## Exponential Growth is REALLY Powerful!

Internet traffic, the Richter scale for measuring earthquakes, the spread of disease, our response to brightness, and Intel co-founder Gordon Moore’s Law all involve exponential growth. Exponential growth even applies to noodle making, as I’ll detail below!

With all these applications, sure-footed intuition about exponential growth is extremely useful. Yet I’ve met many students who have yet to fully grasp the concept. Intellectually, they know that exponential growth beats polynomial growth of any degree. But knowing this fact is far removed from feeling its power.

For example, the function $2^x$ greatly outpaces the function $x^2$, and yet many grossly underestimate even the growth of $x^2$.

So for starters, here’s an exercise that helps to convey the rapid growth of $x^2$: Graph $y = x^2$ with, say, 1/4″ representing one unit over the range of $x$ values from 0 to 30. Your $x$-axis will then be about 7.5″ long and will fit within the width of a piece of standard typing paper.

If you haven’t done the exercise but plan to, don’t read further until you do because there are spoilers.

Spoiler Alert! If you did the exercise, then you discovered that you needed to make your paper almost nineteen feet long to contain the graph! That’s about the height of a two-story home.

Now consider that $2^x$ grows MUCH faster! If you graph $y = 2^x$ instead of $y = x^2$ over the same range of values with 1/4″ representing one unit, standard typing paper will again suffice for the width, but how high would your paper have to be to fully contain the graph? Take your best guess now. I’ll put the answer at the end of this post.

Last spring at the Girls’ Angle club, Jennifer Che of tiny urban kitchen demonstrated the rapidity of exponential growth by making Chinese hand-pulled noodles, also known as lamian. Starting with a handful of specially prepared dough, she stretched it and doubled it over 8 times in less than a minute. The blob of dough became one long noodle doubled over multiple times. When the strands were cut out, the result was $2^8$ noodles – that’s 256 strands! At that rate, she could have made 1,024 strands (which represents 10 doublings; $2^{10} = 1024$) of noodles in just about one minute. Now here’s a question: If, instead of stopping at 1,024 strands, she kept on going at the same pulling rate, how much longer would it take her to make over one million strands? An hour? A day? A year? What do you think?

Exponential functions grow fast! To go from 1,024 to over a million strands is just 10 more doublings. So if she did the first 10 doublings in 1 minute, the next 10 would take another minute. So it would only take 1 minute more to reach over a million strands! (Another way to reckon this is to notice that $1024^2$ is just over a million so the time it takes to make just over a million strands is twice the time it takes to make 1024 strands.)

Actually carrying out 20 doublings is a challenge. It’s hard to do without breaking the noodle midway through the process. However, in 2009, Chinese master chef Li Enhai succeeded. In less than 2 minutes, he set the world record for the longest noodle: 2852 kilometers! The noodle would stretch all the way from Boston to San Antonio, Texas.

Here’s a math question: With each doubling, how does the diameter of the noodle change?

Finally, the answer to the question about the graph of $y = 2^x$: The paper would have to be 268,435,456 inches (over 4,000 miles) high! That height takes you into outer space, more than ten times further than the International Space Station.