3.4 Introduction and Key Insights

Let's return to your aunt's coffee shop. She was already running a steady business, but her popularity really took off when she introduced her delicious breakfast sandwiches! One happy person posted a favorable review online, which caused two more people to stop by and try her food. Each of these two customers told two more people about their delicious breakfast. All four of these new people enjoyed their experience so much that they each recommended the shop to two more individuals.

And now we have questions! Just how many new customers will arrive in the next wave? And how many new customers will those new customers cause to arrive? At how many "steps" will this process repeat itself before the customers reach the 256 seating capacity? (Take a minute to do the math!)

We can organize this coffee shop phenomenon in several ways. For example, we can notate this as a chain of popularity, with symbols 1→ 2→ 4, meaning one review led to two new happy customers, which itself led to four new happy customers, and so on. The chain continues in the following pattern:

1→ 2→ 4→ 8→16

To determine how many new customers will be stopping by your aunt's coffee shop, we simply multiply the previous number by two. If this chain of popularity keeps growing, we will arrive at a seating capacity of 256 in eight steps.

1→ 2→ 4→ 8→ 16→ 32 → 64→ 128 → 256

For the first step in this pattern (represented by the first arrow between one and two), we multiply by 2 one time. To get the number after the second arrow, we multiply by 2 twice, resulting in $2 \times 2=4$. The third, fourth, and fifth steps involve multiplying by 2 three, four, and five times. During the last step, we arrive at 256: we have multiplied two by itself eight times. Rather than write out $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2=256$, we can use exponential notation to organize the symbols in a shorter, more informative way. We can write

$2^{8}=256$

The teeny, tiny eight in the superscript tells us how many times the two is multiplied against itself, while the number two itself tells us which number we are repeatedly multiplying. That teeny, tiny eight is called the exponent (and the 2 is referred to as the base). Check to see if the following equations also make sense:

$3^{2}=9$, $0^{5}=0$, $1^{8}=1$, $10^{3}=1000$