So you think you know how to multiply decimals?

Posted by: Gary Ernest Davis on: February 12, 2010

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In school students are taught how to add , subtract, and multiply, – but usually not divide – decimal numbers.

There is a big catch however. The decimal numbers are not infinite decimals, such as we would obtain from $\frac{1}{3}=0.333333\ldots$ or $\frac{1}{7}=0.142857142857142857\ldots$

The decimal numbers dealt with in school are all finite decimal numbers, which is to say they are decimal numbers that can be written as fractions of the form $\frac{m}{2^p\times 5^q}$.

Arithmetical operations on these finite decimals are no more tricky than they are on whole numbers.

D. Fowler, in his American Mathematical monthly paper, entitled “Dedekind’s Theorem: $\sqrt{2}\times \sqrt{3}=\sqrt{6}$“, 1992, writes:

Many mathematicians have a touching and naive belief that arithmetical operations on decimals pose no problems; or they pretend to believe this, as in some circumstances the most scrupulously honest among us may sometimes some pretend to believe in Father ChristmasÂ  …; or perhaps they have never considered the question to be problematic.”

It seems that middle and high school teachers have generally not thought about the problems involved in arithmetical operations on decimal numbers, although a middle school teacher in one of my classes did say she knew how to do it and that there was no problem.Â  She could not demonstrate just how she, or we, would go about multiplying two decimals, for example.

A colleague – a Ph.D. Â mathematician – also expressed a view that there is no problem.

Another young woman in one of my classes got the issue right away: infinite carrying is involved.

To see the difficulty, try to work out the first digit after the decimal point in $1.222222\ldots \times 0.818181\ldots$

Sneakily converting to fractions …

We could, of course, convert both the numbers in this multiplication into fractions:

Let $a=0.222222\ldots$ so that $10\times a=2.222222\ldots = 2+a$ so $a=\frac{2}{9}$.

Therefore $1.222222\ldots=1\frac{2}{9}=\frac{11}{9}$

In the same way, if we let $b= 0.818181\ldots$ then $100\times b=81+b$ so $b=\frac{81}{99}=\frac{9}{11}$

So now, as fractions, we see that

$1.222222\ldots \times 0.818181\ldots =\frac{11}{9}\times\frac{9}{11}=1 = 1.000000\ldots=0.999999\ldots$

The general problem of infinite carrying

But what if we did not convert the decimals to fractions?

How would you carry out $1.222222\ldots \times 0.818181\ldots$ by a general procedure for multiplying decimals?

Fowler was right, and most mathematicians, and mathematics teachers, are wrong: they do not know how to multiply (or add) infinite decimals ( such as $\frac{11}{9}, \frac{9}{11}, \frac{1}{3}, \frac{1}{7}$ written in decimal form).

It turns out there is a way to multiply infinite decimals, but it is a pretty intimidating procedure, described in the paper:

F. Faltin, N. Metropolis, B. Ross, and G.-C. Rota, The real numbers as a wreath product, Advances in Math. 16 (1975) 278-304.

What are we doing pretending to students – middle school, high school undergraduate, that we, and they, know how to add and multiply infinite decimals when we almost certainly do not?

This basic ignorance, or dishonesty has worried me for a long while.

Shouldn’t we follow Richard Feynman‘s example, and simply say: “I don’t know.”

Wouldn’t our students respect us more, and trust us better, if we showed this basic honesty?

Do we, as teachers, have to pretend to know everything even when we know we don’t?

16 Responses to "So you think you know how to multiply decimals?"

I haven’t noticed people pretending that this is something they know how to do. I’ve only been taught multiplying finite decimals. Sometimes in the division algorithm you can see a repeating pattern, but that’s all I can remember being discussed.

I think it’s because we’re so used to doing such multiplications and divisions with a calculator or computer that we don’t really think about doing them on paper. Indeed, when using a calculator, we wouldn’t encounter these problems (although we don’t deal with infinite precision).

That’s the point really. Arithmetic these days is finite arithmetic – has to be on calculator. But no one discusses that we are only looking at rational numbers of very specific form: $\frac{m}{2^a\times 5^b}$

Finite precision arithmetic is sufficient for the vast majority of circumstances. Sure, there are instances where you might want to do infinite precision multiplication with decimals, but these are usually well beyond the scope of elementary, middle, or high school mathematics.

luckytoilet, so you’re saying that mathematics can be reduced to working only with rational numbers of the form $\frac{m}{2^a\times 5^b}$.

Where does that leave $\frac{1}{3}, \sqrt{2}, \pi$ all of which occur in school mathematics?

And if that’s the case, why bother with decimals at all? Just use rational numbers of the form $\frac{m}{2^a\times 5^b}$. To see this just multiply a finite decimal by an appropriate power of 10.

Okay, maybe this is what the Faltin et al. paper describes, but this is how I would teach it: http://screencast.com/t/ODU0YzY3

To sum it up, use the same procedure we use in Calc II for multiplying two infinite series … also, can be demonstrated pretty easily with an Excel Spreadsheet, see: http://screencast.com/t/MGE0NmZhN

The links didn’t work in Opera, but did in Safari.

No, that procedure is not an algrothm for multiplying decimals. The problems is this:
given two decimal strings $0.a_1a_2a_3\ldots$ and $0.b_1b_2b_3\ldots$ what is a finitary algorithm for determining $0.c_1c_2c_3\ldots$ where $0.a_1a_2a_3\dots \times 0.b_1b_2b_3\ldots =0.c_1c_2c_3\ldots$?
In other words, how is multiplication of decimal strings defined?
The procedure oulined does not define multiplication because it does not tell us how to define the $c_i$ in terms of the $a_i, b_i$, only that we should proceed in some unspecified manner.

Multiplication of decimals cannot be carried out simply by treating the decimals as power series in $x=\frac{1}{10}$ and multiplying. The reason is that the coefficients have to be digits = 0 through 9 – and not general numbers.

Basically the Faltin et al method devises a clever way to “clear” the answer you would get by multiplying as power series. They do it in base 2 for simplicity, but it works in any base. What they do is to make the definiton of decimal string multiplication algorithmic, and, to the best of my knowledge most mathematicians and mathematics teachers are not aware of the problem, let alone an answer.

Maria’s approach seems reasonable to me as long as you only need a specific number of accurate digits in your answer (and therefore are using only a finite power series).

Given the above condition, I don’t understand why a coefficient of one term or another in the power series would need to be a general number if we accept that only a finite number of terms from the power series will be used, and that the answer will be accurate only to a specific number of digits.

However, this would seem to change your original question (from what is “the” answer, to what is a “good enough” answer). So, I see the above discussion (but not your original posting) as being partially an “applied” vs “pure” discussion… and I mostly roam the applied side.

So, I heartily agree that we don’t “know all”, and also concur that we should all strive not to leave our students with the impression that we know more than we do.

http://mathmaine.wordpress.com

Whit,

” if we accept that only a finite number of terms from the power series will be used” means that we are dealing with only finite decimals. Therefore only with numbers of the form $\frac{m}{2^p\times5^q}$.

Such numbers are closed with respect to addition – and subtraction – and multiplication, but not division.

This includes the numbers $\frac{1}{3}, \frac{1}{7}, \frac{1}{11}\ldots$ and generally $\frac{m}{p}$ where $p\neq 2, 5$ is a prime not dividing $m$; most algebraic numbers, including $\sqrt{p}$ where $p$ is prime; all transcendental numbers, including $\pi$. In other words, most numbers that are fundamental to mathematics, and science.

The infinite decimal representation of real numbers was dreamed up by Weierstrass to exhibit a concrete example of what is essentially the unique complete ordered field. That is what constitutes real numbers.

If only allow finite decimals we throw away most of our number system, and the conceptual basis of science, as I understand it, becomes impossible: how can we express the wave equation, for example, using only finite decimals?

Wow, Gary! Thanks for that. It’s exactly these unexamined assumptions which allow us to sometimes step into errors of logic.

I never knew about the problem,and always assumed we’d convert to rational, or simply not attempt the operation. There are probably products that are simply not computable.

It reminds me of functions that are not graphable, yet continuous. Math never ceases to surprise me!

Mike, this is a VERY deep question (both of them). Greg Chaitin has written extensively about the essential algorithimc randomness of most random numbers. In practice this means these numbers are essentially not compressible – cannot be described in essentially shorter terms. So the numbers on which we can do arithmetic are a very limited set indeed.

[…] So you think you know how to multiply decimals? Finite and infinite decimals In school students are taught how to add , subtract, and multiply, â€“ but usually not […] […]

Concept versus reality is the issue I see here. I agree that in concept, we require the set of all Real Numbers. Solutions to conceptual problems will often be expressed as quantities that cannot be expressed as a Rational Number.

The challenge comes when we try to link the concepts to reality. Experimental measurements are always going to be limited in their precision and accuracy, so the conceptual answer is only needed to a certain number of decimal places in order to test whether the theory and experiment are mutually consistent. If they are, that sets the stage for the next experimental setup, which seeks even greater precision and accuracy.

I am not arguing that we are throwing away “most of our number system”, but rather that we are incapable of using it “practically”. We need all of the Reals in order for our conceptual model to hold water. However, in life our ability to perceive and measure beyond a certain number of decimal places is limited at any point in time.

Isn’t this the same sort of dilemma we run into by stating that you will bump into the wall, even though mathematically you never “reach it”, by travelling half the distance to the wall with each step? The practical is not capable of matching the theoretical, because we are incapable of acting beyond a certain degree of precision and/or accuracy at the moment.

Whit,

you raise some intruiging points, as usual!

The only thing I would say at this juncture is that “reality” is a very contentious issue.

Buddhists, for example, have some well worked out views on reality that differ from what many people see as reality.

Form a mathematical perspective the most exreme point of view is that expressed bt Alain Connes who states that we know mathematical reality better than we know physical reality.

I am somewhat drawn to Connes’ point of view.

I believe the problem these days is that society seems to want everything in terms of decimals or percentages, even though this may not be the optimal way of representing the information. Writing 1/9 often carries a lot more information than 0.111111… or 11%.

The required forms of answers in school maths exams has bothered me for a while. Some questions say “give your answers to n decimals place” or “n significant figures,” whereas others require “exact” answers. Bizarrely, the latter questions very rarely use the word “exact,” and instead present the student with a linear combination of irrationals with unknown coefficients to find.

Is the most exact way of writing pi “3.141….” to as many decimal places as you possibly can? No – the most exact way of writing pi is just to use the symbol. Sin(3.141….), to whatever incredibly large finite number of decimal places you like, does not ever equal 0.

While I do appreciate the usefulness of representing real numbers by their decimal expansion, it should be remembered that this is not always as representative as you may like.

Joel,

you raise many interesting points! Too many perhaps to discuss in a reply here.

I disagree that $0.111111\ldots$ carries a lot more information than $\frac{1}{9}$. To me they carry the same information because they unpack one into the other.

Writing the area of a unit circle as $\pi$ tells us nothing at all about the numerical value of this number. Is it bigger than 3? Less than 4? “$\pi$” is just a name (of realtively recent orgins – I do not think Archimedes used that name). But mathematicians over the centuries have discovered many numerical properties of the number this name stands for. What we don’t know is a description of its decimal representation (though much progress has been made in recent years in figuring out significant parts of its base 16 representation).

Decimal representations ae indeed useful in science and engineering, as a number of people point out. But that is beside the point for my concern here: those “decimals” are always finite, and there is no siginificant problem with their arithmetic.

It is only when we move to infinite decimals to describe the totality of real numbers that we find, as we reflect upon it, that arithmetical operations on infinite decimals are rather subtle, and not at all well known.

My point in this post is that we should all wake up and stop deluding ourselves that we have a clear idea of how to multiply infinite decimals if in fact we do not!

Most people – the overwhleinmg majority of mathematically well-educated people – cannot, in my experience, describe an algorithm for multiplying the decimal strings representing $\frac{2}{3}$ and $\frac{5}{7}$ for example.

I personally do not like to say “it can be done”, referring to some vague knowledge, not capable of being demonstrated.