# Ann Varela: Leonhard Euler

An Interview with Ann Varela: Leonhard Euler

Michael F. Shaughnessy –

1) Ann, A great Swiss mathematician was Leonhard Euler, who lived from 1701 to 1783. What do we know about his early life and childhood and his family of origin?

Well, Mike, we know that Leonhard Euler was born in Basel, Switzerland, but spent most of his youth in Riehen.  His father was a Calvinist pastor who had connections with Johann Bernoulli; so Leonhard had the opportunity to study with this distinguished mathematician.  Although Euler’s father wanted him to follow in his footsteps and become a preacher, Euler ultimately pursued a career in mathematics.  By the age of 25, he defended his dissertation and acquired a position at St. Petersburg Academy in Russia.

2) I understand that he is MOST known for mathematical notation. Can you explain what is meant by mathematical notation and please tell us why it is important in the big scheme of things?

In mathematics, symbols and variables are used to denote a common mathematical language, if you will.  For example, Euler used the letters a, b, and c to represent constants, like the sides of a triangle.  Furthermore, the use of x, y, and z are now common variables in an equation originating with Euler.  The creation and use of these common symbols and variables were what made it possible to share and collaborate with other mathematicians and scientists of the time.

Also, since there was now uniformity of symbols, knowledge was more easily disseminated to, and studied by, the masses.

3) Euler’s number, e, is supposedly used in calculus. It has been years since I have taken Calculus, so can you tell us the meaning of this “e” and why it is important?

In mathematical models of natural events, the number e = 2.7182… often appears as the base of an exponential function.  One familiar example of presenting this number is by using the compound interest formula:  A = P(1 + r/n)nt.  If the number of compounding periods, n, becomes particularly large, the value of (1 + r/n)nt comes very close to the value of e.  When the amount invested grows exponentially, according to the formula A = Pert, we say that interest is compounded continuously.

Euler’s number, e, is important because it allows us to model other natural events, such as population growth, the rate at which disease spreads, alcohol and drug absorption rates, and the velocity of free-falling objects, similar to a skydiver, to name a few.

4) Apparently “e” is the base of what is termed the natural logarithm. Can you give us an example as to how this all fits together in terms of math?

You may be interested to know that the natural logarithm is also called Napierian logarithms, after John Napier, who worked extensively with natural logarithms.  There are several examples of how Euler’s number, e, is used in mathematics, specifically with natural logarithmic formulas.  Here are some examples:  1)  Electrical engineers can measure the breakdown voltage that a coaxial cable can withstand; 2)  Chemists measure the concentration of acid to hydrogen ions; 3)  In physiology, experiments propose that the relationship between the loudness and the intensity of sound is a logarithmic one known as Weber-Fechner law; 4)  Doubling time for population growth and isothermal expansion are two additional examples of how Euler’s number, e, is tied into math; and 5)  Even the rate of charging a battery is a logarithmic function.

5) Now–key word—if you will—” function”. What do you mathematicians mean by “function” and apparently Euler was the first to write f(x) to indicate that f was a function of x. Why is this important in mathematical notation?

In its most basic sense, a function describes the relationship between a set of inputs (domain) and a set of allowable outputs (range).  Each input is paired with a unique output.

One example of function notation is f(x), which is read “f” as a function of “x”.  In this case, we use the value of “x” to evaluate a given function, in other words, to find the function’s output value.

Given: f(x) = 5x + 1, let x = 2.  Next, the function is evaluated with an x-value of 2.  Substitute “2” for “x” and you will get your output, or function value.  f(2) = 5(2) + 1.  Therefore, f(2) = 11.  When the input value is 2, the function value is 11.  Now this is an extremely simple example of a function, but you get the idea.

Function notation is important to mathematical notation because it allows for data collection and ultimately interpretation of results.