Math for Grownups

Author: Math Expert

  • The Mother of All Scientific Computing

    The Mother of All Scientific Computing

    Ada Lovelace was probably bound for greatness. The product of the brief marriage between Lord Byron (yes, that Lord Byron) and Anne Isabella (“Annabella”) Milbanke, she was born in 1815. But in true Romantic tragedy, her parents separated soon after her birth , and she never knew her father. Her mother, whom Lord Byron called “the Princess of Parallelograms,” was pretty quick with the calculations, and so Lovelace got a good education in math and science. This approach also served to protect Lovelace from the fiery passions of poetry (according to her prudish mother).

    Seems Ada got the best of both parents. At age 13, she developed a design for a flying machine — quite a feat in 1828, a full 85 years before and an ocean away from the Wright Brothers at Kitty Hawk. But over time, her approach to mathematics was decidedly verbal. She called herself the poetic scientist, and her writings were imaginative and described in metaphors.

    When she was 17 years old, Lovelace met Mary Somerville, the self-taught mathematician and scientist. The two became fast friends, attending lectures, demonstrations and concerts together. And it was Somerville who introduced Lovelace to the man who would help cement her name in history.

    Charles Babbage was the inventor of the Difference Engine, a rudimentary calculator that wasn’t built until more than 100 years after his death. He and Lovelace met in 1834, when he was working out the design of his next invention, the Analytic Engine.

    Unlike his Difference Engine, this new design was programmable, an idea that completely enthralled Lovelace. She and Babbage became good friends and colleagues, and in 1843, Babbage asked Lovelace to translate into English a French summary of a presentation he gave describing the Analytic Engine. And by the way, could she also expand upon the ideas, since she was so familiar with the design?

    What Lovelace wrote was nothing less than prescient:

    The distinctive characteristic of the Analytical Engine, and that which has rendered it possible to endow mechanism with such extensive faculties as bid fair to make this engine the executive right-hand of abstract algebra, is the introduction into it of the principle which Jacquard devised for regulating, by means of punched cards, the most complicated patterns in the fabrication of brocaded stuffs. It is in this that the distinction between the two engines lies. Nothing of the sort exists in the Difference Engine. We may say most aptly that the Analytical Engine weaves algebraical patterns just as the Jacquard loom weaves flowers and leaves.

    (I said she wrote in metaphors!)

    Again Babbage’s machine was not built in his lifetime, but the design — featuring punch cards of the early mechanical computers — is still acknowledged as the precursor to the modern-day computer. And Lovelace is considered the first computer programmer because of what she suggested the machine could do: compute the Bernoulli numbers.

    What the heck are they? Well, first off, Bernoulli numbers are a pretty big deal in number theory and analysis. Basically, they’re a sequence (or list) of rational numbers (or decimals that either repeat or terminate). These numbers show up in a variety of places that won’t matter to you. The important thing here is that they are darned difficult to compute. In the 19th century, folks who needed them typically depended on tables that listed these numbers. But Lovelace developed a program that would generate them automatically.

    Thus, the first computer program program was born.

    Unfortunately for all of us, Lovelace would never see her invention realized. She died of cancer in 1852, before publishing anything more. Still, her contribution is so great that computer geeks around the world still revere her. In 1977, the Department of Defense named its high-level computer programming language Ada. Heck, the IT guy at my last regular job named his first daughter Ada.

    I wonder what Lord Byron would have written about his daughter, the poetic scientist?

    Had you heard of Ada Lovelace? What do you think Lord Byron would have thought of her contributions? Share your feedback below.

  • En Garde! The great calculus duel

    En Garde! The great calculus duel

    Before beginning this story, a little background. There are two really basic ways to think of calculus:

    1. The study of the infinite (extremely large) and the infinitesimal (extremely small).

    Or

    2. The study of limits. Imagine a gnat that is flying from the middle of a room to the doorway. The gnat first moves halfway to the door. Then he takes a little breather and moves half of the remaining distance. Another breather, another jaunt half of the remaining distance. And so on and so on. Will he ever get to the door?

    (Okay, most mathematicians might hate me for boiling things down to this very basic level, but for the average Joe or Jane, these explanations will do the trick. And due to space issues, I need for you to just trust me on why these things matter. Some day, I’ll write about the applications of calculus and other higher-level math.)

    If you think math history isn’t very exciting — in a Batman meets Joker or Clint Eastwood make-my-day kind of way — you’re pretty much right. There are a few life-and-death situations, like Galileo’s (okay, it was his soul in peril, not his physical body), but for the most part, mathematicians were either revered or went unnoticed. Except for Sir Isaac Newton and Gottfried Wilhelm Leibniz.

    I wish I could say that this was an actual duel, not because I love violence or wish ill on one of these fine mathematicians, but because it would make this story even more interesting — especially to high school students or grown ups who think math is BOR-ring. But in the end this story is still pretty fascinating, especially given the fact that these men never met or spoke on the phone or Skyped (because cell phones and the internet didn’t exist).

    It was 1666, around the time of the Apple Incident (you know, when a fallen apple prompted Newton to develop his theories of gravity) that The Sir thought up his ideas of fluxions. Don’t worry, you shouldn’t know what that word is, as it’s never used in modern mathematics. Instead we call his development differential calculus.

