By PATRICK HONNER
Mathematics
Teaching ideas based on New York Times content.
Materials | Computers with Internet access
Warm-Up | Invite students to consider the following two games. In “Name the Biggest Integer,” players take turns choosing integers (whole numbers); the winner is the person who names the largest number. In “Closest to 0,” players take turns choosing rational numbers (which include both integers and fractions, like 3/4 and 7/8); the winner is the player who names the number closest to zero.
Ask students to articulate a strategy for each game, and to compare and contrast the two. It should be clear that the strategy in both games is to go last. But what happens if no one goes last? What if you never stop playing the game? In short, what happens if you infinitely play many rounds?
When comparing and contrasting the two games, discuss the role of infinity in each. In both games, the fact that the player going last can always win is a consequence of the infinitude of the sets of numbers. However, in each game, a different kind of infinity is used: in “Name the Biggest Integer,” we rely on the fact that we can choose numbers to be infinitely large; in “Closest to 0,” we rely on the fact that we can choose numbers to be infinitely small.
Use these simple games and the ensuing discussions to get students thinking about the different kinds of infinity. Then have them read the related article to get them thinking deeper about this important concept.
Related | In “The Life of Pi, and Other Infinities,” Natalie Angier writes:
The popular notion of infinity may be of a monolithic totality, the ultimate, unbounded big tent that goes on forever and subsumes everything in its path — time, the cosmos, your complete collection of old Playbills. Yet in the ever-evolving view of scientists, philosophers and other scholars, there really is no single, implacable entity called infinity.Background Vocabulary: Read the entire article with your class, then answer the questions below. You may wish to introduce students to the following words or concepts before reading: finitude, monolithic, implacable, doppelgänger and subjugate.
Instead, there are infinities, multiplicities of the limit-free that come in a vast variety of shapes, sizes, purposes and charms. Some are tailored for mathematics, some for cosmology, others for theology; some are of such recent vintage their fontanels still feel soft. There are flat infinities, hunchback infinities, bubbling infinities, hyperboloid infinities. There are infinitely large sets of one kind of number, and even bigger, infinitely large sets of another kind of number.
Questions | For discussion and reading comprehension:
- What are some examples of infinite quantities given by the author?
- How could one infinity be larger than another? What example does the author use?
- What does the mathematician Steven Strogatz mean when he says “infinity is your friend”?
- What does the mathematician Ian Stewart mean when he says “You can’t be freewheeling in your use of infinity”?
- What is a doppelgänger, and why does the existence of an infinite universe suggest yours might exist?
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Activity | Have students, either individually or in small groups, research the role infinity has played throughout mathematical history by studying both interesting applications as well as paradoxes that result from accepting the existence of the infinite.
Have each student or group choose a topic; some suggestions are offered below. Have them conduct research online to prepare and ask them to give a short presentation on their use of infinity. After each presentation, facilitate a group discussion about the mathematics and logic behind each topic, and encourage students to argue for, or against, the legitimacy of each idea. Just be careful — these debates can go on forever!
Cardinality of Number Sets
The cardinality of a set is the number of elements it contains. The sets of integers (… -3, -2, -1, 0, 1, 2, 3, and so on), rational numbers (all numbers that can be expressed as the ratio of two integers, like 3/4, -5/18, 2/1, etc.), and real numbers (any number that can be represented as a [possibly infinite] decimal expansion) all have infinite cardinality. But are they the same infinity? Surprisingly, the answer is no!
Read Mr. Strogatz’s excellent description of these different kinds of infinity in his NYT Opinionator article on the infamous Hilbert Hotel to get a sense of why, and how, these infinite sets of numbers have different sizes.
Zeno’s Paradox
Imagine you were standing one meter away from a wall. If you wanted to get to the wall, you wouldn’t expect it to take very long, unless you think of it this way: first, imagine getting halfway there. Then, once you were halfway there, imagine getting halfway from your new spot. And so on, ad infinitum. Before you know it, you’ve got infinitely many moves to make. How long do you think that will take?
This is a version of a paradox of Zeno that has confounded philosophers for thousands of years.
Interesting Infinite Sums
What happens when you add up infinitely many numbers? That depends on which numbers you are adding. Usually if you try to add up infinitely many numbers, the sum will just get bigger and bigger; but if you add up the right kinds of numbers, sometimes you can add up infinitely many numbers and get something finite.
Here’s a visual example of how adding up fractions in a geometric progression can give you 1. And in his piece “Take it to the Limit,” Mr. Strogatz uses an infinite sum to derive the formula for the area of a circle.
0.999999….. = 1Everyone knows that 0.9 is less than 1, and that 0.99 is less than 1, and that 0.999 is also less than 1. But what if you never stop writing 9s? Amazingly, the number you’d get by writing a zero, a decimal point, and infinitely many nines is actually equal to 1!
If you don’t believe this, ask yourself this question: what does one-third plus two-thirds equal? Or check out this explanation at the Math Forum. This one always stirs up an argument, so be prepared for an endless debate.
Infinitude of Primes
Euclid’s famous proof of the infinitude of prime numbers, given over 2,000 years ago, offers an accessible entry into infinite reasoning: if you claim that there are only 10 prime numbers, Euclid will show you an 11th prime; if you claim there are only a million primes, Euclid will show you prime number 1,000,001! Knowing that there are infinitely many primes is especially remarkable, given how hard it is in general to identify large prime numbers.
Going Further | Have students find examples of infinity in their own worlds. Have them make lists of quantities that may or may not be infinite and present their findings to the class. Have students debate whether or not the quantities are finite or infinite, and if infinite, which kind of infinity. Here are some simple examples to get students thinking: the number of planets in the universe; the number of human beings who have ever lived; the number of human beings who will ever live; the number of words in the English language; the number of sentences in the English language.
Read about the crucial role infinity plays in calculus, one of the most important areas of mathematics. The idea of the infinitesimal — an infinitely small quantity — is essential in the mathematical operation of integration, which Mr. Strogatz describes beautifully in “It Slices, it Dices.”
Check out how the mathematician and professionally trained mime Tim Chartier uses movement to explore and express the idea of infinity. Watch the video with students and discuss this portrayal of the infinite. Have students create their own skits or mimes to convey how they see infinity.
Common Core Mathematics Standards, 6-12:
Standards
- Apply and extend previous understanding of numbers to the system of rational numbers
- Know that there are numbers that are not rational, and approximate them by rational numbers.
- Use properties of rational and irrational numbers
- Reason abstractly and quantitatively
- Construct viable arguments and critique the reasoning of others
- Look for and express regularity in repeated reasoning
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