Maths on a Chalk Plain

Far away and in an indeterminate time, a young woman, Integration, sat brushing her hair. The patio in which she rested was raised above the ground, allowing her to look long into the distant reaches of the Calculus Kingdom, nestled in its deep valley.

For some time a dot had been proceeding from those far borders, and now he came into clearer view. His starched shirt, polished boots and disciplined stride were first in focus, before a firm gaze appeared to watch over them. The only defect in his appearance was the sheen of sweat on his brow, but it was unavoidable: the Summer was hot and humid.

Upon dismounting, he gave a shallow bow, which was reciprocated by Integration. She had been awaiting this visit from her husband-to-be, Differentiation, since he departed earlier that day.

Expectation in the edge of her voice, she enquired, "Have you had any further thoughts on what I asked this morning?"

He glanced quizzically at her before dispatching his dismissal, "I don't understand your meaning. All we need to know has been present since birth." And indeed, the stone tablet's inscription did not leave much room for question: 'Differentiation for gradients. Integration for areas.'

She clicked her tongue. "Perhaps you're content. You have a derivation from first principles after all. But I know I'm incomplete: there's something more!" Differentiation's usually clear eyes began to cloud over with impatience and he started to raise his voice. What he said, Integration did not know, nor was she interested. She looked at him angrily before walking away. He did not try to follow.

After travelling up some of the valley's wooded wall, Integration came to a clearing. She sat down and began to think.

Often, people referred to her as just Differentiation's counterpart, his inverse. Then, in the same breath they would ask her to calculate areas! So, she decided to consider the question: 'Why can the same operation that finds a function given its gradient be used to find the area under that gradient function?'

For a while, she pondered despondently, faced with the intricacies of infinitecimals she could make no progress. As she became frustrated, a small breeze snuck through the trees, delivering temporary relief from the persistent heat and carrying an unusually straight stick with it.

After a momentary glance, Integration realised her course of action. While curves were hard to work with, straight lines required only simple geometry!

Following this, she considered that where a function was a straight line, and so had a constant gradient, the change in the function could be given by

While this equation could of course be represented as a triangle, with \(g'(x)\) being the ratio of the sides, she tried to think harder. The expression on the right was a product of two quantities. A product of two sides? That was an area!

Excitedly she drew the figure in the dirt.

At least for a constant gradient, the height and area were just two sides of the same coin. However, she wasn't satisfied yet: she wanted to know about functions more interesting than just straight lines!

Using her sleeve, Integration erased the diagram in the loose soil, and in its place, drew a curve. Her hand then placed the stick on the curve, touching in only one place, like a tangent.

She wondered how she could use the gradient of the tangent to find a change in the curve itself.

Though she couldn't think of a way to find the exact value, she thought that by using the same formula she had for functions with constant gradient, she could obtain an approximation.

Upon comparing the values using her fingers, she nearly wiped the whole diagram away in frustration! It was a poor approximation at best.

Trying to find something more useful, she squinted, looking at the smallest \(\delta x\) that she could. As \(\delta x\) became smaller and smaller, the approximation became better and better. Almost exact!

This was the closest she had ever come to a breakthrough.

Leaping up from the spot, Integration began to speed back down into the valley: she wanted to tell Differentiation as soon as possible!

Whilst she was darting down the road, Integration noticed a man in the distance and began to slow down. Perhaps she had run into Differentiation early!

As she neared him, he started haphazardly pacing about the width of the path, seemingly with no awareness of her approach. Unable to predict where he was moving, she ended up colliding with him.

On the ground before her lay a heap of limbs, clad in dirty, creased clothes and a pair of shoes barely holding themselves together. Suddenly, he sprung to life.

The man noticed Integration and looked around nervously. He gazed longingly into the distance and began to creep away, before spinning on his heel and slowly plodding back. "I am sorry," he said, and began to walk away, with some pace this time.

"Wait!" Integration called after him, "I have not seen you here before."

"I am Summation. I travelled here looking for an answer." he replied reluctantly.

"I am also looking for answers; in fact, I just found one!" she shouted excitedly. Thinking of Differentiation's likely dull reaction, she added, "And you shall be the first to know."