    Leibniz was just 20 years old at that time. Sure, he was a genius — he had already earned degrees in philosophy and law, and that year he published his first book, De Arte Combinatoria or On the Art of Combinations. While this expansion of his philosophy dissertation is obliquely related to mathematics, it was well before Leibniz began formally dabbling in the Queen of the Sciences.

    This timing is pretty darned important. Trust me.

    So Newton farts around with this idea of fluxions, finally getting around to publishing Method of Fluxions in 1736. But along he published a few manuscripts on the subject, sending early copies to some colleagues. Meanwhile, in Germany, Leibniz was jotting down his own discoveries in his journal. In 1675, he noodled around, finding the area under a the graph of y = f(x) using integral calculus.

    In other words, the two men were discovering calculus at the same time and in completely different parts of the world. (Okay, Germany and England weren’t too distant from one another, but in the 17th century, they may as well have been on different planets.)

    I’d bet that given Newton’s stereotypical absent-minded-professor approach to the world around him, he might never have even noticed Leibniz’s publications, which came in 1684 and 1686. Or at the very least, he might have simply acknowledged the great coincidence and moved on. (Apparently, the man could barely be trusted to keep a dinner date, much less worry about a rival in a different country.)

    In fact, it was neither Newton nor Leibniz who lit the fire of the great calculus war. In 1704, an anonymous review of Newton’s fluxions suggested that he borrowed [ie stole] the idea from Leibniz, which of course infuriated Newton. Letters flew back and forth between the two mathematicians and their surrogates. Newton was behind the publication of these letters, called Commercium Epistolicum Collinii & aliorum, De Analysi promota. (I am not kidding.) A summary of this publication was published anonymously in 1714 in the Philosophical Transactions of the Royal Society of London. But everyone knows that Newton wrote it.

    The Swiss mathematician Johann Bernoulli — who later made his own contributions to infinitesimal calculus — attempted to defend Leibniz, but Newton pretty much took him down. In the end Leibniz meekly defended himself, refusing to look through his “great heap of papers” to prove that he had independently discovered calculus at the same time as Newton. When he died in 1716, Leibniz had been pretty well beaten up by Newton and his buddies (metaphorically speaking, of course).

    It wasn’t until much later that everyone came around to the accepted and logical — though really coincidental — truth of the whole ordeal. Both Newton and Leibniz discovered calculus at the same time, using slightly different approaches. To many of us math folks, this is a truly wondrous event.

    But there’s more. Even though Newton enjoys (and did enjoy) a bit of celebrity for his genius, he largely wrote for himself, while Leibniz was a bit obsessive about notation, wanting to be sure that his discoveries could actually be used. This is one of the big reasons that today’s calculus is pretty much Leibniz’s discovery. Newton’s approach turns out to be a bit too clunky for everyday use.

    So whether or not you possess a general (or specific) understanding of calculus, you can certainly appreciate the 17th-century-style drama surrounding the discovery of this critical field of mathematics, right? At the very least, we can thank Newton and Leibniz for that.

    Did you know about the great calculus controversy? What questions does it bring up for you? Ask them in the comments section!

  • A Greek, a Bathtub and an Amazing Discovery

    A Greek, a Bathtub and an Amazing Discovery

    It seems to me that the Greek philosopher and scientist, Archimedes, was like the forgetful scientist. And a few tales of his life support this theory.

    Born in 287 B.C. on the island of Sicily, he had the good fortune — for him and us — to have a wealthy astronomer for a father. He enrolled in an Alexandrian school based on the principles of Euclid — the father of plane geometry. (You know: points, lines, planes, if the corresponding sides of two triangles are congruent then so are the triangles and vertical-angles-are-always-congruent.)

    He must have gotten a good education, because Archimedes went on to apply mathematics to building tools, like the Archimedes screw, which is used to efficiently pump water from one place to another. (Contraptions based on his design are still being used today.) He also explained how levers and pulleys work, developing new ways to move even heavier objects. (“Give me a lever long enough and a fulcrum on which to place it, and I shall move the world,” he said.)

    And speaking of heavy objects, my very favorite math story is about our dear, old, absent-minded Archimedes. Apparently his good buddy King Hiero hired a goldsmith to make him a crown of the shiny stuff. But the king was suspicious that the goldsmith was cheating him — giving him a crown made of a composite of gold and another (cheaper) metal.

    So Hiero took his crown to the smartest man he knew, Archimedes, who gave the problem some deep thought. But it wasn’t until he lowered himself into one of the city’s public baths that the solution hit him like a ton of bricks (or a crown of gold). He got so excited that he ran through the streets naked and shouting, “Eureka! Eureka!” or “I’ve got it! I’ve got it!”

    Nobody  knows for sure if this is a true story, but it sure got the attention of my high school math students back in the day. And Archimedes’ discovery has certainly stood the test of time. See, when he got into his bath, Archimedes noticed that his body caused some of the water to spill over the side. That got him thinking about the relationship between the volume of his body and the amount of water that was displaced. By replicating the experiment with gold and silver, he realized he had discovered the principle of displacement — if an object sinks in water, the amount of water that is displaced (or overflows) is equal to the volume of that object.

    P.S. Apparently the goldsmith was trying to pull one over on the king. The crown was made of iron and covered in gold.

