By Chris Deziel, Updated Aug 30, 2022
Photo credit: Marek Uliasz / iStock / Getty Images
The letter E can carry two distinct meanings in mathematics, depending on whether it is capitalized.
On calculators and in engineering texts, a capital E denotes an exponent of 10. For example, 1E6 means 1 × 106, or one million. This shorthand is handy for numbers that would otherwise overflow a screen or clutter a page. Typically, E is reserved for base‑10 exponents; it is not used with other bases.
When writing a number in scientific notation, the format is xEy, where x is the significant figure(s) and y is the power of ten. Common examples include 5E6 (five million) and 4.27E4 (42,720). Most scientific contexts round to two decimal places for clarity.
Mathematicians use the lowercase e to denote Euler’s constant, an irrational number approximately 2.7182818284 (to ten decimal places). Like π, it has a non‑repeating, infinite decimal expansion. Despite its seemingly abstract nature, e is one of the most essential constants in mathematics and the natural sciences.
The constant e emerged from a financial problem posed by Jacob Bernoulli in the late 17th century. Consider a $1,000 deposit at 100% annual compound interest for one year: the balance becomes $2,000. If the interest rate is halved but applied twice a year, the balance rises to $2,250. At a monthly rate of 8.33% (1/12 of 100%), applied 12 times a year, the balance reaches $2,613.
The general formula for compound interest is:
(1 + r/n)^n, where r is the annual rate (here 1) and n is the number of compounding periods.
As n approaches infinity, the expression converges to the limit e. Euler discovered this limit, showing that the maximum achievable return in one year on a $1,000 investment is approximately $2,718.
Functions of the form y = e^x are called natural exponentials. The graph of this function is unique because, at every point, the slope of the curve equals its value, and the area under the curve up to that point also equals the function’s value. These properties make e indispensable in calculus, differential equations, and modeling growth or decay.
One of the most ubiquitous appearances of e in nature is the logarithmic spiral, described by the equation:
r = a e^(bθ). This spiral shape is found in seashells, fossils, and many flowers.
Beyond geometry, e surfaces in diverse scientific contexts, such as electric circuit analysis, Newton’s law of cooling, and the differential equation governing damped harmonic oscillators.
Even after three centuries since its discovery, Euler’s number continues to reveal new applications across physics, biology, economics, and engineering.