And so, the pair sat together on the side of the road, while Integration explained her revelation.

"That's a wonderful discovery" he praised, "but surely it's incomplete. What use is only being able to find an accurate estimate for very small changes in \(x\)?" She scowled at him, before relaxing into a sigh.

"That may be true," she assented, "but I don't know how to proceed further."

Summation smiled widely. "As repayment for my obstructing the path earlier, I shall try and help you!"

Having chatted whilst walking along, Integration and Summation had come to the conclusion that they needed a way to find changes in a function accurately, when the difference in \(x\) was large. Abruptly, Summation came to a stop. Integration looked back to see he was pointing frantically at the stone staircase before them, leading to the clearing where she had been before.

"If I asked you to make one huge jump to the top of that flight, could you do it?" he demanded.

Integration only needed to look at it for a second before responding, "Of course not?"

"Precisely, that's why you take the stairs!" he said. Seeing Integration still confused, he continued, "With our unwieldy large change in \(x\), we need only split it into lots of smaller ones, then build up the overall change in the function's value step by step."

Returning to her stick diagram from earlier, Integration tested this new idea out.

Summation grabbed a piece of chalk from his pocket, and inscribed the equation describing the current approximation on the stone.

Integration pointed out the odd symbol in the equation. "What does this mean?" she enquired.

"That's a capital sigma" he replied, "it denotes evaluating and summing together the expression inside for \(x=a+\delta x, a+2\delta x, a+3\delta x\), all the way up to \(x=b\). It saves writing a lot of dots..."

The duo continued and generalised this new principle, splitting up the interval into stairs. They realised that to find a difference in a function between \(x=a\) and \(b\), they needed to split it up into \(n\) stairs, so that \[ \delta x = \frac{b-a}{n} \]

Integration took this further, applying the principle she found earlier, that approximations become better and better as they get smaller and smaller, to realise that the larger \(n\) was, the better the approximation would be.

Integration jumped up from the spot.

"Is it possible," she asked Summation, "that if \(\delta x\) becomes small enough, if it approaches \(0\), our sum would become equal to a change in the function?"

Summation stood up as well. "I think you're right!" he exclaimed. Then, he grabbed his chalk and added this new formula to the stone. \[ f(b) - f(a) = \lim_{\delta x \rightarrow 0} \sum_{x=a}^{b}f'(x)\delta x \] Integration beckoned for the chalk, and made one final addition. \[ f(b) - f(a) = \lim_{\delta x \rightarrow 0} \sum_{x=a}^{b}f'(x)\delta x = \int_{a}^{b}f'(x)dx \]

She finally understood who she was.

There remained one thing for her to do though. "If the difference in a curve is simply the sum of the heights of many small triangles, then I think I know a trick that can help us," she said, thinking back to the geometry from earlier. "If we say that \(f'(x)\) gives the height above the \(x\) axis (that \(y=f'(x)\)), then \(f'(x)\delta x\) is the area of many small rectangles, which in turn is the area under \(f'(x)\)," she blurted out, exhausted.

They had done it. They had found why 'The same operation that finds a function given its gradient can be used to find the area under that gradient function.'

As they relaxed under a tree together, tired from their exploration, Summation started fidgeting. "I must admit, I had an ulterior motive in assisting you" he confessed, "I have my own problem whose answer I'm seeking, and I thought you might be able to help."

Integration did not look in the slightest bit perturbed.

"I would be happy to help," she replied, pleased that she could do something for Summation as well, and anticipating what manner of problem he could have.

Excusing himself, Summation wandered into the forest. "The ancient texts," he murmured, "I'll find the answer. Yes, yes, I'll find the answer."

Differentiation strode up and down the castle corridors, his furrowed brow casting a shadow over his face. Though he didn't understand what had upset Integration so severely, he needed to make amends soon: their wedding fast approached.

Arriving at The Royal Library, he swept through the entrance and past the works of fact, fiction and frivolity, eyeing up a cage of books in the far corner. Upon reaching it, he unlocked the gate, and selected a particularly worn looking volume: 'On the Three Fates of Kinematics'. Swiftly locating the chapter on their summoning, he read the forbidden text within.

