Pondering: The Reals from the Rationals

Bablyonian root approximation

Real number theory was perhaps the most significant course I took in college. There were three of us enrolled my last semester, a number small enough that that it could have been cancelled. I’m indebted to the late Prof. Jack Lamoreaux and the math department that it wasn’t. Each student would spend 10-20 minutes at the board pursuing Dedekind’s question, Was sind und was sollen die Zahlen?. Prof. Lamoreaux would guide us and involve the two sitting students with a minimal touch. For the last five minutes of class, he would lay out a direction for the next time we would meet. There were no exams, as we exposed continually all that we knew and all that we lacked knowing.

There were no assigned texts. We started with Peano’s axioms, a few simple statements that define the set of natural numbers. Using logic and set theory, we showed that the set defined by Peano’s axioms have various of the properties we think of the natural numbers having. Rules of arithmetic and comparison can be described that work like those we’ve known since childhood.

Having the set of natural numbers at our disposal, we constructed integers as classes of ordered pairs of natural numbers. We did something similar for the rational numbers, and proved, for instance, that the rational numbers are an infinite, countable set.

The next step was to define the set of real numbers based on the framework we had built up from the natural numbers. This step was qualitatively different from those that produced integers and rational numbers. My main tool for this task was the Cauchy sequence. A Cauchy sequence is an ordered set of numbers that converges. Defining where it converges is the interesting part because it may be a value of a kind that the sequence elements are not. A sequence of rational numbers can converge around something that is not a rational number, so how can one describe where they are going? Remember, the task is to define the real numbers, so their existence can’t be used until we’ve defined what they are. The way the problem is handled with a Cauchy sequence is that for any distance, no matter how small, and often labeled ε, a number N can be found such that any two elements in the sequence beyond the Nth are closer than distance epsilon apart. Classes of converging sequences of rational numbers can be used as a definition of the set of real numbers.

An example is √2, that number which when multiplied by itself equals 2. This number is not a rational number. Sequences of rational numbers can be defined, however, that come as close as desired to √2. One ancient Babylonian left behind 1 + 24/60 + 51/602 + 10/603 as a rational approximation to √2, a value that is good to 1 part in 2 million. The Babylonian method of approximating square roots converges quadratically, so the next approximation in a sequence built by that method would be good to 1 part in 11 trillion. Even if we went past the nineth number in that sequence where the approximations all match one another for the first thousand digits, or past the twentieth where they match for the first million digits, nowhere in a sequence of rational numbers could √2 be found.

None of the elements of the sequence is a real number; the sequence is the real number.

“For my thoughts are not your thoughts, neither are your ways my ways, saith the Lord. For as the heavens are higher than the earth, so are my ways higher than your ways, and my thoughts than your thoughts.”-Isaiah

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About John Mansfield

Mansfield in the desertA third-generation southern Nevadan, I have lived in exile most of my life in such places as Los Alamos, Baltimore, Los Angeles, the western suburbs of Detroit, and currently the northern suburbs of Washington, D.C. I work as a fluid dynamics engineer. I was baptized at age twelve in the font of the Las Vegas Nevada Central Stake Center, and on my nineteenth birthday I received the endowment in the St. George Temple. I served as a missionary mostly in the Patagonia of Argentina from 1985 to 1987. My true calling in the Church seems to be working with Cub Scouts, whom I have served in different capacities in four states most years since 1992. (My oldest boy turned eight in 2004.) I also currently teach Sunday School to the thirteen-year-olds. I hold degrees from two universities named for men who died in the 1870s, the Brigham Young University and the Johns Hopkins University. My wife is Elizabeth Pack Mansfield, who comes from New Mexico's north central mountains and studied molecular biology at the same two schools I attended. We have four sons, whose care and admonition, along with care of my aged father, require much of Elizabeth's time. She currently serves the Church as Mia-Maid advisor, ward music chairman, and choir director, and plays violin whenever she can. One day, I would like to make shoes.

16 thoughts on “Pondering: The Reals from the Rationals

  1. Thanks. This is cool. I hope you post more spiritual insights from math. By the way, do you have a link to how the Babylonians approximated square roots?

  2. Brother Jonathan, here is a description of the Babylonian method, also known as Heron’s method. The approximations to √a are calculated iteratively as xN+1 = (xN + a/xN)/2. This is the same formula that Newton’s method would produce to find roots of the equation x² – 2 = 0 x² – a = 0. [I wrote at first the specific equation where a = 2.]

  3. Small correction to the last sentence you wrote — “None of the elements of the sequence is the real number….” because all of those elemets are real numbers.

    Aside from that, interesting article.

  4. I know what you’re getting at, Brother Blain. I hesitated over that sentence when I wrote the post, but I stand by it. You are pointing out that the rationals are a subset of the reals, and so any element in a sequence of rationals is also a real. That is true when you have a set of reals and identify the subset within that set that is the rationals. It is not true if you have constructed the set of rational numbers that does not have the properties of the real numbers. Your set of rational numbers (a subset of the real numbers) can be mapped to my set of rational numbers (classes of ordered pairs of integers, i.e. ratios), but these sets differ in that your rationals are also reals, and mine are not. My set is the kind needed to construct the reals, since we can’t assume the thing we are defining.

