236 lines
11 KiB
Markdown
236 lines
11 KiB
Markdown
---
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created_at: '2014-05-11T03:57:12.000Z'
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title: Take It to the Limit (2010)
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url: http://opinionator.blogs.nytimes.com/2010/04/04/take-it-to-the-limit
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author: drjohnson
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points: 89
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story_text: ''
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comment_text:
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num_comments: 14
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story_id:
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story_title:
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story_url:
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parent_id:
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created_at_i: 1399780632
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_tags:
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- story
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- author_drjohnson
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- story_7727710
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objectID: '7727710'
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---
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![Steven
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Strogatz](https://static01.nyt.com/images/blogs/opinionator/contributors/strogatz45.jpg)
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[Steven
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Strogatz](//opinionator.blogs.nytimes.com/category/steven-strogatz) on
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math, from basic to baffling.
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In middle school my friends and I enjoyed chewing on the classic
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conundrums. What happens when an irresistible force meets an immovable
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object? Easy — they both explode. Philosophy’s trivial when you’re 13.
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But one puzzle bothered us: if you keep moving halfway to the wall, will
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you ever get there? Something about this one was deeply frustrating,
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the thought of getting closer and closer and yet never quite making it.
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(There’s probably a metaphor for teenage angst in there somewhere.)
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Another concern was the thinly veiled presence of infinity. To reach
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the wall you’d need to take an infinite number of steps, and by the end
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they’d become infinitesimally small. Whoa.
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Questions like this have always caused headaches. Around 500 B.C., Zeno
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of Elea posed a set of paradoxes about infinity that puzzled generations
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of philosophers, and that may have been partly to blame for its
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banishment from mathematics for centuries to come. In Euclidean
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geometry, for example, the only constructions allowed were those that
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involved a finite number of steps. The infinite was considered too
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ineffable, too unfathomable, and too hard to make logically rigorous.
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But Archimedes, the greatest mathematician of antiquity, realized the
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power of the infinite. He harnessed it to solve problems that were
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otherwise intractable, and in the process came close to inventing
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calculus — nearly 2,000 years before Newton and Leibniz.
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In the coming weeks we’ll delve into the great ideas at the heart of
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calculus. But for now I’d like to begin with the first beautiful hints
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of them, visible in ancient calculations about circles and pi.
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Let’s recall what we mean by pi. It’s a ratio of two distances. One of
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them is the diameter, the distance across the circle through its
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center. The other is the circumference, the distance around the
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circle. Pi is defined as their ratio, the circumference divided by the
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diameter.
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![circle with diameter and circumference
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indicated](https://static01.nyt.com/images/2010/04/04/opinion/04strogatz1/04strogatz1-custom3.jpg)
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If you’re a careful thinker, you might be worried about something
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already. How do we know that pi is the same number for all circles?
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Could it be different for big circles and little circles? The answer is
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no, but the proof isn’t trivial. Here’s an intuitive argument.
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Imagine using a photocopier to reduce an image of a circle by, say, 50
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percent. Then all distances in the picture — including the
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circumference and the diameter — would shrink in proportion by 50
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percent. So when you divide the new circumference by the new diameter,
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that 50 percent change would cancel out, leaving the ratio between them
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unaltered. That ratio is pi.
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Of course, this doesn’t tell us how big pi is. Simple experiments with
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strings and dishes are good enough to yield a value near 3, or if you’re
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more meticulous, 3 and 1/7th. But suppose we want to find pi exactly or
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at least approximate it to any desired accuracy. What then? This was
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the problem that confounded the ancients.
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Before turning to Archimedes’s brilliant solution, we should mention one
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other place where pi appears in connection with circles. The area of a
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circle (the amount of space inside it) is given by the formula
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![formula for area of a
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circle](https://static01.nyt.com/images/2010/04/04/opinion/04strogatzfig1/04strogatzfig1-custom3.jpg)
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Here A is the area, π is the Greek letter pi, and r is the radius of the
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circle, defined as half the diameter.
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![Circle, with area filled in, and radius marked with letter r
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](https://static01.nyt.com/images/2010/04/04/opinion/04strogatz2/04strogatz2-custom2.jpg)
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All of us memorized this formula in high school, but where does it come
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from? It’s not usually proven in geometry class. If you went on to
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take calculus, you probably saw a proof of it there, but is it really
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necessary to use calculus to obtain something so basic?
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Yes, it is.
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What makes the problem difficult is that circles are round. If they
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were made of straight lines, there’d be no issue. Finding the areas of
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triangles, squares and pentagons is easy. But curved shapes like
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circles are hard.
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The key to thinking mathematically about curved shapes is to pretend
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they’re made up of lots of little straight pieces. That’s not really
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true, but it works … as long as you take it to the limit and imagine
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infinitely many pieces, each infinitesimally small. That’s the crucial
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idea behind all of calculus.
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Here’s one way to use it to find the area of a circle. Begin by
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chopping the area into four equal quarters, and rearrange them like so.
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![Four quarters of a circle on left, then rearranged on
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right](https://static01.nyt.com/images/2010/04/04/opinion/04strogatz3/04strogatz3-custom2.jpg)
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The strange scalloped shape on the bottom has the same area as the
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circle, though that might seem pretty uninformative since we don’t know
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its area either. But at least we know two important facts about it.
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First, the two arcs along its bottom have a combined length of πr,
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exactly half the circumference of the original circle (because the other
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half of the circumference is accounted for by the two arcs on top).
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Second, the straight sides of the slices have a length of r, since each
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of them was originally a radius of the circle.
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Next, repeat the process, but this time with eight slices, stacked
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alternately as before.
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![Circle showing eight slices
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](https://static01.nyt.com/images/2010/04/04/opinion/04strogatz4/04strogatz4-custom2.jpg)
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The scalloped shape looks a bit less bizarre now. The arcs on the top
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and the bottom are still there, but they’re not as pronounced. Another
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improvement is the left and right sides of the scalloped shape don’t
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tilt as much as they used to. Despite these changes, the two facts
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above continue to hold: the arcs on the bottom still have a net length
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of πr, and each side still has a length of r. And of course the
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scalloped shape still has the same area as before — the area of the
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circle we’re seeking — since it’s just a rearrangement of the circle’s
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eight slices.
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As we take more and more slices, something marvelous happens: the
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scalloped shape approaches a rectangle. The arcs become flatter and the
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sides become almost vertical.
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![Circle with many
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slices](https://static01.nyt.com/images/2010/04/04/opinion/04strogatz5/04strogatz5-custom1.jpg)
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In the limit of infinitely many slices, the shape is a rectangle. Just
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as before, the two facts still hold, which means this rectangle has a
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bottom of width πr and a side of height
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r.
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![rectangle](https://static01.nyt.com/images/2010/04/04/opinion/04strogatz6/04strogatz6-custom1.jpg)
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But now the problem is easy. The area of a rectangle equals its width
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times its height, so multiplying πr times r yields an area of πr2 for
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the rectangle. And since the rearranged shape always has the same area
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as the circle, that’s the answer for the circle too\!
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What’s so charming about this calculation is the way infinity comes to
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the rescue. At every finite stage, the scalloped shape looks weird and
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unpromising. But when you take it to the limit — when you finally “get
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to the wall” — it becomes simple and beautiful, and everything becomes
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clear. That’s how calculus works at its best.
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Archimedes used a similar strategy to approximate pi. He replaced a
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circle by a polygon with many straight sides, and then kept doubling the
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number of sides to get closer to perfect roundness. But rather than
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settling for an approximation of uncertain accuracy, he methodically
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bounded pi by sandwiching the circle between “inscribed” and
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“circumscribed” polygons, as shown below for 6-, 12- and 24-sided
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figures.
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![Circles inscribed in
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polygons](https://static01.nyt.com/images/2010/04/04/opinion/04strogatz7/04strogatz7-custom1.jpg)
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Then he used the Pythagorean theorem to work out the perimeters of these
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inner and outer polygons, starting with the hexagon and bootstrapping
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his way up to 12, 24, 48 and ultimately 96 sides. The results for the
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96-gons enabled him to prove that
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![formula for
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96-gons](https://static01.nyt.com/images/2010/04/04/opinion/04strogatzFig2/04strogatzFig2-custom3.jpg)
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In decimal notation (which Archimedes didn’t have), this means pi is
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between 3.1408 and 3.1429.
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This approach is known as the “method of exhaustion” because of the way
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it traps the unknown number pi between two known numbers that squeeze it
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from either side. The bounds tighten with each doubling, thus
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exhausting the wiggle room for pi.
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In the limit of infinitely many sides, both the upper and lower bounds
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would converge to pi. Unfortunately, this limit isn’t as simple as the
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earlier one, where the scalloped shape morphed into a rectangle. So pi
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remains as elusive as ever. We can discover more and more of its digits
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— the current record is over 2.7 trillion decimal places — but we’ll
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never know it completely.
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Aside from laying the groundwork for calculus, Archimedes taught us the
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power of approximation and iteration. He bootstrapped a good estimate
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into a better one, using more and more straight pieces to approximate a
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curved object with increasing accuracy.
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More than two millennia later, this strategy matured into the modern
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field of “numerical analysis.” When engineers use computers to design
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cars to be optimally streamlined, or when biophysicists simulate how a
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new chemotherapy drug latches onto a cancer cell, they are using
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numerical analysis. The mathematicians and computer scientists who
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pioneered this field have created highly efficient, repetitive
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algorithms, running billions of times per second, that enable computers
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to solve problems in every aspect of modern life, from biotech to Wall
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Street to the Internet. In each case, the strategy is to find a series
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of approximations that converge to the correct answer as a limit.
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And there’s no limit to where that’ll take us.
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NOTES:
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Thanks to Tim Novikoff and Carole Schiffman for their comments and
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suggestions, and to Margaret Nelson for preparing the illustrations.
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Editor’s Note: A correction was made to an earlier version of this
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column, to fix a misspelling of the name of the publisher of Zeno’s
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Paradox.
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Need to print this post? Here is [a print-friendly PDF version of this
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piece](https://static01.nyt.com/packages/pdf/opinion/opinionator/TakeItToTheLimit.pdf),
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with images.
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