    But when it came to mathematics, geometry was his thing. (Duh. It was ancient Greece, after all.) The man had an obsession with circles. In order to better estimate the value of π (or the ratio of the circumference of a circle to its diameter), he drew a 96-side regular polygon. (It was between 3 10/71 and 3 1/7.) He used the same “method of exhaustion” to find the area of a circle and the volume of a sphere.

    Archimedes’ death is a testament to his ability to focus on his studies with no regard to the world around him. Stories say that his last words were, “Do not disturb my circles.” These were said to the Roman soldier who killed him, as Archimedes studied.

    Did anything about Archimedes surprise you? Which of his discoveries have you counted on at home or work? Share your responses in the comments section.

  • Thanksgiving by the Numbers

    Thanksgiving by the Numbers

    It’s turkey time here in the U.S. — the weekend we celebrate family, friends and all of the blessings in our lives. And since I’m certainly thankful for math (seriously!), I thought we should take a look at some interesting Thanksgiving numbers.

    391: The number of years since the first Thanksgiving

    90: The estimated number of Wampanoag tribe members who attended the first Thanksgiving

    1789: The year of the first “national day of Thanksgiving”

    40: The number of years that Sarah Josepha Hale advocated for an annual, national Thanksgiving holiday

    254 million: The number of turkeys expected to be raised in the U.S. in 2012

    6: The number of “home economists” who were hired to answer 11,000 phone calls for the Butterball Turkey Hotline‘s inaugural year in 1981

    50: The number hired to answer more than 100,000 calls last year.

    165: The number of degrees of a safely cooked turkey, according to the Federal Food and Drug Administration (FDA)

    768 million: Projected number of pounds of cranberries expected to be produced in the U.S. in 2012

    50: Percent increase in plumber calls and visits on the day after Thanksgiving, over any other Friday of the year

    43.6 million: Number of Americans expected to travel more than 50 miles during the 2012 Thanksgiving holiday

    90: Percent expected to travel by car

    25: Number of balloon floats in this year’s Macy’s Thanksgiving Day Parade

    How are you spending your Thanksgiving holiday? What math is involved? Share your responses in the comments section.

  • The Cult That Changed Geometry

    The Cult That Changed Geometry

    While the development of numbers continued for many, many centuries, even before the discovery or invention of zero, the Greeks were responsible for a long, long period filled with mathematical advances. By 600 B.C., a fellow named Thales of Miletus brought Babylonian mathematical discoveries to Greece, which were used to calculate distance and other measurements.

    But the big player in Greece was Pythagoras. (Yes, you should recognize that name.) Born in 580 B.C. in Samos, he met old-man Thales when he was but a young lad. Perhaps Thales convinced him to travel to Egypt so that he could learn the mathematics of the Babylonians. At any rate, when Pythagoras returned from his journey, he settled in Croton (which is on the eastern coast of Italy) and this is where things get strange — at least by our modern standards.

    Pythagoras established a philosophical and religious school that was made up of two societies: the akousmatikoi (hearers) and mathematikoi (learned). And while his followers look much like a cult to us, Pythagoras was in fact developing the world’s first intentional, philosophical society. Members — both men and women — were intent on living a contemplative and theoretical life, and as such divorced themselves from the culture at large, becoming completely devoted to philosophical and mathematical discovery.

    But in order to do this, they had to follow a very strict set of rules, which included vegetarianism, giving up all personal possessions and absolute secrecy. And then there are the really strange orders: do not pick up something that has fallen; do not touch a white rooster; do not look in a mirror beside the light.

    That’s not all. Mysticism infused almost all the Pythagoreans did, which led to some really off-the-wall mathematical ideas, like their understanding of numbers.

    1. Nothing exists without a center, and so the circle is considered the parent of all other shapes. It was called the monad or “The First, The Essence, The Foundation, and Unity” — or according to Pythagoras, “god and the good.”
    2. The dyad was a line segment and considered to be the “door between One and Many.” It was described as audacity and anguish, illustrating the tension between the monad and something even larger.
    3. And then there’s the triad, which of course represents the number 3. Continuing in their pseudo-anthropomorphism of numbers, the triad is considered the first born, with characteristics like wisdom, peace and harmony.

    I could go on. Seriously. But while the ideas of the Pythagoreans were kind of kooky, this band of deep-thinking brothers and sisters advanced mathematics in some pretty significant ways. First of all, they began classifying numbers as even and odd, prime and composite, triangular, square, perfect and irrational. Through their strange ideas of numbers, they popularized geometric constructions. They are also attributed with the discovery of the five regular solids (tetrahedron, hexahedron, octahedron, iscosahedron and dodecahedron).

    But their biggest discovery is the theorem named for Pythagoras. The Pythagorean Theorem states that the in a right triangle, the square of the longest side is equal to the sum of the squares of the remaining two sides. In other words:

    This is more than just a silly formula you needed to memorize in high school. Carpenters use it to be sure that they have right angles (in other words that their door frames, decks, and walls are “square”). It’s useful to find the diagonal of a television set (which is how those contraptions are measured for some reason), if you only know its length and width. And it’s the basis of a great deal of additional math discovery, like the distance formula and various area formulas.

    It’s a big, honkin’ deal. And in some ways, we’re lucky it survived the secrecy of the Pythagoreans. Pythagoras wrote nothing down. (If tin foil had been invented, he might have been wearing a hat of the stuff.) But despite its closed society, this cult of nutty mathematicians and philosophers is considered one of the most important influences in all of history.

    What do you remember of Pythagoras from your high school geometry class? Have you used the Pythagorean theorem in your everyday or work life? If so, how?

  • The Number that Changed the World: History of numbers, part 3

    The Number that Changed the World: History of numbers, part 3

    Things were moving right along in the invention and use of number systems. The Sumerians started things off sometime during the 3rd millenium, when their budding commerce system helped them invent the first set of written numbers. The Egyptians systematically engineered a formal base-ten system that morphed from hieroglyphics to the much-easier-to-write hieratic numbers.

    But something was missing. Something really important — and really, really small.

    The Greeks advanced geometry considerably. (More on that next week.) But in the Roman Empire, mathematical invention and discovery virtually stopped — with the exception of Roman numerals. These were widely used throughout Europe in the 1st millenium, but like the number systems that came before, it was positional and did not use place value.

    But why weren’t these systems using place value? It all comes down to zero. Up to this point, this seemingly inconsequential number was absent.

    There is some debate about this, of course. Some historians assert that sometime around 350 B.C. Babylonian scribes used a modified symbol to represent zero, which astronomers found useful to use this placeholder in their notations. And on the other side of the world, the Mayans used a symbol for zero in their “Long Count” calendar. But there is no evidence that zero was used for calculations.

    Along came the Indian mathematician and astronomer, Brahmagupta, who was the first person in recorded history to use a symbol for zero in calculations. But India’s relationship with zero started well before that.

    In ancient and medieval India, mathematical works were composed in Sanskrit, which were easily memorized because they were written in verse. (I am not kidding.) These beautiful sutras were passed down orally and in written form through the centuries. Thus the idea of zero — or śūnya (void), kah (sky), ākāśa (space) and bindu (dot) — was first introduced with words. Eventually, an actual dot or open circle replaced these words, as Indians began using symbols to represent numbers.

    Brahmagupta used zero in arithmetic — adding, subtracting, multiplying and even dividing using the all-important number. All of that was well and good, except for division. It wasn’t until Sir Isaac Newton and his German counterpart Gottfried Wilhelm Leibniz came along that it was established that dividing by zero is undefined.

    But really, the big deal here was not doing arithmetic. Nope, it was place value. This is so important that we all take it for granted. It’s the difference between $65 and $605 or the difference between 0.02% and 2%. See, zero isn’t just a place holder — in our number system it can represent a place value. You think math is hard now? Imagine doing calculations with Roman Numerals! Without place value and our humble zero, this work is exceedingly difficult.

    This is a relatively new idea in the scheme of things. Almost 3,000 years had passed, since the Sumerians developed the first written number. Zero was introduced in India sometime around 400 A.D., though it didn’t show up in a text until around 600 A.D. Through trade routes, zero began showing up in the Middle East and China, but it took a very long time — the middle of the 12th century! — for Europeans to begin using zero and place value.

    And that’s pretty much it — the very long history of our current number system, without which most other major discoveries, like calculus, trigonometry or geometry, could not be developed.

    Of course there is much, much more to say about numbers themselves. For example, they’re arranged in a system based on their particular characteristics, kind of like the way we categorize animals or plants. Positive whole numbers are called natural numbers;positive and negative numbers are called integers; fractions and terminal decimals are rational numbers, and so on. This is connected to a fascinating (to me) branch of mathematics, called abstract algebra. But that’s a story for another day.

    What surprised you about the history of numbers? And how about that zero? Ask your questions or make comments here.

  • Count Like an Egyptian: A history of numbers, part 2

    So the Sumerian system of numbers — as far as we know, the first in the world — came into being rather naturally and out of necessity. But the Egyptians took things one step further, and they did it very systematically. Priests and scribes invented a system of numbers that included tally marks and hieroglyphics. In doing so, they developed a base-ten system featuring different symbols for different numbers.

    The Egyptian people were very fortunate. With few neighbors, they didn’t have spend time worrying about war or defending themselves from attack. They also lived in a very fertile area, making agriculture less troublesome than it might have been. All of this freed up their time to do things like develop a numerical system and make big advances in mathematics. (You know, the ordinary stuff we do when we live in peace and have lots of food and water.)

    Hieroglyphics could be used to express a wide variety of numerical values — all the way to one million! The symbol for one was a tally mark, so four tally marks represented 4, and so on. But 10 was expressed as a horseshoe shape and 100 a coiled rope. A little tiny prisoner begging for forgiveness was the hieroglyphic for 1,000,000. (I’d love to know the story behind that one.)

    Yes, I drew these myself. No, I am not an artist or an ancient Egyptian. But you probably knew that.

    While these characters could be arranged to represent an almost endless set of whole numbers and even fractions, the Egyptians were missing a critical numeral: zero. This meant that with all of their advances, Egyptian numbers had no place value system.

    All of this allowed the Egyptians to take huge steps in the development of arithmetic, including the four basic operations — addition, subtraction, multiplication and division — and using numbers for measurement. Without these advances, we would have no great pyramids.

    As the ancient society moved to the much more portable and easier-to-use papyrus and ink to record words and numerals, hieroglyphs gave way to hieratic numerals. These are more akin to brush strokes, and allowed the Egyptians to write larger numbers with fewer symbols. It’s pretty easy to see that this sped things up quite a bit.

    On Friday, we’ll visit ancient India, where the most amazing creation/discovery revolutionized the system of numbers. (Seriously, this was a big, big deal!)

    Can you imagine having to use hieroglyphics to balance your checkbook? If you have questions about the Egyptian system of numbers, ask them in the comments section.

  • The World’s First Numbers

    The World’s First Numbers

    When the world began 4.54 billion years ago, it didn’t come with numbers. They didn’t appear with the dinosaurs or first mammals or even the first homo sapiens. That’s because numbers were createdas a way to describe the world. And that is a big-honkin’ deal.

    Think about it: Numbers make our daily lives much, much easier — from knowing how much time you have before you must get out of bed to setting the table with the correct number of plates at dinner time. You simply cannot get through your day with encountering numbers — not just once, or twice or a dozen times, butthousands and thousands of times. (Do you see what I did there?)

    So if numbers haven’t been with us since the beginning of time, where the heck did they come from? Well, that history is pretty challenging to tell, but this week I’ll give you a little overview, starting with the Sumerians.

    Sumer was a region of Mesopotamia, roughly where Iraq is today. The Sumerians made so many discoveries and inventions that the region is often called the Cradle of Civilization. Before this time, people used tallies to count things and geometric figures showed up in art and decoration. But these representations were not really mathematical, and they weren’t used widely and systematically.

    It was the rise of cities that really set things in motion. As Sumerians developed commerce, they developed one of the world’s first system of numbers. To keep things fair, people needed a way to keep track of sales and barters. First, they counted on tallies. But there were no numerals associated with the hatch-marks they were using to show the number of sheep in a herd or eggs in a basket.

    (Here is a good time to underscore the difference between a number and a numeral. It’s a teeny-tiny distinction, but an important one. A numeral is a character or symbol that describes a number. A number is the actual value of the numeral. So 3 is a numeral. But if I say I have three kittens, well, I’m talking about the number of sweet, little, purring balls of fur curled up on my lap.)

    The Sumerians took things a little further with their whole commerce thing — they started systematically subtracting. See, if I had five goats, I’d be given five special tokens. If I sold off one of them, I’d have to give back one of my tokens. To keep track of this natural back-and-forth of trading and selling, merchants began to keep clay tablets of tallies that showed not only the number of baskets or cows or whatever they had at any moment, but a sales history.

    And so, arithmetic was born. Oh, and writing. Ta-da! (Those Sumerians were smart and resourceful.)

    Now, as this process developed over time, the Sumerians settled on a base 60 system of numbers. We have a base ten system, which in very, very basic terms means two things: we have ten basic numerals that are used to write all other numbers (0-9) and our numbers are described in sets of 10 or multiples of ten.

    But not the Sumerians. They liked 60, a number that should be very familiar to us, since it’s the basis of our system of time. That’s probably no accident, right?

    Eventually, the Sumerians developed their own set of numerals, called cuneiform numbers. They looked like the inscription in the photo above.

    So there you have it. The world’s first numerals — near as we can tell. Next up: The Egyptians.

    (Disclaimer: I’ll be the first to admit that this history is a lot more complex than can be described here. And I’d bet my last dollar that there are a few historians out there who disagree with the generally accepted history of Sumerians and mathematics. There’s so much we don’t know about his ancient history.)

    Got questions about the Sumerians or the development of numbers? Ask them below. Was anything in this story surprising or particularly interesting? I’d love to hear what you think.

  • A Mathematical Time Machine

    A Mathematical Time Machine

    Was mathematics invented or discovered?

    (I’ll give you a second or two to really think about that.)

    Most non-mathematicians have never really given that question much thought. Math has just always been there. An isosceles triangle has always had two congruent sides, and 3 + 8 has always equalled 11. But the reality is this: since the beginning time, human kind has struggled to find ways to describe its world. And one important outcome of this struggle is what I call the language of mathematics. Whether math was invented or discovered, the people involved were fascinating and scary and funny and sometimes sad. And that’s why I’ve decided to devote the remainder of November to the history of mathematics. Here are a few of the stories I hope to share with you.

    1. There was the 1st Century Roman who, while taking a bath, figured out the idea of displacement. What did he do? Well, naturally, he shouted “Eureka!” and went running down the streets in his birthday suit. (Or so the story goes.)

    2. Then there was the 5th century, mystical cult that demanded complete loyal and secrecy from its members. And by the way one of its members discovered one of the most useful and important facts about right triangles.

    3. In the 1600s, the surrogates of two mathematicians — one in England and the other in Germany — held heated debates over who had actually invented (or discovered) calculus.

    4. A child prodigy born in 1777 was confounding his teachers and managing his father’s business accounts at the tender age of five. He went on to make a staggering number of contributions in number theory, statistics and algebra, including normal distribution and the bell curve. He also apparently chose work over being at his wife’s deathbed, saying, “Ask her to wait a moment; I’m almost done.”

    5.  A girl (gasp!) made significant contributions to the fields of abstract algebra and physics in 19th and 20th century Germany.

    6. After cracking World War II German codes for the Brits and playing a major role in the birth of computer science, one fellow was arrested for the crime of homosexuality, chose chemical castration over prison and is said to have killed himself by cyanide poisoning at the age of 42.

    Clearly, the history of mathematics is full of comedy and tragedy. The stories weave in and out of major world developments and the histories of other sciences. At the least, some of these stories are entertaining. Others help us make connections between ideas that lead to our own personal revelations. Still others remind us that while these contributions have provided the underpinning of how we understand our world today, the people behind them were just that — people.

    So climb aboard this mathematical time machine. I’m still trying to decide whether to take it chronologically or by subject or perhaps even with a more random approach. Let’s just see what happens, shall we?

    Do you have a question about the history of mathematics? If so, please share it in the comments section. I’m happy to take suggestions of topics I should consider.

  • Exit Polling: A statistics refresher

    Exit Polling: A statistics refresher

    Most of you are probably sick to death of Political campaign polls. But these numbers have become a mainstay of the American political process. In other words, we’re stuck with them, so you might as well get used to it — or at least understand the process as well as you can.

    Last Friday, I wrote about how the national polls really don’t matter. That’s because our presidential elections depend on the Electoral College. We certainly don’t want to see one candidate win the popular vote, while the other wins the Electoral College, but it’s those electoral votes that really matter.

    Still, polls matter too. I know, I know. Statistics can be created to support *any* cause or person. And that’s true. (Mark Twain popularized the saying, “There are lies, damned lies, and statistics.”) But good statistics are good statistics. These results are only as reliable as the process that created them.

    But what is that process? If it’s been a while since you took a stats course, here’s a quick refresher. You can put it to use tomorrow when the media uses exit polls to predict election and referendum results before the polls close.

    [laurabooks]

    Random Sampling

    If I wanted to know how my neighbors were voting in this year’s election, I could simply ask each of them. But surveying the population of an entire state — or all of the more than 200 million eligible voters in the U.S. — is downright impossible. So political pollsters depend on a tried-and-true method of gathering reliable information: random sampling.

    A random sample does give a good snapshot of a population — but it may seem a bit mysterious. There are two obvious parts: random and sample.

    The amazing thing about a sample is this: when it’s done properly (and I’ll get to that in a minute) the sample does accurately represent the entire population. The most common analogy is the basic blood draw. I’ve got a wonky thyroid, so several times a year, I need to check to see that my medication is keeping me healthy, which is determined by a quick look at my blood. Does the phlebotomist take all of my blood? Nope. Just a sample is enough to make the diagnosis.

    The same thing is true with population samples. And in fact, there’s a magic number that works well enough for most situations: 1,000. (This is probably the hardest thing to believe, but it’s true!) For the most part, researchers are happy with a 95% confidence interval and a ±3% margin of error. This means that the results can be trusted with 95% accuracy, but only outside ±3% of the results. (More on that later.) According to the math, to reach this confidence level, only 1,000 respondents are necessary.

    So we’re looking at surveying at least 1,000 people, right? But it’s not good enough to go door-to-door in one neighborhood to find these people. The next important feature is randomness.

    If you put your hand in a jar full of marbles and pull one marble out, you’ve randomly selected that marble. That’s the task that pollsters have when choosing people to respond to their questions. And it’s not as hard as you might think.

    Let’s take exit polls on Election Day. These are short surveys conducted at the voting polls themselves. As people exit the polling place, pollsters stop certain voters to ask a series of questions. The answers to these questions can predict how the election will end up and what influenced voters to vote a certain way.

    The enemy of good polling is homogeneity. If only senior citizens who live in wealthy areas of a state are polled, well, the results will not be reliable. But randomness irons all of this out.

    First, the polling place must be random. Imagine writing down the locations of all of the polling places in your state on little strips of paper. Then put all of these papers into a bowl, reach in and choose one. That’s the basic process, though this is done with computer programs now.

    Then the polling times must be well represented. If a pollster only surveys people who voted in the morning, the results could be skewed to people who vote on their way home from their night-shift or don’t work at all or who are early risers, right? So, care is made to survey people at all times of the day.

    And finally, it’s important to randomly select people to interview. Most often, this can be done by simply approaching every third voter who exits the polling place (or every other voter or every fifth voter; you get my drift).

    Questions

    But the questions being asked — or I should say the ways in which the questions are asked — are at least as important. These should not be “leading questions,” or queries that might prompt a particular response. Here’s an example:

    Same-sex marriage is threatening to undermine religious liberty in our country. How do you plan to vote on Question 6, which legalizes same-sex marriage in the state?

    (It’s easier to write a leading question asking for intent rather than a leading exit poll.)

    Questions must be worded so that they illicit the most reliable responses. When they are confused or leading, the results cannot be trusted. Simplicity is almost always the best policy here.

    Interpreting the Data

    It’s not enough to just collect information. No survey results are 100 percent reliable 100 percent of the time. In fact, there are “disclaimers” for every single survey result. First of all, there’s a confidence level, which is generally 95%. This means exactly what you might think: Based on the sample size, we can be 95 percent confident that the results are accurate. Specifically, a 95% confidence interval covers 95 percent of the normal (or bell-shaped) curve.

    The larger the random sample, the greater the confidence level or interval. The smaller the sample, the smaller the confidence level or interval. And the same is true for the margin of error.

    But why 95%? The answer has to do with standard deviation or how much variation (deviation) there is from the mean or average of the data. When the data is normalized (or follows the normal or bell curve), 95% is plus or minus two standard deviations from the mean.

