Steps in a mile
If I were to start a new exercise program where I double my previous day’s steps each new day, on which day of the program will I first walk a full mile? I will start with 1 step on Day 1 and all of my steps are 2 feet long.
Bonus: On that day, how many steps beyond a mile will I go? What fraction of a mile is this? How many steps in a mile?
Day 1 is 1 step, Day 2 is 2 steps, Day 3 is 4 steps, etc. How many of these steps equals 1 mile? At two feet per step, we can divide the length of a mile by this distance to get the number of steps needed. 5280/2 = 2640 steps. By doubling the number of steps each day, when do you have more than 2640? Multiplying each day’s steps by 2 and keep track of how many times you do this until you have more than 2640.
Day 1 |
Day 2 |
Day 3 |
Day 4 |
Day 5 |
Day 6 |
Day 7 |
Day 8 |
Day 9 |
Day 10 |
Day 11 |
Day 12 |
Day 13 |
1 |
2 |
4 |
8 |
16 |
32 |
64 |
128 |
256 |
512 |
1024 |
2048 |
4096 |
On Day 13, I walk 4096 steps. This is 1456 steps beyond a mile (4096 – 2640 = 1456) and is 55.15% of the second mile (1456/2640 = .55151515).
Although learning how many steps in a mile is cool, this crazy exercise program may not be practical – but it does help show how quickly numbers can grow when there is an exponent (greater than 1) involved. Doubling every day is the same as saying multiply by 2, so the 2 x 2 x 2 x 2 x 2 … until you reach 13 2s. This is also written 2^{13} and every additional increase of the exponent results in much bigger numbers. The first few days don’t change so much, but imagine if you followed this program for an entire month. 2^{30} = 1073741824 steps, multiply by 2 = 2147483648 feet, divide by 5280 to get 406720.39 miles, or more than 16 times around the earth!