# Lesson 12

The Number $e$

• Let’s learn about the number $$e$$.

### 12.1: Matching Situations and Equations

Match each equation to a situation it represents. Be prepared to explain how you know. Not all equations have a match.

$$f(t) = 400 \boldcdot (0.5)^{0.1t}$$

$$g(t) = 400 \boldcdot (1.25)^{0.1t}$$

$$h(t) = 400 \boldcdot (0.75)^{0.1t}$$

$$j(t)=400 \boldcdot (2)^{10t}$$

$$k(t) = 400 \boldcdot (2)^{0.1t}$$

1. A scientist begins an experiment with 400 bacteria in a petri dish. The population doubles every 10 hours. The function gives the number of bacteria $$t$$ hours since the experiment began.
2. A patient takes 400 mg of a medicine. The amount of medicine in her bloodstream decreases by 25% every 10 hours. The function gives the amount of medicine left in her bloodstream after $$t$$ hours of taking the medicine.
3. The half-life of a radioactive element is 10 years. There are 400 g of the element in a sample when it is first studied. The function gives the amount of the element remaining $$t$$ years later.
4. In a lake, the population of a species of fish is 400. The population is expected to grow by 25% in the next decade. The function gives the number of fish in the lake $$t$$ years after it was 400.

### 12.2: Notice and Wonder: Moldy Growth

A spot of mold is found on a basement wall. Its area is about 10 square centimeters. Here are three representations of a function that models how the mold is growing.

time (weeks) area of mold (sq cm)
0 10
1 27
2 74
3 201
4 546

$$a(t) = 10 \boldcdot e^t$$

What do you notice? What do you wonder?

### 12.3: $(1 + \text{tiny})^{\text{huge}}$

1. Here are some functions. For each function, describe, in words, the outputs for very tiny, positive values of $$x$$ and for very large values of $$x$$.

$$a(x) = 1^x$$

$$b(x) = \text-x$$

$$d(x) = \frac{1}{x}$$

$$f(x) = \left(\frac{1}{x}\right)^x$$

$$g(x) = \left(1 + \frac{1}{x}\right)^x$$

$$h(x) = e^{x}$$

$$k(x) = 1 + x$$

2. Remember that $$e \approx 2.718$$. What does the function $$g$$ have to do with the number $$e$$?
3. What do you notice about the relationship between $$h$$ and $$k$$ for very small, positive values of $$x$$?

Complete the table to show the value of each expression to the nearest hundred-thousandth. Two entries have already been completed as an example.

$$x$$ $$2^x$$ $$e^x$$ $$3^x$$
0.1 1.07177 1.10517 $$\phantom{1.11612}$$
0.01
0.001
0.0001

What do you notice about the values in the table?

### Summary

Scientists, economists, engineers, and others often use the number $$e$$ in their mathematical models. What is $$e$$?

$$e$$ is an important constant in mathematics, just like the constant $$\pi$$, which is important in geometry. The value of $$e$$ is approximately 2.718. Just like $$\pi$$, the number $$e$$ is irrational, so it can’t be represented as a fraction, and its decimal representation never repeats or terminates. The number is named after the 18th-century mathematician Leonhard Euler and is sometimes called Euler’s number.

$$e$$ has many useful properties and it arises in situations involving exponential growth or decay, so $$e$$ often appears in exponential functions. In upcoming lessons, we will work with functions that are expressed using $$e$$.

### Glossary Entries

• $e$ (mathematical constant)

The number $$e$$ is an irrational number with an infinite decimal expansion that starts $$2.71828182845\ .\ .\ .$$, which is used in finance and science as the base for an exponential function.