    This isn’t the same thing as the margin of error, which represents the range of possibly incorrect results.

    Let’s say exit polls show that Governor Romney is leading President Obama in Ohio by 2.5 percentage points. If the margin of error is 3%, Romney’s lead is within the margin of error. And therefore, the results are really a statistical tie. However, if he’s leading by 8 percentage points, it’s more likely the results are showing a true majority.

    Of course, all of that depends — heavily — on the sampling and questions. If either or both of those are suspect, it doesn’t matter what the polling shows. We cannot trust the numbers. Unfortunately, we often don’t know how the samples were created or the questions were asked. Reliable statistics will include that information somewhere. And of course, you should only trust stats from sources that you can trust.

    Summary

    In short, there are three critical numbers in the most reliable survey results:

    • 1,000 (sample size)
    • 95% (confidence interval or level)
    • ±3% (margin of error)

    Look for these in the exit polling you hear about tomorrow. Compare the exit polls with the actual election results. Which polls turned out to be most reliable?

    I’m not a statistician, but in my math books, you’ll learn math that you can apply to your everyday lives and help you understand polls and other such things.

    P.S. I hope every single one of my U.S. readers (who are registered voters) will participate in our democratic process. Please don’t throw away your right to elect the people who make decisions on your behalf. VOTE!

  • Why National Polls Don’t Matter: Electoral college math

    Why National Polls Don’t Matter: Electoral college math

    This post makes me scared. Not because the math is challenging or because I’m worried about the election. I’m afraid of looking partisan or being accused of ideology. (It’s happened before!) But I can’t avoid election math any longer, so I’ve decided to take the plunge — today and Monday — into these shark-infested waters, trusting that my readers (and new guests) will put away their partisan differences if only for a few hours. Do for the sake of the math.

    There’s no denying the math that goes on in elections. There are polls, ad buys, the number of minutes each candidate has spoken during debates — and yes, the electoral college. Whatever you may think of our dear map, it is how elections are decided in this country — for the most part.

    There’s no reason to expect a repeat of Election Day (and the weeks following) 2000 this year. So I thought it would be a good idea to review the electoral map — from a mathematical perspective — so that we can better understand its power. First some history.

    During the Constitutional Convention in 1787, the founding fathers quickly rejected a number of ways to select the country’s president: having Congress choose the president, having state legislatures choose and direct popular vote. The first two ideas were tossed based on fears of an imbalance of power — giving Congress or the states too much control. They also worried that a direct popular vote would be negatively influenced by the lack of consistent communication. In other words, without information about out-of-state candidates, voters would simply choose the candidate from their own states. And then there was the very real fear that a candidate without a sufficient majority would not be able to govern the entire nation.

    So, these fine men drew up a fourth option: a College of Electors. The first design, which is outlined in Article II of the U.S. Constitution, was pitched after four Presidential elections, after political parties emerged. Much of the original system remained, but the 12th Amendment to the Constitution instituted a few changes to reflect the country’s new party system. Here what the electoral college looks like today:

    • The Electoral College consists of 538 electors.
    • Each state is allotted the same number of electors as it has Congress members (Senators and Representatives)
    • Therefore, representation in the Electoral College is dependent on each state’s population. More populous states have more electoral votes; less populous states have fewer electoral votes.
    • The 23rd Amendment to the Constitution gives the District of Columbia 3 electoral votes, event though it is not a state.
    • Each state has its own laws governing how electors are selected. Generally, electors are selected by the political parties themselves.
    • Most states have a “winner takes all” system, which means that the candidate with the majority of the direct popular votes in the state gets all of the electoral votes.
    • However, Maine and Nebraska have a proportional system, which means the electoral votes can be divided between candidates.

    Whew!

    Some basic calculations allow the media and election officials and the candidates themselves to make really good predictions on election night in most situations. But the electors don’t officially cast their votes until the first Monday after the second Wednesday in December. Then, on January 6 of the following calendar year in a joint session of Congress, the electoral votes are counted, and the President and Vice-President are declared. (Got all that?) Almost always, though, the losing candidate concedes the election on election night or the next day, making the electoral vote and counting a mere formality.

    The thing that makes this complex is that each state has a different number of electoral votes. In order to win the presidential election, a candidate must secure at least 270 electoral votes. And that’s why you’re probably seeing a red and blue (and purple?) map in your newspaper, on television and online.

    In my state, there is no question which candidate will take all of the electoral votes. Maryland has been staunchly Democratic for several decades. And there’s no mystery about Texas, which is about as red as a state can get. But if it were a contest between Maryland’s and Texas’ electoral votes, Governor Romney would win. That’s because Texas has 38 electoral votes, while Maryland has 10.

    Right now, there are lots and lots of predictions out there concerning how the electoral college will vote. (Personally, I think Nate Silver0 of the New York Times is the most reliable source. Dude has a killer math brain, correctly predicting the electoral college outcomes in 49 of the 50 states in the 2008 election. In that same election, he correctly predicted all of the 35 Senate races.) But there’s little doubt about many of the states. A few swing states will certainly claim this election: Colorado, Florida, Iowa, New Hampshire, Ohio, Virginia and Wisconsin. Mathematically speaking, we’re talking about 89 votes:

    • Colorado: 9
    • Florida: 29
    • Iowa: 6
    • New Hampshire: 4
    • Ohio: 18
    • Virginia: 13
    • Wisconsin: 10

    Now out of those, which states would you guess the candidates really want to win? Yep, the ones with the highest number of electoral votes. So to them, the most important states in these last days of the campaign are Florida, Ohio and Virginia. (Where do you draw the line? I chose more than 10 electoral votes.)

