developed tables of them that made arithmetic computations much easier. It turns out that you can quickly multiply two numbers by adding their logarithms (to any base - the tables that used to be widely used were base 10), then finding the anti-logarithm of your answer. Division required subtraction of logarithms. Other operations such as square roots, cubes, etc. could also be dramatically simplified. Of course, the accuracy of your answer relied on the precision of the calculations in the table!

Answer: Google

Of course, computer folk love e! The IPO filed by Google represents e billion dollars (to the nearest dollar). IPOs usually use nice round figures, so this one stood out from the crowd. Google also set up a billboard designed to lure people into solving a number of puzzles relating to e.

The billboard read: "{first 10-digit prime found in consecutive digits of e}.com"; going to 7427466391.com (don't try, it's long gone) led to another puzzle which led to another, eventually taking the successful puzzler to a job application page.