Immediately after, he set off to conduct the ritual. He made such haste, in fact, that he overlooked the warning at the end: 'Wise words have the three, but take their message in fraction and high will be the fee.'

Leaving the clearing, Integration and Summation began travelling further up the valley's edge. After a while in silence, Summation brandished a manuscript. He leant over and handed it to Integration.

"My purpose is to add things together," he explained, "and I love finding closed form expressions, those composed of only basic functions, to do this." He continued, "The scroll I just handed you should serve as a fun little introduction before I show you my real goal."

Integration looked over the figures scribbled on the paper. It described a process for deriving the formula for the sum of the first \(n\) square numbers.

Let \(P(n)\) be a polynomial such that \[ P(n) - P(n-1) = n^2 \quad \textrm{and} \quad P(1)=1 \textrm{.} \] We know that \(P(n)\) will satisfy \[ P(n) = \sum_{x=1}^{n}x^2 \] and that it will have order \(3\).

Integration turned to Summation. "How do we know that \(P(n)\) will be equal to the sum of squares, and that \(P(n)\) has order \(3\)?" she queried.

"A difference of \(n^2\) between consecutive integer inputs is the definition of the sum of square numbers. For example, \[ 1^2 + 2^2 + 3^2 - (1^2 + 2^2) = 3^2 \textrm{.} \] However, since \(P(n)\) is defined on all the reals, \(P\) can be considered a continuation of \[ \sum_{x=1}^{n}x^2 \] which is defined for natural number (positive integer) values of \(n\) only."

"As to why \(P(n)\) has order \(3\)," he continued, "The order of a polynomial is the highest power of the variable that appears in it. So, if \[ P(n) \quad \textrm{has order} \quad m \textrm{,} \] then \[ P(n) - P(n-1) \quad \textrm{has order} \quad m-1 \textrm{,} \] as the highest power cancels in the subtraction. Since \(n^2\) has order \(2\), \[ m-1=2 \] \[ m=3 \textrm{."} \]

Integration returned to reading the script, where it coninued with evaluating some algebra. \[ \textrm{let} \quad P(n) = an^3 + bn^2 +cn + d \textrm{.} \] \[ P(n-1) = a(n^3 -3n^2 +3n -1) +b(n^2 -2n +1) +c(n-1) +d \] \[ = an^3 + (b-3a)n^2 + (3a-2b+c)n +(b+d-a-c) \] \[ P(n) - P(n-1) = 3an^2 + (2b-3a)n + (a+c-b) \]

Having done this, it continued to compare coefficients of powers of \(n\), with the RHS, \(n^2\). \[ 3a=1 \therefore a = \frac{1}{3} \] \[ 2b-3a=0 \therefore b=\frac{1}{2} \] \[ a+c-b=0 \therefore c=\frac{1}{6} \]

Finally, in order to find the constant term in \(P(n)\), \(d\), it used the fact that \(P(1)=1\). \[ \frac{1}{3}(1)^3 + \frac{1}{2}(1)^2 - \frac{1}{6}(1) + d = 1 \therefore d=0 \] \[ P(n)=\frac{1}{3}n^3 + \frac{1}{2}n^2 + \frac{1}{6}n \]

Taking back the scroll in clammy hands, Summation looked towards Integration. In a low voice he began to speak, "My true aim is to find a closed form expression for partial sums of the harmonic series."

Intrigued, Integration asked, "What kind of series is that?"

"It's the sum of all the unit fractions down to a certain number," he responded, "for example \[ H_4 = \sum_{x=1}^{4}\frac{1}{x} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} \textrm{.} \] It's deeply mysterious and appears all over the place: in music, stacking problems, even prime numbers! Wise as they were, it stumped the ancients."

For a while, they continued on in silence. Occasionally, Summation would glance over at her hopefully, and she was beginning to feel the pressure. Eventually, she had an idea. "Since the terms keep getting smaller and smaller, perhaps eventually, the sum approaches a constant!" she blurted out.

