Ben Vitale (@BenVitale on Twitter) recently made the elementary and lovely observation that

x

Some people asked why this is so, and the answer, simple as it is, makes a nice middle or high school problem

First let’s think about what’s happening here.

The numerators of these fractions are sums of the first few odd numbers.

The denominators are sums of the same number of odd numbers, starting where the numerator leaves off.

To use variable notation – something some middle schoolers are still struggling with – the numerators look like:

as the positive integer increases by 1.

The denominators, notationally a little more complicated, look like:

We can find short algebraic expressions for these numerators and denominators, and it is these simple expressions that will show us where the fraction comes from.

Let’s write

and

x

Then

and

Therefore, .

For the denominator we have:

Therefore,

independent of .

The situation for similar sums of even numbers is not quite so simple.

while

x

Therefore,

x

as increases.

A lovely observation, some simple algebra, and a challenging yet rewarding problem for middle and high school students.

On the last Twitter #mathchat there seemed to me to be a fair amount of confusion about what constitutes a fraction.

Commonly, people were treating any expression of the form as a fraction, no matter what were .

Confusion about fractions is something I’ve experienced in many places, in many contexts. The confusion seems to stem from the ‘bar’ notation; anything that has two numbers separated by a horizontal or sloping bar is a fraction it would seem.

Why do we need fractions anyway?

The problem lies in the divisibility properties of integers.

Technically, the integers form a commutative ring and not a field; there is no integer .

So, if divide 2 liters of water between 3 people, each person will not get an integer number of liters of water. We could do with numbers other than integers to describe how many liters each person gets.

We could adopt the ancient Greek practice of just saying 2 liters for every 3 people. However Muḥammad ibn Mūsā al-Khwārizmī introduced the Hindu idea of fractions as numbers to represent ratios around 830 AD, and ever since fractions have been part of arithmetic.

The  basic idea of fractions is that if we cannot divide 2 by 3, for example, then we invent a new “number” as if we could. So we invent the symbol which stands for the result of dividing 2 things into 3 equal parts.

When we invent expressions for all pairs of integers we get the field of fractions of the ring of integers, usually denoted by .

Strictly speaking, does not consist of all expressions , because we recognize, for example, from our liquid division problem, that we should treat as the same “number” as .

So, strictly, consists of expressions .

Alternatively, we can take to consist of all expressions and we re-define “equality” between these expressions to mean .

With this understanding every fraction can be written in the form thanks to Euclid’s algorithm for the greatest common divisor , since .

The integers reconstructed

The fractions are are constructed from integers , yet the integers now appear, in disguise as it were, as special cases of fractions: fractions of the form or more generally .

So, now integers appear as special cases of fractions.

We use the same name for the integer, as we do for the fraction . These are essentially different objects, but we give them the same name for obvious reasons, and then we confound them – we act as if they are the same thing.

This situation, in which previous entities appear in new guises, is common in mathematics: it happens again, for example, in the construction of complex numbers from the real numbers. The real numbers appear as special cases of complex numbers.

Real numbers

The real numbers are tricky. We can think of them as decimal strings where the are not all 9′s from some point on.

A big problem here is that addition, subtraction, multiplication and division are not easy to define due to the need for infinite carrying. It can be done, but is not so simple.

The problem is avoided in school mathematics by only dealing with decimals that are finite: that is, those for which from some point on. Unfortunately, these so-called decimal fractions do not even cover the ordinary fractions such as .

But the decimals do form a field: division, except by 0, is always possible, and it makes sense to divide 0.4 by 0.19 for example:  . This is not a fraction: 0.4 and 0.19 are not integers. The expression indicates a division in the field of decimal numbers.

Similarly an expression like is not a fraction: it is a division, in the field of decimal numbers.

But now we observe that the fractions, and so the integers, are hiding in disguise in the decimal numbers: ,  so we can ask of the fraction , does this mean the same as dividing the decimal 2 by the decimal 3? And the answer is “yes”.

So even though the bar notation for fractions was just a notation, it ends up representing actual division when we move to the larger field of fractions. A notation has become an operation.

Postscript

This might not be, as Paul Solomon (@lostinrecursion) says, how students think. But the reason I wrote the post was to ask the question of teachers of mathematics: how do you think about fractions?

Vi Hart brilliantly illustrates the spirit of mathematical thinking

Ian Carpenter (@IanMathmogician on Twitter) came up with the following very cute dynamical process, based on whole numbers:

1. Pick a number between 1 and 99.

2. Write the number as words.

3. Count the number of letters in the words to get a new number.

4. Repeat.

What happens?

Suppose we start with 42.

We write this as “forty two”, which has 8 letters.

We now write 8 as “eight” which has 5 letters.

We write 5 as “five” which has 4 letters.

We write 4 as “four” which again has 4 letters, so we’re stuck on 4.

Turns  this always happens, no matter where you start.

 How about French?

What if we did the same process in French?

42 -> quarante deux ->12 -> douze -> 5 -> cinq -> 4 -> quatre -> 6 -> six -> 3 -> trois ->5

so we get a cycle in French: 5-> 4 -> 6 -> 3 -> 5

What if we start at another number?

How about your language?

What happens in your language?

Hungarian?

Vietnamese?

Try it, and let us know.

 Matt Henderson, @matthen2 on Twitter, who is a very clever fellow, has produced some lovely pictures of this dynamical process.

What happens if you choose numbers bigger than 99? Explore!

 Postscript

Jim Wilder, @wilderlab on Twitter, pointed to the Web page Mathemagical Black Holes by Dr. Mike Ecker which has this and two other similar dynamical processes for whole numbers.

Also see the references to Ecker’s processes given by Alex Bogomolny, @CutTheKnotMath on Twitter, here and here.

Math teachers need “to know more than the text book,”says Alfred Posamentier, Professor Emeritus of City College. There are mathematical wonders beyond the curriculum that can make the subject come alive.

If only the accomplished, clever James Simons (see previous video) were this entertaining.

James Grime stands a serious chance of turning people onto mathematics.

Jon Meacham sits down with Jim Simons, founder of Math for America, to find out why Simons has devoted so much of his time – and money – to improving math education in this country.

Watch the full episode. See more Need To Know.

James Grime – mathematician, juggler and comedy nerd, Cambridge, UK – ( @jamesgrime on Twitter) has a lovely, and funny, video on the connect the towns by the shortest route problem. I think this is an outstanding example of mathematical exposition, and is EXACTLY what kids should be thinking abut in mathematics classes. Bravo James!

Completing the square

Evariste Galois

A former colleague wondered if the quadratic formula held when are complex numbers.

He could, of course, have figured this out by the process of completing the square:

If then exactly when (thanks to James Tanton for this trick).

This happens exactly when or, equivalently when .

This leads us to the equivalent form from which we deduce the quadratic formula.

What allows this to work?

What did we need to get this line of reasoning to work?

1. We need to know that if then exactly when .  This means we can multiply and divide by non-zero quantities (as well as add and subtract).
2. In exchanging for we are using commutativity of multiplication. We use this again in expanding the square

In current mathematical parlance, this means the quantities come from a field. Examples are the field of rational numbers, the field of real numbers, and the field of complex numbers.

Square roots

To take the final step from to we need to be able to extract square roots. In the field of real numbers we can always do this for non-negative numbers, but already the field of rational numbers presents difficulties.  For the complex field we can always solve for but the square root function does not exist as it does for positive real numbers.

A finite field

The numbers   where we add and multiply modulo 7 – keeping only the remainder after division by 7 – form a field, denoted

The only point that might be in doubt is division by a non-zero quantity, but we can see this can always be carried out from the following table of multiplicative inverses:

x		reason
1	1	mod 7
2	4	mod 7
3	5	mod 7
4	2	mod 7
5	3	mod 7
6	6	mod 7

So to solve a quadratic equation where the quantities come from we can reach the step and then wonder if we can always solve this equation.

For example if try to solve the quadratic equation where come from we reach the step mod 7.

What solutions, if any, are there to mod 7?

x
1	1
2	4
3	2
4	2
5	4
6	1

So there are two solutions for to : .

This gives or so that or

Quadratic residues

We can replace the number 7, above, by any prime number to get a field denoted .

The existence of multiplicative inverses follows from Euclid’s algorithm, and when is not a prime number it’s easy to see that division by non-zero quantities in is not always possible.

When the equation has a solution in , the number is called a quadratic residue mod p.

Determining which numbers are quadratic residues  mod p is a solved, but interesting, problem that is an excellent investigation for students, and it comes simply out of trying to solve quadratic equations over these fields.

Finite fields

But this does not exhaust the finite fields, and for every prime number and positive integer there is a field – known as a Galois field – with elements.

So now we want to know under what conditions we can solve where come from a finite field.

Rational numbers

When the quantities are rational numbers we again have the question of whether the equation has a rational number solution for
This, of course boils down to when has a rational number solution when is a rational number – that is, to when square roots of positive rational numbers are again rational numbers.

The moral?

Thinking about exactly what we need to solve a quadratic equation leads us straightforwardly to solving quadratic equations over less often encountered number fields, which in turn leads us directly to interesting and deep questions of number theory, which, nevertheless, are capable of being investigated by school students.

Investigating quadratic residues mod p, p prime, might also lead to more respect for the subtle intricacies of the square root as a function. The answer to is not resolved by simply writing .