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April 25 - July 3, 2019
Looking at numbers as groups of rocks may seem unusual, but actually it’s as old as math itself. The word “calculate” reflects that legacy—it comes from the Latin word calculus, meaning a pebble used for counting. To enjoy working with numbers you don’t have to be Einstein (German for “one stone”), but it might help to have rocks in your head.
At a deeper level, however, subtraction raises a much more disturbing issue, one that never arises with addition. Subtraction can generate negative numbers.
people have concocted all sorts of funny little mental strategies to sidestep the dreaded negative sign. On mutual fund statements, losses (negative numbers) are printed in red or nestled in parentheses with nary a negative sign to be found. The history books tell us that Julius Caesar was born in 100 B.C., not –100. The subterranean levels in a parking garage often have designations like B1 and B2. Temperatures are one of the few exceptions: folks do say, especially here in Ithaca, New York, that it’s –5 degrees outside, though even then, many prefer to say 5 below zero. There’s something
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Perhaps the most unsettling thing is that a negative times a negative is a positive.
But if you’re a hard-boiled pragmatist, you may be wondering if these abstractions have any parallels in the real world. Admittedly, life sometimes seems to play by different rules.
Perhaps the most familiar parallel occurs in the social and political realms as summed up by the adage “The enemy of my enemy is my friend.”
Even more unbalanced is a triangle with a single negative relationship.
This causes psychological stress all around. To restore balance, either Alice and Bob have to reconcile or Carol has to choose a side.
In a balanced triangle, the sign of the product of any two sides, positive or negative, always agrees with the sign of the third. In unbalanced triangles, this pattern is broken.
in a close-knit network where everyone knows everyone, what’s the most stable state?
Perhaps even more surprisingly, these polarized states are the only states as stable as nirvana.
The point is not that this theory is powerfully predictive. It isn’t. It’s too simple to account for all the subtleties of geopolitical dynamics.
By sorting the meaningful from the generic, the arithmetic of negative numbers can help us see where the real puzzles lie.
EVERY DECADE OR SO a new approach to teaching math comes along and creates fresh opportunities for parents to feel inadequate. Back in the 1960s, my parents were flabbergasted by their inability to help me with my second-grade homework. They’d never heard of base 3 or Venn diagrams.
It’s the same convention as in Lionel Richie’s immortal lyrics “She’s once, twice, three times a lady.” (“She’s a lady times three” would never have been a hit.)
Is this commutative law of multiplication, a × b = b × a, really so obvious? I remember being surprised by it as a child; maybe you were too.
My point is that if you regard multiplication as being synonymous with repeated counting by a certain number (or, in other words, with repeated addition), the commutative law isn’t transparent. But it becomes more intuitive if you conceive of multiplication visually.
Yet strangely enough, in many real-world situations, especially where money is concerned, people seem to forget the commutative law, or don’t realize it applies. Let me give you two examples.
think multiplicatively, not additively. Applying an 8 percent tax followed by a 20 percent discount amounts to multiplying the sticker price by 1.08 and then multiplying that result by 0.80. Switching the order of tax and discount reverses the multiplication, but since 1.08 × 0.80 = 0.80 × 1.08, the final price comes out the same.
if you have a pile of money to invest and you have to pay taxes on it at some point, is it better to take the tax bite at the beginning of the investment period, or at the end? Once again, the commutative law shows it’s a wash, all other things being equal (which, sadly, they often aren’t).
Please don’t mistake this mathematical remark for financial advice. Anyone facing these decisions in real life needs to be aware of many complications that muddy the waters:
Maybe we’re wired to doubt the commutative law because in daily life, it usually matters what you do first. You can’t have your cake and eat it too. And when taking off your shoes and socks, you’ve got to get the sequencing right.
As a quantum physicist he would have been acutely aware that at the deepest level, nature disobeys the commutative law.
you might suppose irrationality is rare. On the contrary, it is typical. In a certain sense that can be made precise, almost all decimals are irrational.
Roman numerals may look impressive, but they’re hard to read and cumbersome to use.
The Babylonians were not nearly as attached to their fingers. Their numeral system was based on 60—a clear sign of their impeccable taste, for 60 is an exceptionally pleasant number. Its beauty is intrinsic and has nothing to do with human appendages.
Because of its promiscuous divisibility, 60 is much more congenial than 10 for any sort of calculation or measurement that involves cutting things into equal parts.
When we divide an hour into 60 minutes, or a minute into 60 seconds, or a full circle into 360 degrees, we’re channeling the sages of ancient Babylon.
With place-value systems, you can program a machine to do arithmetic. From the early days of mechanical calculators to the supercomputers of today, the automation of arithmetic was made possible by the beautiful idea of place value.
These changing numbers are called variables, and they are what truly distinguishes algebra from arithmetic.
The formulas in question might express elegant patterns about numbers for their own sake. This is where algebra meets art. Or they might express relationships between numbers in the real world, as they do in the laws of nature for falling objects or planetary orbits or genetic frequencies in a population. This is where algebra meets science.
a formula for the socially acceptable age difference in a romance. According to some sites on the Internet, if your age is x, polite society will disapprove if you date someone younger than x/2 + 7.
Historically, this step was the most painful of all. The square root of –1 still goes by the demeaning name of i, for “imaginary.”
It’s true that i can’t be found anywhere on the number line. In that respect it’s much stranger than zero, negative numbers, fractions, or even irrational numbers, all of which—weird as they are—still have their places in line.
Complex numbers are magnificent, the pinnacle of number systems. They enjoy all the same properties as real numbers—you can add and subtract them, multiply and divide them—but they are better than real numbers because they always have roots.
a grand statement called the fundamental theorem of algebra says that the roots of any polynomial are always complex numbers.
So multiplying by i produces a rotation counterclockwise by a quarter turn. It takes an arrow of length 3 pointing east and changes it into a new arrow of the same length but now pointing north.
The 90-degree rotation property also sheds light on what i² = –1 really means. If we multiply a positive number by i², the corresponding arrow rotates 180 degrees, flipping from east to west, because the two 90-degree rotations (one for each factor of i) combine to make a 180-degree rotation.
I’m not sure, but it seems to be a case of faulty pattern recognition.
The silver lining is that even wrong answers can be educational . . . as long as you realize they’re wrong.
Keith Devlin raises an interesting criticism in his essay “The problem with word problems.” His point is that these problems typically assume you understand the rules of the game and agree to play by them, even though they’re often artificial, sometimes absurdly so.
It’s only when you understand what the quadratic formula is trying to do that you can begin to appreciate its inner beauty.
Tools transform things. So do functions. In fact, mathematicians often refer to them as transformations because of this.
Just as every office worker needs both a stapler and a staple remover, every mathematician needs exponential functions and logarithms. They’re inverses.
Notice something magical here: as the numbers inside the logarithms grew multiplicatively, increasing tenfold each time from 100 to 1,000 to 10,000, their logarithms grew additively, increasing from 2 to 3 to 4.
In every place where they arise, from the Richter scale for earthquake magnitudes to pH measures of acidity, logarithms make wonderful compressors.
(By the way, this is where geometry started, historically—in problems of land measurement, or measuring the earth: geo = “earth,” and metry = “measurement.”)
This way of looking at the Pythagorean theorem makes it seem like a statement about lengths. But traditionally it was viewed as a statement about areas.
For thousands of years, this marvelous fact has been expressed in a diagram, an iconic mnemonic of dancing squares:
Euclid had begun with the definitions, postulates, and self-evident truths of geometry—the axioms—and from them erected an edifice of propositions and demonstrations, one truth linked to the next by unassailable logic.
