Connecting the constants
The final number comes from theoretical mathematics. It is Euler's (pronounced "Oiler's") number: eπi. This number is equal to -1, so when the formula is written eπi+1 = 0, it connects the five most important constants in mathematics (e, π, i, 0, and 1) along with three of the most important mathematical operations (addition, multiplication, and exponentiation).
These five constants symbolize the four major branches of classical mathematics: arithmetic, represented by 1 and 0; algebra, by i; geometry, by π; and analysis, by e, the base of the natural log. eπi+1 = 0 has been called "the most famous of all formulas," because, as one textbook says, "It appeals equally to the mystic, the scientist, the philosopher, and the mathematician."
The reason for this wide-ranging appeal is its utter serendipity. First, there is the ubiquitous number e, which pops up in the most unexpected places. It was first discovered in an attempt to make multiplication easier. In 1614, John Napier figured that adding exponents was easier than multiplying multi-digit numbers, so he (and others) calculated the logarithms of all integers from 1 to 100,000, expressing these numbers as powers of 10. Later mathematicians found it more convenient to express logarithms as powers of the natural log e, a number close to 2.71828.
This number also appears in banking, because it is the limit for growth of compound interest. Let's say one invested $1,000 in a very generous bank that paid an annual interest of 100%. If interest were compounded annually, at the end of the year, the money would have grown to $2,000. If, however, the bank compounded interest four times a year, the money would grow to $2,441.41. If the bank compounded interest continually, the deposit could grow to $2,718.28, which just happens to be the value of e times the original investment.
Finally, e turns up at the origin of calculus, where it is the function equal to its own derivative (if y = ex then dy/dx = ex), and it equals the limit of (1+ 1/n)n as n approaches infinity. e is irrational, so it can never be written exactly in decimal form, but it is a very useful and fascinating number in its own right.
When we combine e with π, we are introducing the oldest irrational number. Two thousand years before Christ, the Greeks knew that π was the ratio of the circumference of a circle to its diameter and that it could not be expressed as the ratio of any two integers. It is essential in geometry, but it also turns up in waves of air, water, electricity, and light, and it even helps actuaries calculate how many 50-year-old men will die this year.
The number i is a relative latecomer, proposed in the 1600s as an imaginary number and defined as the square root of -1. It was proposed to help solve equations like x2+ 1 = 0, but today it is useful in science and engineering. George Gamow, in his book One, Two, Three … Infinity, even uses i to locate buried treasure with an outdated map.
The idea that these two irrational numbers should combine with an imaginary one to yield so utilitarian a result is breathtaking. It is like deconstructing a chemical necessary for life (salt) and finding that it consists of two deadly poisons (sodium and chlorine). That these three strange numbers with such diverse origins should work together to produce a result so basic to mathematics argues that there is a profound elegance or beauty built into the system.
The discovery of this number gave mathematicians the same sense of delight and wonder that would come from the discovery that three broken pieces of pottery, each made in different countries, could be fitted together to make a perfect sphere. It seemed to argue that there was a plan where no plan should be.
Because of the serendipitous elegance of this formula, a mathematics professor at MIT, an atheist, once wrote this formula on the blackboard, saying, "There is no God, but if there were, this formula would be proof of his existence."
Today, numbers from astronomy, biology, and theoretical mathematics point to a rational mind behind the universe. To be sure, they do not point to the personal God of the Bible as such. Yet they are not inimical to the biblical God, either. The apostle John prepared the way for this conclusion when he used the word for logic, reason, and rationality—logos—to describe Christ at the beginning of his Gospel: "In the beginning was the logos, and the logos was with God, and the logos was God." When we think logically, which is the goal of mathematics, we are led to think of God.