  5. Small quibble — wouldn’t it be the series that is the real number, not the sequence?

  6. John,

    When I took the class from Dr. Lamoreaux (spring 1974), it was called “Number Theory.” There were three or four of us in the class. No one wore shoes; we walked around in stocking feet in the class to write our proofs on one of the blackboards; sometimes two of us at the blackboard at a time. It was completely informal. It was one of the most stimulating and fun classes I ever took. A similarly informal (though shoes were required), stimulating and fun class was Dr. Yearout’s “Set Theory.”

    I did not realize Dr. Lamoreaux had passed away. What a wonderful man and teacher.

  7. I believe I had Calc 2 from Dr. Lamoreaux, if memory serves he was enamored of delta-epsilon proofs. Ugh. Other than that complaint he seemed to be a pretty good guy, I was unaware of his passing as well.

    Did you have to prove the existence of imaginary numbers? I find them extremely useful, at least as much so as reals.

  8. Number theory was the one class I wanted to take but didn’t. I’ve read a bit on it but always kind of wished I’d taken it instead of graph theory or numeric methods. Graph theory was fun, but not quite as exciting as one would have wished. (Although finding the relationship between matrices and graphs was useful) Numeric methods was ultimately kind of a waste, all things considered. I’ve read a bit on Number theory, but it’s one of those classes I want to find a good textbook and work through once the chocolate business gets going a bit more. That and review my Misner, Thorne and Wheeler.

  9. Doug D, imaginary numbers didn’t come up in that class, but I agree with you about their usefulness. As someone put it, if you want to understand a function, look at it in the complex plane. In the 18th Century, when complex numbers were being developed, and being dismissed by some, such as Descartes, as “imaginary numbers,” the status of negative numbers was in question. Chuquet, the first European to use negative numbers, called them “absurd numbers,” negative roots where considered spurious and meaningless, and it wasn’t clear what the product of -1 and -1 should be. It appears that work with imaginary numbers did a lot to solidfy the concept of negative numbers.

    DavidH, I am happy to share a good memory with you. I took the class fall of 1991. I had forgotten the shoelessness.

    Bryce, it’s the class of sequences that is the real number. Some of the sequences are things like continued fractions or recursion formulas that are not expressible as a series. Even with the series formulations, it’s the sequence of finite series that is part of the class defining a real number. (There are also other constructions besides Cauchy sequences that can be used to define the reals.)

    Here is a series: aN = 1 – Sum(k=1 to N) of 22-3k(2k-3)!/k!. Working in the real numbers, the limit of aN as N goes to infinity is √2. I think this is the point you are making; the limit of this series of rational terms is an irrational number. Working only with the arithmetic of rational numbers, however, the limit of aN doesn’t exist. The set of reals is complete, meaning that any Cauchy sequence of reals converges to a real number. The set of rational numbers is not complete since a Cauchy sequence of rationals may not converge to an element of that set.

  10. Sorry to be so late to the discussion (again), I enjoyed the post. I’m curious if you have particular uses of imaginary numbers in mind (comments #7 & #9). I don’t use them very much in my research, at least directly, and am curious what some useful applications are….

  11. Reading up about complex numbers here, I understand better why imaginary numbers are useful in engineering and physics (where there are often regular periods to analyze with trig functions), and a bit less so in economics and finance. That being said, Laplace transforms and Fourrier transforms can sometimes be pretty handy when analyzing time series data, which seem to rely heavily on imaginary numbers….

  12. Robert,

    Aside from the uses you pointed out, they are used extensively in RF and microwave engineering (radio frequency); where they are used to characterize scattering parameters. S-parameters are used to characterize the reflected power and transmitted power for a device in a high frequency application. Many useful things can be derived from this data for different devices. For example, s-parameters for resistors can be used to show resistance versus frequency and parasitic capacitance versus frequency for a device. For a capacitor it will demonstrate the capacitance versus frequency as well as the quality factor (Q) versus frequency. These all help designers to better understand the device’s performance at high frequency for their applications. This is how I use them daily.

  13. To add to the list, I use complex numbers for hydrodynamic potential flow around 2-D sections of thin bodies of revolution, and if they’re not bodies of revolution they get conformally mapped to another complex plane where they are. There are similar problems with other systems (electrical, magnetic, thermal) that involve a flow vector along the divergence-free gradient of a scalar field. Complex numbers are useful when the real and imaginary parts are not independent, but are two entwined aspects of a single quantity.

  14. “None of the elements of the sequence is a real number; the sequence is the real number.”

    Whoa! I seem to remember this from one of my books, but I’ve never heard it stated this way. It makes perfect sense.

  15. Well, when I say “perfect sense” really mean “I might sort of kinda understand, but if I do, I really like it.”

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