Our decimal number system is so wonderful that it’s often taken for granted. (If you doubt me, try multiplying and or finding an tip on a restaurant bill using only Roman numerals.)
However, there’s one little quirk about our numbering system that some students find quite unsettling:
If a number has a terminating decimal representation, then the same number also has a second different terminating decimal representation. (However, a number that does not have a terminating decimal representation does not have a second representation.)
Stated another way, a decimal representation corresponds to a unique real number. However, a real number may not have a unique decimal representation.
Some (perhaps many) students find such equalities to be unsettling at first glance, and for good reason. They’d prefer to think that there is a one-to-one correspondence to the set of real numbers and the set of decimal representations. Stated more simply, students are conditioned to think that if two number look different (like and ), then they ought to be different.
However, there’s a subtle difference between a number and a numerical representation. The number is defined to be the multiplicative identity in our system of arithmetic. However, this number has two different representations in our numbering system: and . (Not to mention its representation in the numbering systems of the ancient Romans, Babylonians, Mayans, etc.)
As usual, let be the set of real numbers from to (inclusive), and let be the set of decimal representations of the form . Then there’s clearly a function , defined by
If I want to give my students a headache, I’ll ask, “In Calculus II, you saw that some series converge and some series diverge. So what guarantee do we have that this series actually converges?” (The convergence of the right series can be verified using the Direct Comparsion Test, the fact that , and the formula for an infinite geometric series.)
In the language of mathematics: Using the completeness axiom, it can be proven (though no student psychologically doubts this) that maps onto . In other words, every decimal representation corresponds to a real number, and every real number has a decimal representation. However, the function is a surjection but not a bijection. In other words, a real number may have more than one decimal representation.
This is a big conceptual barrier for some students — even really bright students — to overcome. They’re not used to thinking that two different decimal expansions can actually represent the same number.
The two most commonly shown equal but different decimal representations are . Other examples are
In this series, I will discuss some ways of convincing students that . That said, I have met a few math majors within a semester of graduating — that is, they weren’t dummies — who could recite all of these ways and were perhaps logically convinced but remained psychologically unconvinced. The idea that two different decimal representations could mean the same number just remained too high of a conceptual barrier for them to hurdle.
Method #1. This first technique is accessible to any algebra or pre-algebra student who’s comfortable assigning a variable to a number. We convert the decimal representation to a fraction using something out of the patented Bag of Tricks. If students aren’t comfortable with the first couple of steps (as in, “How would I have thought to do that myself?”), I tell my usual tongue-in-cheek story: Socrates gave the Bag of Tricks to Plato, Plato gave it to Aristotle, it passed down the generations, my teacher taught the Bag of Tricks to me, and I teach it to my students.
Let . Multiply by , and subtract: