Republic of Mathematics

A lovely observation

by Gary Ernest Davis

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

$\frac{1+3}{5+7}=\frac{1+3+5}{7+9+11}=\frac{1+3+5+7}{9+11+13+15}= \ldots = \frac{1}{3}$

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:

$1+3+5=\ldots + 2k-1$

as the positive integer $k$ increases by 1.

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

$(2k+1)+(2k+3)+\ldots + (2k+2k-1)$

We can find short algebraic expressions for these numerators and denominators, and it is these simple expressions that will show us where the fraction $\frac{1}{3}$ comes from.

Let’s write $S(k)=1+3+5+\ldots +2k-1$

and

$T(k)=2+4+6+\ldots +2k-2$

Then

$S(k)+T(k)= 1+2+3+\ldots + 2k-2 + 2k-1= \frac{1}{2}(2k-1)\times 2k=k(2k-1)=2k^2-k$

and

$T(k)=2+4+6+\ldots +2k-2= 2(1+2+3+\ldots + (k-1))=2\times \frac{1}{2}(k-1)k=k^2-k$

Therefore, $S(k)=(2k^2-k)-(k^2-k)=k^2$ .

For the denominator we have:

$(2k+1) + (2k+3) + \ldots + (2k + 2k-1) = (2k+2k+\ldots +2k) + (1+3+\ldots +2k-1)= 2k^2+k^2=3k^2$

Therefore,

$\frac{1+3+5+\ldots +2k-1}{(2k+1)+(2k+3)+\ldots + (2k+2k-1)}=\frac{k^2}{3k^2}=\frac{1}{3}$

independent of $k$ .

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

$2+4+6+\ldots + 2k = 2(1+2+3+\ldots + k)=2\times \frac{1}{2}k(k+1)=k^2+k$

while

$(2k+2)+(2k+4)+(2k+6)+\ldots +(2k+2k) = (2k+2k+2k+\ldots +2k)+(2+4+6+\ldots +2k)= 2k^2+k^2+k=3k^2+k$

Therefore,

$\frac{2+4+6+\ldots +2k}{(2k+2)+(2k+4)+\ldots + (2k+2k)}=\frac{k^2+k}{3k^2+k}=\frac{1+1/k}{3+1/k}\to \frac{1}{3}$

as $k$ increases.

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

{ 2 comments }

Using the same notation for different things, and then treating them as the same anyway

by Gary Ernest Davis

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 $\frac{a}{b}$ as a fraction, no matter what were $a \textrm{ and } b$ .

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 $x\textrm{ for which } 3\times x=2$ .

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 $\frac{2}{3}$ which stands for the result of dividing 2 things into 3 equal parts.

When we invent expressions $\frac{a}{b}$ for all pairs of integers $a\textrm{ and } b \textrm{ except } b=0$ we get the field of fractions of the ring of integers, usually denoted by $\mathbf{Q}$ .

Strictly speaking, $\mathbf{Q}$ does not consist of all expressions $\frac{a}{b}\textrm{ with }b\neq 0$ , because we recognize, for example, from our liquid division problem, that we should treat $\frac{4}{6}$ as the same “number” as $\frac{2}{3}$ .

So, strictly, $\mathbf{Q}$ consists of expressions $\frac{a}{b}\textrm{ with }b\neq 0 \textrm {and } a,b \textrm{ co-prime}$ .

Alternatively, we can take $\mathbf{Q}$ to consist of all expressions $\frac{a}{b}\textrm{ with }b\neq 0$ and we re-define “equality” between these expressions to mean $\frac{a}{b}=\frac{c}{d}\textrm{ when } a\times d = b\times c$ .

With this understanding every fraction $\frac{a}{b}\textrm{ with }b\neq 0$ can be written in the form $\frac{a'}{b'}\textrm{ with }b'\neq 0 \textrm{ and } a',b' \textrm{ coprime}$ thanks to Euclid’s algorithm for the greatest common divisor $\textrm{GCD}(a,b) \textrm{ of integers } a,b$ , since $\frac{a}{b}=\frac{a}{\textrm{GCD}(a,b)}/\frac{b}{\textrm{GCD}(a,b)}$ .

The integers reconstructed

The fractions are $\frac{a}{b}$ are constructed from integers $a,b$ , yet the integers now appear, in disguise as it were, as special cases of fractions: fractions of the form $\frac{a}{1}$ or more generally $\frac{ac}{c}$ .

So, now integers appear as special cases of fractions.

We use the same name $a$ for the integer, as we do for the fraction $\frac{ac}{c}$ . 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 $b_kb_{k-1}\ldots b_0.a_1a_2\ldots$ where the $a_i$ 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 $a_i=0$ from some point on. Unfortunately, these so-called decimal fractions do not even cover the ordinary fractions such as $\frac{1}{3} \textrm{ and } \frac{1}{7}$ .

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: $\frac{0.4}{0.19}$ . This is not a fraction: 0.4 and 0.19 are not integers. The expression $\frac{0.4}{0.19}$ indicates a division in the field of decimal numbers.

Similarly an expression like $\frac{\sqrt{2}}{\pi}$ is not a fraction: it is a division, $\sqrt{2}\textrm{ divided by }\pi$ 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: $2=2.000\ldots \textrm{ and } 3=3.000\ldots$ , so we can ask of the fraction $\frac{2}{3}$ , 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?

{ 4 comments }

Spirit of Mathematics: squiggle inception by Vi Hart

by Gary Ernest Davis

Vi Hart brilliantly illustrates the spirit of mathematical thinking

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