Key takeaway: Euler’s number is the irrational constant e ≈ 2.71828, the value approached by (1 + 1/n)^n as n grows without bound and the natural base for continuous growth and decay.
Grade 11 – Algebra 2F-LE.A.4Functions and Graphs
Definition of Euler's Number
Euler’s number, written \(e\), is a constant a little larger than 2.7. It appears when growth happens continuously instead of in jumps. Imagine a balance that earns 100% interest per year. Paying the interest in more and more installments helps, but the balance never passes a ceiling of about 2.71828 times the start.
That ceiling is \(e\). Like \(\pi\), it is irrational: its decimal expansion never ends and never repeats. It is the base of the natural exponential function \(e^x\), which grows at a speed equal to its own value.
Formula and notation
Euler’s number is defined by a limit:
\[e=\lim_{n\to\infty}(1+\frac{1}{n})^{n}\approx 2.71828.\]
For continuous growth or decay, the amount after time \(t\) is \(A=Pe^{rt}\), where \(P\) is the start and \(r\) the rate per unit of time as a decimal.
Key facts
- Euler’s number starts 2.718281828, but its digits never settle into a repeating pattern.
- The function e^x equals its own rate of change at every point.
- The inverse of e^x is the natural logarithm ln x.
- Continuous compounding always earns slightly more than any fixed number of payments per year.
Euler's Number example
Mateo invests \$2,400 at 3.5% annual interest compounded continuously. How much is in the account after 8 years?
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Use \(A=Pe^{rt}\) with \(P=2400\), \(r=0.035\), \(t=8\).
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The exponent is \(rt=0.035\times 8=0.28\).
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Compute \(e^{0.28}\approx 1.32313\), then \(2400\times 1.32313\approx 3175.51\).
The balance is about \$3,175.51.
Common mistake
Watch out
Hana argues that as \(n\) gets huge, \(\frac{1}{n}\) goes to 0, so \((1+\frac{1}{n})^n\) becomes \(1^n=1\).
Why it is wrong
Two things happen at once: the base shrinks toward 1 while the exponent grows without limit. Their tug of war ends at 2.71828, not 1. At \(n=1000\) the value is already 2.717.
Related terms and lessons
Browse the glossary: A–Z list · Algebra 2 vocabulary · Functions and Graphs