Summation looked over at her with a look of disappointment, which quickly receded to a small smile. "It's a good idea," he answered, "but I'm afraid not." "The concept you're referring to is called convergence; we say a series converges on a particular value. One of the mathematicians in the old scriptures, Oresme, proved that this series doesn't converge. Consider the harmonic series as

now we can compare it with another series

Since adding one together forever gives infinity, that means this series diverges, and does not approach any number. The harmonic series is greater than the prior one, and so must also diverge." he said.

As they gained more and more height, heading out of the valley, Integration began to slow. "I must head back soon," she said, "my wedding is not far away now."

"I see," responded Summation, and the two began to halt.

"The man I am to marry, Differentiation, has his issues, but he is intelligent, and may be able to help with your problem!" added Integration.

Summation started to look excited at the prospect of meeting the mysterious fiancé, and they decided to go on for just a little longer before turning back. Clouds gathered behind them.

Differentiation arrived at a hidden location and carried out the unholy ceremony. Slowly, a hot, foul mist spread across the gound, and in its midst appeared three hooded figures.

"Please, three fates, tell me where my wife-to-be has gone," he petitioned them.

Displacement was the first to speak: "Far from here she is, closer to the sun than ever she was with you."

Then spoke Velocity: "And still she moves away, another at her side."

At this Differentiation started. Before another word could be spoken, he dashed off.

To the empty space spoke Acceleration: "She accelerates towards you my Lord; she will yet return."

The first, Displacement, turned to watch the vanishing Differentiation: "Though such a storm as is coming is poor weather indeed for a wedding." The three fates melted back into the fog from which they had come.

Tired from travelling, Integration and Summation stopped to rest in the clearing. They could see rain falling over the Kingdom in the distance. Though they were both put out, Integration made a suggestion, "Since we don't know how to approach the series, why don't we start with just the graph of \(1/x\) itself?" Seeing no other way forward, Summation obliged and drew the curve on the stone. "Ah! Surely the sum can be represented as the area under lines which meet \(y=1/x\) at its integer inputs." Integration suggested.

As Summation progressed in scrawling the additions onto the diagram, he started speeding up and a smile emerged on his face. "Yes! The difference between the sum and the area under the curve gets smaller and smaller. It converges!" he shouted. Immediately though, his face sunk back to a scowl. "What the area under the curve is is anyone's guess." he muttered.

Integration quickly interjected, "You seem to have forgotten who you're here with. \[ \int_1^n \frac{dx}{x} = ln(n) \textrm{.}" \]

Brightening again, Summation got to work, using this to approximate the difference in area. Fortunately, his previous explorations had led him to calculate partial sums of the series to very high values, and he always kept his trusty log tables with him. Finishing, he started to read the digits out.

\(0\)

\(.\)

\(5\)

\(7\)

He began to sweat profusely. A look of terror crept onto his face.

\(7\)

4

"No!" he screamed.

He had seen this number before. It appeared frequently in the ancient texts, referred to as Euler's constant, or \(\gamma\) (gamma). Many had sought meaning in it but none had found it. No one even knew whether it was irrational! What hope did he have.

He would never understand the harmonic series.

Abruptly, he dashed into the woods, screeching horribly. Though Integration tried to chase him, he quickly disappeared from view. She sat down to catch her breath as the rain began to fall.

Thunder sounded as Differentiation smashed into the clearing. He spotted Integration in the corner looking distraught.

"How could you run from me, with another!" he shouted. "In the week of our wedding!"

Integration began to cry, her tears invisible in the dense rainfall.

Summation suddenly appeared in front of Differentiation, shaking him by the shoulders. "You, you're Differentiation, yes? You can help me!" he screamed. Before he could continue, a look of surprise entered his face. Differentiation's sword had been driven through his chest. "Scum!" Differentiation decried.

He did not have long to express his anger though. Before he could react, Integration pushed him down the steep, stone stairs leading down from the clearing. He lay at the bottom, dead.

Integration stared down at the royal guard that had just arrived from the Kingdom. her face was devoid of expression.