Fractions, Roots and Decimals

This note addresses confusions about fractions, decimal numbers, polynomials etc. and is intended for teachers rather than students.

Objective and Overview

As students learn they develop internal representations of mathematics. If these representations are faithful and correct then they support further learning. Incorrect representations impede learning and the problems grow at higher levels. Correcting a deeply-seated misunderstanding is difficult and many such students eventually quit in frustration.

Development of correct internal representations is heavily dependent on the way material is taught. The reason is that errors in development at a given level frequently do not manifest themselves in testable work until a later level, so do not get corrected when they occur.

Students learning arithmetic, for instance, are internalizing the behavior of addition, multiplication, division and so on. If their internal representation is accurate then the generalization to algebra goes smoothly. But a representation that will not support algebra may still be good enough to get a student through arithmetic.

The conclusion is that it is vital for teachers to have a solid and correct (in a long-term sense) internal representation of the material they are teaching. They should know when a teaching method will support understanding for the long run, and more important avoid simplifications or misrepresentations that work at a given level but are counterproductive in the long run. Good leadership is essential when corrective feedback is weak.

This note provides material for teachers to use in solidifying their own understanding.

• Standard notational conventions are compact and effective at the cost of being ambiguous. Problems result when these ambiguities are not properly resolved. For instance:
  • Fractions are used to indicate both an operation and the outcome of the operation.
  • Polynomials are used to denote both an element of the “polynomial ring” and a function of a real number.
• Mathematical objects have contexts that strongly influence their properties. For instance ‘3’ can refer to a natural number, integer, integer modulo 7, rational number, real number, complex number, etc., and it behaves differently in all these contexts. To write ‘3’ without specifying the context - at least implicitly - is ambiguous and invites errors.
• The ‘root’ approach to radicals and exponentials is mathematically incorrect and causes serious problems.
• Exact and ‘approximate’ numbers (decimals) are different in basic ways that should be acknowledged.

Named numbers and integers

There are a few basic numbers with individual names: the integers 0, 1, 2, … , 9, 10; a few irrationals , and the complex number .

• Other “named” numbers have compound names that indicate how they are obtained from basic numbers by various operations.

For instance is not a brand-new name but rather shorthand for “the integer obtained by evaluating the expression ”. Similarly is shorthand for “the integer obtained by evaluating the expression ”, or “the integer that gives 0 when added to 5”.

• The number 10 is a special case. It is given a compound name for consistency but belongs among the basic numbers because it appears in expansions.

Which numbers are considered basic and given individual names is very much a matter of convenience and convention. Binary (base two) and Roman numerals have fewer basic integers than the Arabic system but expressions for other integers are longer and harder for people because they involve more, or more elaborate, operations. The Babylonians used a base 60 system. In this system large integers have relatively compact names (few operations), but there are a lot of individually-named numbers to be learned.

The Arabic system strikes a good balance—for humans—between a lot of basic numbers and a lot of operations. Machine-language programmers often use hexadecimal (base 16) because it translates instantly into binary (preferred by machines) but has about the same balance between basic numbers and operations.

Polynomials

The most fundamental view of polynomials is that they are a number system (the polynomial ring) obtained by introducing a new basic “number”, say . Elements are denoted by expressions like . Exactly as with integers this has a double meaning: it is both as the name of an element of the polynomial ring and a sequence of operations that produces the element. The ambiguity is clearer and more problematic here than for integers.

There is another ambiguity in this notation. can also denote a function, with a placeholder for a real number. For elementary purposes this confusion is harmless and even useful because polynomials-as-generalized-numbers and polynomials-as-functions are manipulated the same way and the number version can be used as a function by substituting a number and evaluating.

The two views diverge at deeper levels. The ring of formal power series is obtained by allowing infinite sums . These cannot be interpreted as functions because substituting a number for almost always gives a divergent series.

The history of formal power series illustrates the hard experience that lies behind the distinctions described here. When the theory of series was being developed in the nineteenth century even professional mathematicians did not distinguish between a series and it's limit. Divergent series were regarded as failed attempts to represent numbers rather than objects in their own right. When divergent series started being used in proofs and calculations there was a great deal of anxiety - clearly recorded in the professional literature - about whether they were legitimate and, if so, what rules are necessary to avoid making mistakes. This anxiety went on for several decades before the ambiguity was recognized and resolved. The fact that it can be described here in a few sentences certainly does not mean it is easy or obvious.

Mathematica syntax

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The function and extended-number views of polynomials have to be distinguished in computer-algebra systems. Mathematica uses the symbol # as a placeholder in a function. The expression (23 #^2 + 76)& denotes the function that squares whatever you feed it, multiplies by 23 and adds 76. In contrast the expression 23 x^2+76 is treated essentially as an element of the polynomial ring.

