Follow one remarkable number through history, calculation, growth, calculus, engineering, physics, and the complex plane.
Euler's number was not invented in a single moment. It slowly emerged as people searched for faster calculations, studied interest, and tried to describe continuous change.
Logarithms give astronomers and navigators a practical way to turn difficult multiplication into easier addition.
Edmund Gunter and William Oughtred help turn logarithms into the sliding scales that become the slide rule.
He notices that compounding more and more often approaches a fixed value near 2.71828.
Euler adopts the symbol e and shows that the same number belongs naturally to growth, calculus, infinite series, and complex numbers.
Euler's number appears in probability, circuits, heat transfer, motion, population growth, quantum physics, and countless other systems.
Astronomers, navigators, engineers, and merchants repeated long calculations by hand. A method that could replace multiplication with addition could save hours of work—and reduce costly mistakes.
Now imagine doing thousands of calculations like this without a calculator.
A logarithm table pairs each ordinary number with a second number—its logarithm. To multiply, people looked up two logarithms, added them, then used the same table in reverse to recover the answer.
Some numbers are already easy to recognize as powers. Since \(32=2^5\) and \(64=2^6\), multiplying them becomes \(2^{5+6}=2^{11}=2048\). A logarithm table applies that same exponent shortcut to less convenient numbers.
Logarithms can use different bases. Changing the base changes the scale—but one strange base keeps showing up when math describes smooth, continuous change.
A logarithm is really asking one question: “What power of this base gives me my number?”
ln means natural logarithm. Why “natural”? Because later mathematicians discovered that this base makes the math of continuous growth unusually clean. Other bases work—but this one fits without extra conversion factors.
For now, just remember that number. We are about to leave logarithms behind and find it again somewhere completely different.
Imagine investing one dollar at 100% interest for one year. The more often the interest is added, the more the balance grows—but the year's total rate must still remain 100%.
We want every example to promise the same total yearly rate: 100%. The only thing we change is how often that year's interest is placed into the account.
So one payment adds 100%, two payments add 50% each, and four payments add 25% each. Leaving every payment at 100% would quietly increase the total yearly rate, so it would no longer be the same investment.
The number \(n\) tells us both how many payments occur and how small each payment becomes.
The logarithm room and a completely different problem about money have led to the same decimal: 2.718281828… This is the clue that the number is more than a calculating trick.
A number first noticed in repeated growth became the natural language of continuous change.
Leonhard Euler did not discover every piece of the story himself, but he connected the pieces, developed the mathematics around them, and gave the constant the symbol we still use today.
In calculus, the slope at one exact point is called a derivative. It is shown by the straight line that just touches the curve there.
Now we can finally name its base. That mysterious \(2.71828\ldots\) is \(e\), so:
These two operations undo each other. Think of them as two opposite machines:
Give it an exponent, and it tells you the resulting number.
Give it the result, and it tells you what exponent was used.
There is an even deeper calculus reason this pair is so clean. We will save that derivation for the Calculus exhibit rather than dropping a wall of new symbols on you here.
A Taylor series builds a complicated curve from simple pieces. Begin with a flat line, then add one piece at a time. Each new piece makes the white curve look more like the true curve of \(e^x\).
The exclamation mark means factorial. For example, \(4!=4\times3\times2\times1=24\). Calculators use approximation ideas like this to turn functions into arithmetic they can actually compute.
The same series from the previous room changes character when the imaginary number \(i\) enters the exponent. Instead of simply growing, it begins to move around a circle.
Replace \(x\) with \(i\theta\). The equation still has the same pieces, but now those pieces contain powers of \(i\).
They move through \(i,-1,-i,1\), then repeat. That naturally divides the series into ordinary real pieces and imaginary pieces.
The real pieces match the series for cosine. The imaginary pieces match the series for sine.
Cosine controls the horizontal position. Sine controls the vertical position. Together they place the glowing point on a circle.
Whenever the speed of change depends on how much is already present, \(e\) tends to appear. Choose a real system and adjust one plain-language control to see how its behavior changes.