This is what Theoni Pappas writes on page 108 of her Magic of Mathematics:
"A perfect square is a number which can be written as a whole number times itself. For example, 36 is 6 * 6 and 49 is 7 * 7."
Chapter 11 of Eugenia Cheng's Beyond Infinity is "Things That Are Nearly Infinity." In this chapter, she writes about very large numbers. She begins:
"I am almost certain that I will never go to the top of Mount Everest. I will optimistically leave open the possibility of teleportation, but apart from that, I am sure I will never go."
Cheng gives the height of the world's tallest peak as an example of a quantity that is finite, yet seems to be infinite (at least to novice climbers). She also mentions another large number -- the quantity of grains that can fill a chessboard if there is one on the first square, two on the second, four on the next square, eight on the next square, and twice as many of each square as the previous square for the entire board. (This is a classic puzzle.) Cheng performs the following calculation:
1 g = 50 grains of rice
1 bowl = 100 g = 5000 grains
1 person = 4 bowls of rice = 20,000 grains
the world = 7 billion people = 140,000,000,000,000 grains
a year = about 500 days = 70,000,000,000,000,000 grains
The total of grains on the chessboard is 2^64-1, which is about three orders of magnitude larger. So the grain could feed the world's population for a millennium.
By the way, this reminds me of a STEM project I did with my eighth graders. They were supposed to estimate 100,000 grains of rice in order to learn about large numbers such as 100,000 and one million. Based on Cheng's calculations, we see that this would be about 20 bowls of rice, or enough to feed five people in one day.
Cheng moves from bowls of rice to puff pastry. I know that she is an excellent cook, and so she often writes about recipes in her math books. I specifically recall her mentioning puff pastry -- and how difficult it is to make -- in her first book, How to Bake Pi.
"Puff pastry relies on the same principle, that repeated multiplication makes things grow extremely fast. Puff pastry has an apparently miraculous number of tiny layers in it, and they are created by folding the dough in three just six times."
A quick calculation reveals that a single puff pastry would have 3^6 = 729 layers! So we can easily see why puff pastry is so tough to prepare.
Cheng finally provides an example that isn't edible -- the iPod Shuffle. She writes:
"I remember when the iPod Shuffle first came out, I saw this big advertisement on the Tube with the slogan '240 songs. A million different ways.'"
As it turns out, there are many, many more than a million different permutations of 240 songs that we can play. She draws diagrams to demonstrate that with three songs, there are six different orders in which they can be played. With four songs, she calculates:
-- 4 possibilities for the first song,
-- 3 possibilities for the second song,
-- 2 possibilities for the third song, and
-- 1 possibility for the last song.
So Cheng calculates 4 * 3 * 2 * 1, so 24 different orders. She explains that this can also be written as 4!, for four factorial. In general, n factorial is:
n * (n - 1) * (n - 2) * ... * 4 * 3 * 2 * 1,
She writes we can also define n! by induction, which is how a computer programmer would most likely do it:
-- 1! = 1
-- (n + 1)! = (n + 1) * n!
Cheng uses this definition to provide the first few factorials:
1! = 1
2! = 2
3! = 3 * 2 = 6
4! = 4 * 6 = 24
5! = 5 * 24 = 120
6! = 6 * 120 = 720
7! = 7 * 720 = 5040
8! = 8 * 5040 = 40,320
9! = 9 * 40,320 = 362,880
10! = 10 * 362,880 = 3,628,800
She points out that with a mere ten songs, we already have millions of possible orders in which to play them. Next, we continue with the highest factorial found on a spreadsheet:
17! = 355,687,428,096,000.
Here is the highest factorial found on Cheng's phone:
18! = 6,402,373,705,728,000.
Beyond this, both her phone and spreadsheet give approximations for large factorials:
103! = 9.9 * 10^163
170! = 7.3 * 10^306
Cheng mentions a webpage that gives the exact value of 200! I won't list all the digits here, but I will mention something Cheng notices about the digits:
"You might be wondering if it's a weird fluke that there are so many 0's on the end of that number: 49 of them in fact."
As it turns out, it's not a fluke -- there are so many zeros because we're multiplying many multiples of two and five together. She points out that since there are more even numbers than multiples of five, it's the latter that controls how many zeros there are.
One thing I find interesting is that in base 10 it's clear that it's the multiples of five that control the zeros, but in a handful of other bases it's not as clear-cut. One such base is the ever popular dozenal, or base 12. The base factors as 2^2 * 3, and so neither two nor three dominates. We must keep track of both to determine the number of zeros in the dozenal factorials.
Cheng moves on to comparing the growth rate of various functions. She writes:
"Children grow very quickly when they're little, and it seems even quicker because they're so small."
Likewise, an exponential function may appear to grow faster than a factorial function at first, but in the long run, the factorial always wins out. She reminds us:
"We are only thinking about positive powers of n because negative powers don't actually grow at all -- they get smaller as n gets bigger."
Her final example is the inverse of an exponential function -- a logarithmic function. As exponential functions grow quickly, logarithmic functions grow slowly. She writes:
"It's possible to grow so slowly that it looks like you're not growing at all, and yet you're actually still growing."
Here is the worksheet on tangents to circles, with a review question on square roots: