Tag Archives: mathematics

Introduction to Mathematics

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If you have 3 apple pies and 19 people, how should you slice the pies so that each person gets an equal share? Each person should get 3/19 ≈ 0.1579 pies, but if you make each pie into 6 slices, that’s only 18 slices for 19 people. You have to slice each pie into 6 and 1/3 slices, with each slice being equal except the 1/3 being smaller, and then give the three 1/3 slices to the 19th person.

What if you have 1 pie and the Half-Blood Prince from Harry Potter and the Half-Blood Prince is the only person dining? Then you have 2 pies per person, assuming he’s half a person. In the expression 1/(1/2), move the denominator to the numerator and flip the ex-denominator, making the reciprocal and thereby converting division to multiplication. 1/(1/2) becomes 1*(2/1) which is just 2, because any number without a denominator has a denominator of 1.

Your neighbor lends you $8000 at 3.75% interest compounded annually, with no payments being required for 25 years and the full balance and interest being required to be repaid at that time. What is the payment? $8000*1.0375^25 ≈ $8000*2.5102 = $20,081.34.

What if you want to make a graph of the increasing amount owed on the Cartesian coordinate system where y is the number of years and x is the dollar amount in thousands? Use the equation y = 8*(1.0375^x).

PROBLEM: Your truck gets 15 miles per gallon in the city and 19 mpg on the highway. Your destination is 38 miles away as the crow flies, 42 miles away through the cities, and 49 miles away if you take Interstate-95. However, taking the interstate requires 5 miles of city driving. How many gallons of gas will be used on each route, which one uses the least gas, and how much does each trip cost if gas is $4.15 per gallon?

SOLUTION: First, note that there are only two routes: 42 miles through the cities and 5 miles through the cities PLUS 44 miles on I-95. Then, do this:
42/15 = 2.8 gallons –> 2.8*4.15 = $11.62 in gas
5/15 + 44/19 ≈ 2.65 gallons –> 2.65*4.15 = $10.99 in gas

NOTE that with the variables provided, the longer route is in fact the cheaper route. However, in reality there would be an inordinate number of variables, such as your engine’s efficiency, time of day, traffic patterns, traffic lights, and unforeseen events. For example, traveling an extra 7 miles may necessitate an earlier oil change or some other maintenance. If you get into an accident on the interstate at 80 miles per hour, you might be instantly killed rather than being only wounded if crashing at a much lower speed in a city. Highway driving vs. city driving requires different mental concentration and one may appeal over the other depending on your upbringing and psychological makeup. In the city, you are more likely to be pulled over by policemen and ticketed. The road surface may be smoother on the interstate, which will prevent your tires from wearing down as quickly. If you break down on the interstate, you may be stranded if you don’t have a cell phone. All math problems simplify.

PROBLEM: The 2012 Presidential election is coming up, and observing that Ron Paul (R) and Barack Obama (D) have won the primaries, General Electric is deciding how much to donate to each candidate’s campaign. GE’s budget is $4,000,000, and they estimate Paul has an 8% chance of winning and Obama has a 92% chance of winning. GE estimates donations to Paul have a political worth three times greater than donations to Obama to secure support from the Constitutionalist movement. How much will GE donate to each campaign?

SOLUTION:
3*0.08 = 0.24
1*0.92 = 0.92
0.24+0.92 = 1.16
4,000,000/1.16 = 3,448,275.862
3,448,275.862 * 0.24 = $827,586.21 to Ron Paul
3,448,275.862 * 0.92 = $3,172,413.79 to Barack Obama

PROBLEM: Steve Jobs is developing the iPad 3 for release November 28, 2012 and must choose between Foxconn’s 64GB isolinear-NAND flash chip and Foxtrot’s 59GB neo-EEPROM flash chip. Foxconn’s chip costs $28.78 and has a five-year failure rate of 2.8%. Foxtrot’s chip costs $26.55 and has a five-year failure rate of 2.1%. Both chips are functionally identical in form factor, read/write speed, power consumption, resiliency, and failure potential, both technologies are equally reliable, both companies use slave labor, and both companies are of equal capacity, reputability, and geographic location.

Apple estimates the market value of an extra 5GB (64GB vs. 59GB) of storage capacity to be $14.50 per unit, and estimates that each five-year failure will have an effective cost of $895.88 on Apple’s image, future sales, and support network. Which chip should Steve Jobs choose?

SOLUTION:
($28.78 – 14.50) = $14.28 effective cost per 64GB Foxconn chip
$26.55 = $26.55 relative cost 59GB per Foxtrot chip
$895.88 * 0.028 = $25.08 failure cost per 64GB Foxconn chip
$895.88 * 0.021 = $18.81 failure cost per 59GB Foxtrot chip
$14.28 + 25.08 = $39.36 total cost per 64GB Foxconn chip
$26.55 + 18.81 = $45.36 total cost per 59GB Foxconn chip

Jobs should choose the 64GB Foxconn chip, even though it is 0.7% more likely to fail in the first five years, because it’s easier to market a 64GB device than a 59GB device so the 64GB Foxconn chip has an effective cost of $6.00 less than the 59GB Foxtrot chip, given the variables.