    If you live in one of these three states, you are acutely aware of this fact. Unless you don’t have a television set or listen to the radio or have a (really) unlisted phone number.

    So what does this mean? Right now, it means that President Obama is likely to win the election. There are scenarios that show the opposite outcome — and there are even a few that produce a tie. However, most political analysis says that it’s Obama’s to lose at this point. This is despite the fact that most polls show the popular vote at a statistical dead heat (in other words, any lead by either candidate is well within the margins of error).

    Because our founding fathers made a decision that we wouldn’t elect our presidents with a direct popular vote. What matters in these last days are the popular votes in the swing states — most importantly Florida, Ohio and Virginia — though there are scenarios that give Mitt Romney the edge without winning all of the swing states.

    If you are a complete geek about election numbers, do visit Silver’s FiveThirtyEight blog at the New York Times. His math is good, regardless of what some conservative pundits have claimed in recent weeks.

    EDITOR’S UPDATE: Sam Wang of the Princeton Election Consortium also has great analysis. Hurricane Sandy has messed with his servers, so the site looks pretty rudimentary, but he is updating his site regularly. It’s pretty cool to compare Silver’s and Wang’s conclusions — especially on a day-by-day basis.

    I also highly recommend a really slick interactive tool put out by the New York Times. It graphically illustrates ways in which the electoral votes could swing the election in either way, based solely on the math. Unlike Silver’s blog, this section does not offer a prediction of who will will win, but describes the various scenarios for each candidate.

    Whatever you think of the candidates and the issues, vote. No matter what, vote. Our votes — even outside swing states — matter. It’s our responsibility as U.S. citizens to declare our preferences. And in my mind, if you don’t vote, you can’t complain.

    Coming on Monday… a look at the polls themselves. What makes a good poll? How should we average folks interpret polls? Can they really tell us what’s going on?

    What are your thoughts on the math of the electoral college? (I get it. These discussions can get heated. Please be respectful in your comments. I will not approve or will delete any comments that I deem outside the bounds of civility. Thank you for playing nice.)

  • Welcome Sandy! Meteorology and math

    Welcome Sandy! Meteorology and math

    Things are looking bad for those of us in Hurricane Sandy’s path. Like most of my neighbors I spent the weekend cleaning up the yard and cleaning out the local grocery stores. But one thing is certain: In a short while, my electricity will be out, and I can expect to be living like Laura Ingalls Wilder in the city for at least a few days. That means no computer, no internet.

    So for part of this week, at least, I’m bringing you some topical (not tropical!) highlights from posts past. First up is my interview with on-air meteorologist, Tony Pann. Here’s how he uses math in his work. (I’m betting he’s pretty darned busy this morning!)

    Math at Work Monday: Tony the on-air meteorologist

    Tony Pann is an on-air meteorologist for WBAL-TV 11 in Baltimore, Maryland.

    I have been a television meteorologist for 22 years. Since 2009, I’ve been working as part of the morning team at WBAL TV.

    When do you use basic math in your job?

    I use math everyday! The computer models that we use to forecast the weather, are based on very complicated formulas derived from fluid dynamics. The atmosphere acts very much like a body of water, so the same mathematics can be applied to both. Each day, over a dozen different computer models are run predicting the state of the atmosphere at different time frames. An initial set of data is entered at a specific starting time, then the model shows us it’s interpretation of what the state of the atmosphere will be at certain time intervals. For example, the data might be entered at 7 a.m., then the model will predict the temperature, wind speed, and barometric pressure at 10 a.m., 1 p.m. and 4 p.m. Some of these models are short range, and only extend out to 48 hours, while others go all the way out to 365 hours from the starting point!

    So let’s say there are 13 models that do this same thing each and every day, two or three times a day. It’s my job as a meteorologist to interpret all of that data, and translate it into the very understandable and reliable seven-day forecast that you see on TV. With so much data out there, the intuition and experience of the forecaster is very important. Since each model takes in the same starting data, but is run on a different formula, they all come up with different answers. For example, one model might say the high temp for today is going to be 45 and another could say 50. Or one could predict 6 inches of snow and the other says 1 inch. It’s my job to decide which one is right and why.

    Sometimes I don’t trust any of them, and I’ll do a quick calculation on my own.  Here’s an equation that I can use to calculate the high temp for the day by hand:

    I then go on TV, and try and explain it all in an interesting manor — at least that’s the goal.

    Did you have to learn new skills in order to do the math you use in your job? 

    In order to get a degree in meteorology, you actually have to learn all of the math that the computers are doing to give us those answers. It’s not easy! By the time we are finished, we’re just a class or two short of having a minor in mathematics. It’s great to know what the computers are doing, but I’m glad we don’t have to work it out by hand anymore. If not for the wonderful training in the world of mathematics, I most certainly would not be doing this job.

    Do you have questions for Tony? Ask them in the comments section, and I’ll let him know to peek in! He’ll be a bit busy for a while, so be patient!