The difference between these two views is illustrated by how roots are handled. Functions have roots (values of the argument that yield 0) and this is reflected by straightforward syntax. Root[(23 #^2 + 76)&, 1] returns , the number-one root according to some ordering convention.

An element of a polynomial ring is not officially a function so doesn't have roots. The mathematical formulation of “find roots” is “find (complex) numbers that satisfy ”. However the equal sign has several different meanings and the one we want here is is the same as . This form of ‘equal’ is denoted == in Mathematica. The maneuvering required to get a precise polynomial formulation of “roots” is reflected in the syntax Solve[ 23 x^2+76 == 0, x ].

Internally Mathematica translates this polynomial circumlocution into the straightforward function form. To see this we feed it a polynomial whose roots can't be described using standard functions. Entering Solve[ x^5 + 5 x^2 - x + 11 == 0, x ] returns x -> Root[11 - #1 + 5 #1^2 + #1^5 &, 1]}, …, {x -> Root[11 - #1 + 5 #1^2 + #1^5 &, 5]. This translates as “the function obtained by evaluating the polynomial has five roots that have been located and named. For some reason you named the variable ‘’; you can get this symbol to take the desired values by making the indicated substitutions.”

Fractions

Fractions are things like or .

As usual the notation is sloppy, indicating both a division operation and the result of performing the operation. This ambiguity causes confusion: is different from as algebraic expressions but they evaluate to the same number.

Fractions provide the first instance in which names provide important “membership” information. Rational numbers are ones obtained by dividing integers. Rational functions are obtained by dividing polynomials, though calling them “functions” is misleading. The objects specified by and are obviously a rational number and a rational function respectively. Such membership information can be hard to get when not obvious from the name. There are difficult and sophisticated theorems that assert some weird combination of trig functions gives a rational number, or a power series actually specifies a rational function.

At a deeper level fractions provide general approach to defining “rational” things. The general process starts with a “number system” (ring) such as the integers, a polynomial ring, etc. Then consider formal symbols . These are not yet names because we don't know for sure there is anything for them to refer to. Standard operations (addition, multiplication) are defined for these formal symbols and it turns out the symbols themselves form a number system called the ring of fractions. Once this system and its properties are firmly established we usually get sloppy and think of as an operation that yields an element rather than as the definition of the element.

Understanding this deeper level can clarify a number of issues. When forming a ring of fractions it is necessary to specify which elements are to be inverted. We always exclude 0, for instance, because if it is inverted (division by 0 is allowed) then everything in the ring of fractions is equivalent to 0. When mathematicians say “division by 0 is undefined” they don't mean it is impossible or they haven't yet figured out how to make sense of it. They really mean that it is unwise because if you insist on inverting 0 then the whole number system collapses. In general it can be tricky to figure out what can be usefully inverted and what should be left alone.

Examples besides rational numbers and functions are:

• the ring of integers with (and therefore divisors of powers of ) inverted. This is a subring of the rationals.
• the integers localized at in which all primes different from are inverted. This is useful in isolating the number-theoretic properties of a single prime.
• Laurent polynomials are obtained by starting with polynomials in and inverting . is an element of this ring.
• Laurent series are obtained by starting with the formal power series ring and inverting the variable. Elements of this ring are infinite linear combinations of integer powers of the variable, but only finitely many terms can have negative exponent.

Teaching

A mathematically sound description of fractions is “ indicates a division operation. In cases like we can partly carry out this operation using our standard numbers. When the division cannot be completed using standard numbers we use the uncompleted operation (fraction) to specify the number rather than introducing a new name.”

The usual introduction in terms of cutting up pies is special to rational integers, is computationally ineffective, and offers lots of opportunities for confusion. As a definition it has to be unlearned even for middle-school algebra. Mathematically it would be better if it could be presented as an application of fractions rather than as a definition.

Another problematic idea is “now that we have calculators we don't have to put up with fractions as stuck division operations. Just carry out the division.” However this destroys a lot of useful mathematical structure, removes rational fractions as a starting point for generalization e.g. to rational functions, and is technically incorrect.

Radicals, exponents and roots

Radicals are expressions of the form or , with a positive integer. They are generally accepted as numbers in the sense that an expression involving them is thought of as “solved”.

Radicals lie at the juncture of exponentials and roots. They are usually introduced as roots:

• If is an integer and then is the largest real root of the function .

The most serious problems come from extrapolation from the root version into the context of exponentials.

Radicals as exponentials

After exponents are introduced as roots they are extended to rational exponents by taking integer powers. The usual functional rules for manipulating exponents follow from this description.

Then there is a swindle: somehow becomes a function defined for real numbers . This does not cause problems in elementary math: students can manipulate correctly without realizing they are in a new context because they only use the functional rules. It causes severe problems in calculus and even with precalculus rate problems.