PROBLEM: General Motors Company is developing a new type of engine that improves fuel economy by 100%, but has discovered that 0.000097% of the engines blow up when reaching a speed of 88 miles per hour, instantly killing everyone in the vehicle and seriously wounding everyone in a 100 foot radius. GMC is considering including this engine in its new SUV, the ThinkNeighbor Plus, which will get 42 highway miles per gallon instead of the standard 21 highway mpg, will be manufactured in a quantity of 5 million, and will sell for $38,000. GMC estimates only 0.5% of ThinkNeighbor Pluses will ever reach a speed of 88 miles per hour, and estimates the Public Relations costs of each explosion will be $8 million. The U.S. State Department has pledged to blame the explosions on domestic terrorist attacks, but only to a limit of three. Should GMC manufacture the ThinkNeighbor Plus?

SOLUTION:
NO. Since the question is “should GMC manufacture the ThinkNeighbor Plus?,” it isn’t even a math question because “should” is completely subjective.

More next time.

A Series of Near-Hits

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In life, it’s easy to go through a process called a series of near-hits, where you get close to the mark many times in a row without ever succeeding. An invisible wall stops you from reaching the goal, but you expend an increasing amount of effort for ever-reducing gains.

Sometimes, this is the story of a person’s whole life: a series of failures which were almost successes. “Failure,” of course, is a relative term. Perhaps he succeeded in supporting his family, but failed as a businessman. Perhaps he was a successful businessman who ignored his family. For my purposes, the shortcoming can be anything.

More often, the leech attacks you for just a day in your life, or perhaps in a minor hobby over a period of months. It could be just a few minutes. I had one of these experiences last week.

The night sky in my front yard was flashing with bolts of lightning; not a common sight in this area. Usually the sky flashes, but there are no bolts. As impressive as that is, it looks like nothing on film. This was different. I ran out with my camera, and, not owning a tripod, I braced the camera against the fence and took dozens of two-second exposures of the sky. These were the 15 best:

Mediocre lightning series

Click above to enlarge the thumbnails. If you can’t tell at this size, there are a bunch of shots of lightning; usually just a couple small branches across the clouds, or light in absence of a trunk. The top-right one looks good, but it’s blurry because I slipped with the camera. There is no bolt, so you can’t tell what it is except in this context. None of the photos are particularly good. They are a series of near-hits.

At this point, my excitement from seeing the awesome flashes had disappeared, and I was thoroughly ravaged by the Florida mosquitoes. There were a lot of good shots to be had, but magically I’d missed all of them. My outing was a complete waste of time.

It wasn’t really a complete waste of time; it just seemed it. Since I’m a photographer, nothing I do with my camera can be a complete waste of time, unlike for you non-photographers. I have an in-built advantage. However, if my next several shots are like this, the series of near-hits may become a self-fulfilling prophecy. I’ll subconsciously sabotage my efforts because I’ll become afraid of taking a good photograph, or I’ll want to prove to my mind that the world is against me. Either way, it becomes a repetitive cycle.

Personally developed people enjoy exponential progress, because it’s so much faster than advancing linearly. While with 5 + 5 + 5 you have 15, with 5 * 5 * 5 you’re way ahead at 125. It’s a lot nicer to have your income double each month than increase by 2 dollars (unless you’re making pennies!). If you’ve ever played with a calculator, though you’ve noticed a strange thing happens south of 1. Instead of increasing, the numbers decrease. .5 * .5 * .5 is .125. If each multiplication is a day, continuing the trend, your progress goes down, each and every day. You can never quite hit rock bottom (zero), because you’re experiencing inverse exponential growth. Even if your calculator, after a few more operations, reads zero, you know it’s a lie. A “rounding error,” we call it. If your goal is zero, it’s forever fleeting.

This sickly kind of growth is exactly what a series of near-hits looks like. You get closer while never succeeding. It’s called logarithmic growth, and it’s the opposite of exponential gains. Whereas exponential operations race toward infinity without ever meeting, logarithmic operations reach for zero but never make it. There is an asymptote at zero; an invisible wall which can never be touched. The numbers get closer and closer, with more and more decimal places, but they’ll never match in a million years.

Evil, logarithmic growth

That’s a graph of logarithmic growth. And you want to avoid it. If your stuck in a cycle of logarithmic progress, a.k.a. a series of near-hits, something is wrong. You’ve got to try something new, because you’ve reached a dead end. In the game of life, every turn you waste in this downward spiral is one more turn off your life. It feels comfortable, because you can never crash and you’ll never hit zero, but it’s ultimately a waste because there’s no growth to be had in it. Your in a worse situation than stagnation, because you’ve been tricked into thinking you’re making progress. You could expend years stuck in logarithmic growth. Don’t.

It’s really easy to have “near-hits” in gambling. What they really are is wishful thinking. If the lottery numbers were 4-8-12-16-20, and you picked 5-9-13-17-21, you could say you were really close to winning a million dollars. You “almost” won. Surely, you’re on a lucky streak. You can’t stop; quitting gambling now would be foolish. But in truth, you were no closer to winning than if you had picked 1-2-3-5-6. Either set of numbers lost, unequivocally and irrevocably. There is no middle ground. Either you won, or you lost.

When you’re stuck in a series of near-hits, redefine life in black and white terms (binary) instead of shades of gray (analog). Analog has it’s place, but it’s a poor substitute for definiteness. You didn’t have a “near-hit,” you had a “complete miss.” You won’t hit till you do, and when you do, there will be no modifiers; it will be a “hit” and nothing else. Once you start thinking like this, you might give up entirely on some goals. It’s okay; that proves those goals weren’t important to you anyway. If you can survive many cycles of utter failure, then you know you are on the right path, because you have the strength to persevere through all hardships. “Near-hits” are just an illusion. They may try to waste your time and muddle your vision, but you shall triumph.