Exponential Growth: Explained and Visualized

Explore the fascinating world of exponential growth through a classic story of a clever mathematician and a king's chessboard. We'll break down how small, consistent increases can lead to surprisingly big changes, using real-world examples like bacterial growth, investment returns, and population trends.

Video Transcript:
Legend tells of a clever mathematician who performed a great service for his king. When asked to name his reward, he made what seemed like a modest request:

"Place one grain of rice on the first square of a chessboard. On the second square, put two grains. On the third square, put four grains. Simply double the grains on each square until the board is full."

The king laughed at such a humble request. After all, how much rice could it be?

By the 8th square, there were 128 grains. By square 16, over 65,000 grains. By square 43, there would be more grains than could fill the entire volume of a palace. And by the final 64th square? More rice than has been produced in human history and enough to cover all of Manhattan in rice 7 kilometers deep.

The king had fallen victim to one of the most powerful forces in mathematics: exponential growth.

While doubling every interval is more extreme, exponential growth often appears with smaller growth rates for each interval. Let's look at a typical example: bacteria growing in a petri dish.

Imagine a bacterial colony that grows by 30% every hour. Let's break down what happens:
Starting amount: 1,000 bacteria
After 1 hour: 1,300 bacteria (up by 300)
After 2 hours: 1,690 bacteria (up by 390)
After 3 hours: 2,197 bacteria (up by 507)
After 4 hours: 2,856 bacteria (up by 659)

Notice something interesting? Even though the growth rate stays at 30%, the actual number of new bacteria gets larger each time. That's because each generation builds on the one before it.

This reveals the three key elements to understand exponential growth:
1. The initial amount - our 1,000 bacteria
2. The growth rate - 30% per hour
3. The time interval - every hour

The mathematical formula looks like this:
f(x) = a(1 + r)^x
Where:
f(x) is the final amount
a is the starting amount
r is the growth rate (0.30 for 30%)
x is the number of time periods

We see similar patterns in many places:
A retirement account earning 7% annual interest
A city's population growing by 2% each year
Social media followers increasing by 15% each month

But here's where people often get tripped up: we're naturally wired to think linearly, not exponentially. If your city grows by 9,000 people this year, you might expect 9,000 new people next year. [show bar chart] But with 15% exponential growth, each year brings significantly more new residents than the last. [show bar chart behind other bar chart and have it follow exponential curve]

This is why exponential growth often catches us by surprise. It's why many people:
Start saving for retirement too late
Underestimate population growth
Get caught off guard by technological change
Misjudge how quickly small changes compound

The next time you hear about something growing exponentially, remember: it's not just about getting bigger - it's about getting bigger faster and faster. Even modest growth rates like 7% or 2% can lead to dramatic changes over time.

While our king's chessboard story shows an extreme example, the real power of exponential growth often lies in these smaller, steady increases that compound over time. An 8% annual return might not sound exciting, but over 30 years, it can turn $100,000 into more than $1,000,000.

In summary, exponential growth is about the surprising power of persistent growth over time. Whether it's your savings, your city's population, or a bacterial colony, small percentage increases can lead to big changes when given enough time to compound.

Thanks for watching. See you next time.