The exponential approach is:

• The exponential function or is defined separately;
• the logarithm is defined as the inverse of the exponential; and
• for general and we define .

This definition must be used in calculus and most other analytic settings. Exponentials are used extensively in teaching at higher levels because they are relatively quick and easy to manipulate when the right definition is used. Unfortunately many students show no sign of having seen the general definition and have difficulty giving up the root approach. These students are at a significant disadvantage.

Radicals as roots

The following are now generally known:

• it is possible to express roots of polynomials up to degree four in terms of radicals, but above degree 2 the expressions are too complicated to be useful;
• it is impossible to express roots of general fifth degree polynomials in terms of radicals; and
• in principle a real polynomial can be written as a product of quadratic and linear factors, but because of the above this is not considered possible in practice.

To some extent these are artificial problems coming from our conventions about what is acceptable as a “known exact number”. Perhaps, as computer algebra systems become more widely available, it would be reasonable to accept “the third root of ” as the name of an exact number. If we denote this by then the corresponding quadratic factor of the polynomial is . This manipulates as an exact expression. It can be approximated at any time, preferably after error analysis to determine the precision needed. A few keystrokes give an approximation, e.g. .

Teaching

For long-term purposes it might be better to avoid introducing radicals other than square roots as roots. Instead introduce the exponential function as a miraculous thing to be explained later; then the logarithm; then give the definition of general exponentials.

Functional properties of exponents follow quickly from properties of the exponential function. A corollary is that some roots can be expressed in terms of exponentials. Radicals are an application not a definition.

When complex numbers are available students can be told that the exponential function extends to a complex function of a complex variable (another miracle!) and the functional rules still work. Using this one can express all the roots of using exponentials, not just the largest real root.

Decimal numbers

Following common practice, decimal numbers are considered as approximations: 3.1415 represents the range of real numbers whose decimal representation rounds off to the given one. In principle it could also represent the rational number but nobody does this. The presumption that a decimal is approximate is so strong that if someone wants to specify that something is rational then it is always written as a fraction.

Approximate numbers are vulnerable to a host of difficulties that do not arise with exact numbers. On a machine of known precision, for instance, it is easy to make up numerical examples where the associative and distributive properties fail. Some phenomena, such as quadratics with double roots, are numerically unstable and effective numerical treatments can be elaborate and tricky. Errors accumulate and grow in extended computations, and are unavoidable in applications using approximately-known data.

Numerical failures are nearly always “silent” in that the procedure gives an outcome, but it is garbage and there is no straightforward way to know this. In principle approximate calculations should be accompanied by error analysis that would give warning of such failures.

Approximate numbers can be thought of as a context similar to rational or real numbers. Each context has advantages and disadvantages:

• Rationals are relatively simple to write exactly and manipulate, but for many purposes there are not enough of them. Few rationals have rational square roots, for instance.
• There are plenty of real numbers so e.g. every positive real has a square root. However most of them can only be specified indirectly and can be hard to manipulate.
• Approximate numbers are both plentiful and easy to manipulate but are vulnerable to errors. The tradeoff for ease of calculation is the need for error estimation and control.

A consequence is that it is not completely honest to say . These symbols specify not only numbers but contexts. Writing something as a fraction identifies it as a rational number. Writing something as a decimal identifies it as an approximate number. The contexts are different so elements in them can “correspond” but cannot be the same.

We note that distinctions of this sort are sometimes true but pointless. For instance as a rational number is technically different from as a real number, but the distinction is not important in elementary work. The distinction between exact and approximate numbers is important and should be carefully maintained.

Summary

• “Exact” and “approximate” numbers are substantially different contexts;
  • Exact calculation is often hard or (in applications) inappropriate, but reliable;
  • Approximate calculation can be unreliable, and in principle should be accompanied by error analysis.
• A careful distinction should be made between the two.

Teaching

In the current K-12 math curriculum it is entirely possible to ignore the distinction between exact and approximate computation.

• Most problems are short and simple enough that obvious numerical glitches are highly unlikely.
• Outcomes are not checked or used in a way that would expose errors. The fact that an outcome is garbage can be overlooked as long as everyone gets the same garbage.
• The few numerically unstable problems (e.g. double roots) can easily be contrived or treated in a way that hides difficulties.

It would make a lot of sense to add some error analysis to the curriculum, to bring out this difference and provide some tools to deal with it. See Numerical Error Analysis for a discussion. At present, however, it is a matter of leadership and concern for down-the-road success. Teachers can make the distinction and hint at the eventual significance, or they can ignore it in the assurance that the defect in student understanding will not cause trouble until well after their own period of responsibility.